2.1. Derivatives and polynomials

We identify $\mathfrak{g}$ with the left invariant vector fields $\mathfrak{g}_L$ on $\mathbb{G}$ and write $\mathfrak{g}_R$ for the right invariant vector fields. We write X_i for the the basis elements as in (2.2) seen as elements of $\mathfrak{g}_L$ and Y_i for the basis of $\mathfrak{g}_R$ satisfying $Y_i |_e= X_i |_e$. Thus we can write

\begin{equation*}X_jf(y)= \partial_t f(y\exp(tX_j))|_{t=0}\quad \text{and } \quad Y_jf(y)= \partial_t f(\exp(tY_j)y)|_{t=0} \ \end{equation*}

for any smooth function $f\in C^\infty(\mathbb{G})$.

A map $P: \mathbb{G} \to \mathbb{R}$ is called a polynomial if $P\circ \exp : \mathfrak{g} \rightarrow \mathbb{R}$ is a polynomial on $\mathfrak{g}$.Footnote ² Let ζ_i be the basis dual to the basis X_i of $\mathfrak{g}$. We set

(2.3)\begin{equation} \eta_j := \zeta_j\circ \exp^{-1}:\mathbb{G}\to \mathbb{R}. \end{equation}

Note that $\eta= (\eta_1,...,\eta_d)$ forms a global coordinate systemFootnote ³, and furthermore any polynomial map on $\mathbb{G}$ can be written in terms of coefficients $a_I \in \mathbb{R}$ as

\begin{equation*}P= \sum_I a_I \eta^I,\end{equation*}

with the sum running over a finite subset of $\mathbb{N}^d$ and where for a multi-index $I=(i_1,...,i_d)\in \mathbb{N}^d$ we write $\eta^I= \eta_1^{i_1}\cdot...\cdot \eta_d^{i_d}$. Define $d(I)=\sum_j \mathfrak{s}_j i_j$ and $|I| = \sum_j i_j$, we call $\max \{d(I)\ : \ a_I \neq 0\}$ the homogeneous degree and $\max\{|I| \ : \ a_I \neq 0\}$ the isotropic degree of P. For a > 0, we denote by $\mathcal{P}_a$ the space of polynomials of homogeneous degree strictly less than a and define $\triangle= \{d(I)\in \mathbb{R} \ : \ I\in \mathbb{N}^d \}$.Footnote ⁴ We can rewrite the group law on $\mathbb{G}$ explicitly in terms of $\eta= (\eta_1,...,\eta_d)$.

Proposition 2.5.

For $j\in \{1,\ldots,d\}$and multi-indices $I,\,J$ s.t. $d(I)+d(J) = \mathfrak{s}_j$, there exist constants $C^{I,J}_j\in \mathbb{R}$ such that the following formula holds:

\begin{equation*}\eta_j (xy)=\eta_j (x)+\eta_j (y) + \sum_{I,J\neq 0,\ d(I)+d(J)=\mathfrak{s}_j} C_j^{I,J}\eta^I(x)\eta^J(y),\end{equation*}

for all $x,\,y \in \mathbb{G}$, where η_j is given by the expression (2.3).

Proof. Applying the Baker–Campbell–Hausdorff formula (2.1) there exist constants $C^{I,J}_j$ which depend only on the choice of basis for $\mathfrak{g}$ made in (2.2) but are independent of $x,\, \& \, y \in \mathbb{G}$ such that

\begin{equation*}\eta_j (xy)=\eta_j (x)+\eta_j (y) + \sum_{|I|+|J|\geqslant 2} C^{I,J}_j \eta^I(x)\eta^J(y), \qquad \text{for all } x,\, y\in \mathbb{G}.\end{equation*}

By setting either x = e or y = e, we find that $C^{I,J}_j=0$ if I = 0 or J = 0. Since furthermore, $\eta_j((rx)(ry))= r^{\mathfrak{s}_j}\eta_j(xy)$ the claim follows.

Remark 2.8. If follows from Proposition 2.5 that one can write

\begin{equation*}\eta^K(xy)=\eta^K (x)+\eta^K (y) + \sum_{I,J\neq 0,\ d(I)+d(J)=d(K)} C_K^{I,J}\eta^I(x)\eta^J(y),\end{equation*}

where the constants $C_K^{I,J}$ can be written in terms of the constants $C_j^{I,J}$.

Lemma 2.9. For $i,\,j \in \{1,\ldots,d\}$,

(2.4)\begin{equation} X_i \eta_j = \delta_{i,j} + \sum_{I\neq 0,\ d(I)= \mathfrak{s}_j-\mathfrak{s}_i} C_j^{I, e_i} \eta^I, \end{equation}

where e_i denotes the multi-index $(0,\ldots,0,1,0,\ldots,0)$, with 1 being in the i-th slot.

Proof. This follows directly from Proposition 2.5, applying X_i to the function

\begin{equation*}y\mapsto \eta_j(xy), \end{equation*}

evaluating at y = 0 and using the fact that $X_i|_0= \frac{\partial}{\partial\eta_i}|_0$.

Proposition 2.10 ([Reference Folland and Stein32, Prop. 1.26])

One has $X_j= \sum P_{j,k} \left(\frac{\partial}{\partial \eta_k}\right)$, where

\begin{equation*}P_{j,k}=\left\{ \begin{array}{ll} 1 & \mbox{if } k=j \\ 0 & \mbox{if } \mathfrak{s}_k\leqslant \mathfrak{s}_j, k\neq j \ \end{array} \right. \end{equation*}

and $P_{j,k}$ is a homogeneous polynomial of degree $\mathfrak{s}_k-\mathfrak{s}_j$ if $\mathfrak{s}_k \gt \mathfrak{s}_j$. The analogous statement holds for the vector fields Y_j,

For a multi-index $I=(i_1,...,i_d)\in \mathbb{N}^d$ we introduce the notation $X^I = X_1^{i_1}... X_d^{i_d}$. Note that the order of the composition matters since $\mathfrak{g}$ is not in general Abelian. It is a well-known fact that any left invariant differential operator on $\mathbb{G}$ can (uniquely) be written as a linear combination of $\{X^I \}_{I \in \mathbb{N}^d}$. The next proposition follows as a direct consequence.

Remark 2.12. One observes that for $I\in \mathbb{N}^d$ and $a \gt d(I)$ one has that

\begin{equation*} X^I \operatorname{P}^a_x[f]= \operatorname{P}^{a-d(I)}_x[X^If] \end{equation*}

for all $f\in C^\infty(\mathbb{G})$. Indeed, this follows from the fact that for any $d(J) \lt a-d(I)$ one has $ X^J (X^I \operatorname{P}^a_x[f])(e)= X^J(X^I f)(x)$, since $X^J X^I$ can be written as a linear combination of $\{X^K\}_{d(K) \lt a}$.

Theorem 2.13 (Taylor’s Theorem)

For each a > 0 and every $f\in C^\infty(\mathbb{G})$ it holds that

\begin{equation*} f(xy)-\operatorname{P}^a_x[f](y)= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(xz) Q^I(y, \text{d} z),\end{equation*}

where for each multi-index I and $y\in \mathbb{G}$ the measure $Q^I(y, \,\cdot\,)$ is supported on $B_{\beta^{[a]+1} |y|}(e)$ for some β > 0 depending only on $\mathbb{G}$ and satisfies $\int_{\mathbb{G}} |Q^I(y, \text{d} z)| \lesssim |y|^{d(I)}$.

Remark 2.14. Note that there is a more common version of Taylor’s Theorem on homogeneous Lie groups, cf. [Reference Folland and Stein32, Thm. 1.37], which states that the remainder satisfies the estimate

(2.5)\begin{equation} |f(xy)-\operatorname{P}^a_x[f](y)|\lesssim_a \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} |y|^{d(I)} \sup_{|z|\leqslant \beta^{[a]+1} |y|} |X^{I}f(xz) | \ \end{equation}

and which follows as a trivial consequence of Theorem 2.13. For us the explicit formula of the reminder is needed in order to establish Schauder estimates in Section 3.4.

Remark 2.15. The analogous claim for the right Taylor polynomial, $(\operatorname{P}_R)^a_x[f]$, holds. In this case the analogue of (2.5) reads

(2.6)\begin{equation} |f(yx)-(\operatorname{P}_R)^a_x[f](y)|\lesssim_a \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} |y|^{d(I)} \sup_{|z|\leqslant \beta^{[a]+1} |y|} |Y^{I}f(zx) |. \end{equation}

Remark 2.16. If we define $\tilde{\operatorname{P}}^a_x[f](y):= \operatorname{P}^a_x[f](x^{-1}y)$, then it follows that

\begin{equation*} f(y) - \tilde{\operatorname{P}}^a_x[f](y)= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(xz) Q^I(x^{-1}y, dz) \end{equation*}

and in particular that

\begin{equation*} |f(y) - \tilde{\operatorname{P}}^a_x[f](y)| \lesssim_{a}\sum_{\substack{|I|\leqslant [ a ]+1\\d(I)\geqslant a }} |x^{-1} y|^{d(I)} \sup_{|z| \leqslant C|x^{-1} y|} |X^I f(xz)|. \end{equation*}

In the particular case when f is compactly supported, one can rewrite this for the rescaled function $f^{\lambda}(x)= \frac{1}{\lambda^{|\mathfrak{s}|}} f (\lambda\cdot x)$ as

(2.7)\begin{align} |f^\lambda (y) - \tilde{\operatorname{P}}^a_x[f^\lambda](y)| & \lesssim_{a} \sum_{\substack{|I|\leqslant [ a ]+1\\d(I)\geqslant a }}\lambda^{-d(I)-|\mathfrak{s} | }|x^{-1} y|^{d(I)} \sup_{z\in \mathbb{G}} |X^I f(z)| \\ & \lesssim_{a,f } \sum_{\delta\geqslant 0}\lambda^{-(a+\delta)-|\mathfrak{s}|}|x^{-1} y|^{a+\delta} ,\nonumber \end{align}

where the sum over δ runs over some finite subset of $[0,\infty)$.

Remark 2.17. Given $f\in C^\infty(\mathbb{G})$, if we define $F\in C^\infty(\mathbb{G}\times \mathbb{G})$ by setting $F(y,z)=f(z^{-1}y)$, we find that

\begin{equation*}\tilde{\operatorname{P}}^a_x[F(\,\cdot\,,z)](y)=\tilde{\operatorname{P}}^a_{z^{-1}x}[f](z^{-1}y).\end{equation*}

Remark 2.18. For any $p \in \mathcal{P}_{\gamma}$ one has

\begin{equation*}\tilde{\operatorname{P}}_x^{\gamma} [p](y)= p(y).\end{equation*}

Proof of Theorem 2.13

Define for $i\in \{1,...,d\}$ the following measure on $\mathbb{R}^d$ with support on $B^\mathfrak{s}_{|t|_\mathfrak{s}}(0)$

\begin{equation*}\tilde{Q}^{e_i}(t, ds)=\prod_{j \lt i} \delta_{t_j} (ds_j) \cdot \mathbf{1}_{[0,t_i]} (ds_i) \prod_{j \gt i} \delta_{0} (ds_j)\end{equation*}

and define ${Q}^{e_i}(y, \,\cdot\,)= \Phi_* \tilde{Q}^{e_i}(\Phi^{-1}(y), \,\cdot\,) $ to be the push-forward measure. First we show that

(2.10)\begin{equation} f(xy)- f(x)=\sum_{i=1}^n \int_{\mathbb{G}} (X_i f)(xz) {Q}^{e_i}(y, dz). \end{equation}

Indeed one can write for $y=y_1y_2\ldots y_d$, where $y_i= \exp(t_i X_i)$ and $t=(t_1,...,t_d)= \Phi^{-1}(y)$

\begin{align*} f(xy)- f(x)&= \sum_{i=1}^d f(xy_1\ldots y_{i-1} y_{i})- f(xy_1...y_{i-1})\\ &= \sum_{i=1}^d \int_0^{t_i} \partial_s f(xy_1 \ldots y_{i-1} \exp(s X_i))|_{s=s'} ds'\\ &= \sum_{i=1}^d \int_0^{t_i} (X_i f)(xy_1\ldots y_{i-1} \exp(s X_i)) ds\\ &= \sum_{i=1}^d \int_{\mathbb{R}^d} (X_i f)(x\Phi(s)) \tilde{Q}^{e_i}(t, ds) \\ &= \sum_{i=1}^d \int_{\mathbb{G}} (X_i f)(xz) {Q}^{e_i}(y, dz). \end{align*}

Using the fact that Φ is a diffeomorphism one easily checks that ${Q}^{e_i}(y, dz)$ is supported on $B_{\beta |y|}(0)$, where β is as in (2.8), and satisfies $\int_{\mathbb{G}} |Q^{e_i}|(y, dz) \lesssim |y|^{d_i}$, and thus we have proved the theorem in the special case $a\in (0,1]$, also known as mean-value theorem. We turn to the proof in the general case.

Claim 2.20.

Set $g(y)= f (xy)- \operatorname{P}^a_x[f](y)$, we note that $X^Jg(e)=0$ for $d(J) \lt a$, while $X^Jg(y)=X^Jf(xy)$ for $d(J) \geqslant a$. We shall prove that for any multi-index J it holds that one can write

\begin{equation*}X^Jg(y)= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(xz) Q^{I,J}(y, dz),\end{equation*}

where the measures $Q^{I,J}(x,\cdot)$ satisfy the following properties:

• • $Q^{I,J}(x,\cdot)$ is supported on $B_{\beta^{m(J)} |y|}(e)\subset \mathbb{G}$, where $m(J)= \min\{m\in \mathbb{N}:\ a-m \lt d(J) \},$

• • $\int_{\mathbb{G}}|Q^{I,J}|(x,dy)\lesssim |x|^{d(I)-d(J)}$.

We shall prove by induction on $m\in \{1,..., \lceil a \rceil \}$ that for multi-indices J satisfying $a-m\leqslant d(J) \lt a$ the claim holds.

• • The case m = 1 follows from (2.9) and (2.10),

  \begin{align*} X^Jg(y)&= X^J g(y)-X^J g(e) = \sum_{i=1}^d \int_{\mathbb{G}} (X_i X^Jg)(z) {Q}^{e_i}(y, dz) \\ &= \sum_{i=1}^d \int_{\mathbb{G}} \sum_{I \ : \ d(I) = d(J)+\mathfrak{s}_i} \lambda_{i,J, I} (X^{I} g)(z) {Q}^{e_i}(y, dz) \\ &= \sum_{i=1}^d \int_{\mathbb{G}} \sum_{I \ : \ d(I)= d(J)+\mathfrak{s}_i} \lambda_{i,J, I} (X^{I} f)(xz) {Q}^{e_i}(y, dz) \\ &= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(xz) Q^{I,J}(y, dz). \end{align*}

  The properties of $Q^{I,J}(y, dz)$ follow from the corresponding properties of ${Q}^{e_i}(y, dz)$, completing the case m = 1.

• • Suppose the claim holds for m − 1. Then by the same argument as above

  \begin{align*} X^Jg(y) &= \sum_{i=1}^d \sum_{d(K)= d(J)+\mathfrak{s}_i} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz)\\ &= \sum_{i, K : d(K)= d(J)+\mathfrak{s}_i \lt a} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz) \\ &\quad +\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i\geqslant a} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz). \end{align*}

  We can apply the induction hypothesis to the terms in the first sum, since $d(K)= d(J)+\mathfrak{s}_i\geqslant d(J)+1\geqslant a-(m-1)$ and find

  \begin{equation*}(X^{K} g)(xz)= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(x\tilde{z}) Q^{I,K}(z, d\tilde{z})\end{equation*}

  and therefore

  \begin{align*} X^Jg(y) &=\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i \lt a} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz)\\ &\quad +\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i\geqslant a} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz) \\ &=\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i \lt a} \lambda_{i,J, K} \\ &\quad \times \sum_{|I|\leqslant k+1, d(I)\geqslant a}\int_{\mathbb{G}} \int_{\mathbb{G}} X^{I}f(x\tilde{z}) Q^{I,K}(z, d\tilde{z}) {Q}^{e_i}(y, dz) \\ &\quad + \sum_{i, K : d(K)= d(J)+\mathfrak{s}_i\geqslant a} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz). \\ &=\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i \lt a} \lambda_{i,J, K} \\ &\quad \times \sum_{|I|\leqslant k+1, d(I)\geqslant a} \int_{G} X^{I}f(x\tilde{z}) \big(Q^{I,K} *{Q}^{e_i}\big)(y, d\tilde{z}) \\ &\quad +\sum_{i, K : d(K)= d(J)+\mathfrak{s}_i\geqslant a} \lambda_{i,J, K} \int_{\mathbb{G}} (X^{K} g)(xz) {Q}^{e_i}(y, dz) \\ &= \sum_{|I|\leqslant [a]+1, d(I)\geqslant a} \int_{\mathbb{G}} X^{I}f(x\tilde{z}) Q^{I,J}(y, d\tilde{z})\, \end{align*}

  where $Q^{I,J}$ is defined such that the last line holds true. Concerning the measures $Q^{I,J}(y, \cdot)$, we note that

  1. (i) ${\text{supp}} Q^{I,J}(y, \cdot)\subset B_{\beta^{m(J)} |y|} $ since

     \begin{equation*}{\text{supp}} \big( Q^{I,K} *{Q}^{e_i}(y, \cdot) \big)\subset B_{\beta^{m(K)+1} |y|} \end{equation*}

     as ${\text{supp}} Q^{I,K}(y, \cdot)\subset B_{\beta^{m(K)} |y|}(e)$ and ${\text{supp}} {Q}^{e_i}(y, dz)\subset B_{\beta |y|}(e)$.

  2. (ii) $\int_{\mathbb{G}} |Q^{I,J}(y, \cdot)| \lesssim |x|^{d(I)-d(J)}$ since

     \begin{align*}\int |Q^{I,K} *{Q}^{e_i}|(y, dz)& \leqslant \int |Q^{I,K}|(y, dz) \, \cdot \, \int |{Q}^{e_i}|(y, dz)\\ &\lesssim |y|^{d(I)-d(K)}\cdot |y|^{\mathfrak{s}_i}.\end{align*}

This completes the proof.

2.2. Distributions and convolution

We define $\mathcal{D}(\mathbb{G}):= C^\infty_c(\mathbb{G})$ to be the space of compactly supported smooth functions on $\mathbb{G}$ equipped with the canonical LF topology. The space of distributions on $\mathbb{G}$ is given by the dual $\mathcal{D}'(\mathbb{G})$ of $\mathcal{D}(\mathbb{G})$, and for $\xi \in \mathcal{D}'(\mathbb{G})$ and $\phi \in \mathcal{D}(\mathbb{G})$ we either write $\langle \phi,\xi\rangle$ or $\xi(\phi)$ for the canonical pairing. Given $r \in \mathbb{R}_{\geqslant 0}$ we introduce the following useful space of test functions:

\begin{equation*}\mathfrak{B}^{\lambda}_{r}(x):= \left\{\phi\in C^\infty_c (B_\lambda(x)) \ : \ |X^I \phi | \leqslant \frac{1}{\lambda^{|\mathfrak{s}|+d(I)}} \ \forall I : d(I)\leqslant r \right\}. \end{equation*}

We will often use the shorthand $ \mathfrak{B}_{r}:= \mathfrak{B}^{1}_{r}(e)$. Given $\phi \in \mathcal{D}(\mathbb{G})$ we extensively use the following notations:

\begin{equation*}\phi^\lambda_x(z):=\frac{1}{\lambda^{|\mathfrak{s}|}}\phi \left(\frac{1}{\lambda}\cdot (x^{-1}z)\right).\end{equation*}

Recalling that we use the notation $\text{d} x$ for the Haar measure on $\mathbb{G}$ we define the norms,

\begin{equation*} \|f\|_{L^p} := \begin{cases} \left(\int_{\mathbb{G}} |f(x)|^p\text{d} x\right)^{1/p}, & \text{for } 1\leqslant p \lt \infty,\\ \text{ess\,sup}_{x \in \mathbb{G}} |f(x)|, & p=\infty. \end{cases} \end{equation*}

By left and right translation invariance of the Haar measure, for $f\in L^1(\mathbb{G})$ it holds that

\begin{equation*}\int_{\mathbb{G}} f(yx)\text{d} y = \int_{\mathbb{G}} f(xy)\text{d} y = \int_{\mathbb{G}}f(y)\text{d} y = \int_{\mathbb{G}}f(y^{-1})\text{d} y.\end{equation*}

See the discussion after [Reference Fischer and Ruzhanksy28, Thm. 1.1.1].

Remark 2.21. From the scaling properties of $\operatorname{D}$ it follows that

\begin{equation*} \mathfrak{B}^{\lambda}_{r}(x) = \left\{\phi^{\lambda}_x \ : \phi\in \mathfrak{B}^{1}_{r}(e) \right\}.\end{equation*}

One defines the convolution of two functions $\phi, \psi: \mathbb{G}\to \mathbb{R}$ as

(2.11)\begin{equation} \psi*\phi(x)= \int \psi(y) \phi(y^{-1}x) dy = \int \psi(xy^{-1}) \phi(y) dy. \end{equation}

Note that in general $\psi* \phi(x)\neq \phi * \psi(x)$ but still many of the usual inequalities hold, in particular the Young inequality, cf. [Reference Folland and Stein32, Prop. 1.18]. One can write

\begin{equation*}\psi*\phi(x)= \int \psi(y) \phi_y(x) dy. \end{equation*}

We extend convolution to generalized function, i.e. elements of $\mathcal{D}'(\mathbb{G})$, by duality in the usual manner. From now on we will use the notation $\tilde{\phi}(z):= \phi(z^{-1})$; one notes that the following identities hold:

(2.12)\begin{equation} \langle f, g*\phi \rangle = \langle \tilde{g} * f, \phi \rangle = \langle f * \tilde{\phi}, g \rangle, \end{equation}

(2.13)\begin{equation} \widetilde{f*g} = \tilde{g}*\tilde{f}, \end{equation}

\begin{equation*} \psi^\lambda*\phi^\lambda = (\psi*\phi)^\lambda\quad \text{and}\quad (\psi \ast \phi)_x = \psi_x \ast \phi. \end{equation*}

Recalling the notation $X^I = X_1^{i_1}... X_d^{i_d}$ for any multi-index $I=(i_1,...,i_d)\in \mathbb{N}^d$, we introduce the analogous notation $Y^I = Y_d^{i_d}....Y_1^{i_1}$ for the right invariant basis vectors. Note the reversal in the composition. Since for any right invariant vector field Y and left invariant vector field X such that $X|_e = Y|_e$ it holds that $\langle X f,g\rangle = \langle f, Y g\rangle $ for $f,\,g \in C^\infty_c(\mathbb{G})$, it follows from the definition of the convolution that

• • $X^I(\psi*\phi) = \psi*( X^I \phi) $,

• • $Y^I(\psi*\phi) = (Y^I\psi)* \phi $,

• • $(X^I\psi)*\phi= \psi*Y^I \phi \ $.

The following proposition gives some useful properties that also mirror those of Euclidean Hölder spaces.

Note that this proposition in particular implies that $C^\infty(\mathbb{G})= \cap_{\alpha \gt 0} \mathcal{C}^\alpha(\mathbb{G})$.

Proof. Points i and ii follow directly. For Point iii, denote by $\tilde{P}_x$ the polynomial in Definition 4, we set $F=f- \tilde{P}_{\cdot}(\cdot)$, then for any $\psi\in \mathfrak{B}_{0}$ such that $\int_{\mathbb{G}} \psi = c^{-1} \gt 0$ and smooth compactly supported function g we find

\begin{equation*} \langle F, g\rangle = c\lim_{\lambda\to 0} \langle F, g* \psi^\lambda \rangle = c\lim_{\lambda\to 0} \left \langle F, \int_{\mathbb{G}} g(y) \psi_y^\lambda dy \right\rangle = c\lim_{\lambda\to 0} \int_{\mathbb{G}}\langle F, \psi_y^\lambda \rangle g(y) dy =0, \end{equation*}

where in the last step we used that since $y\to \tilde{P}_y$ is continuous we have

\begin{equation*} \langle F, \psi_y^\lambda \rangle = \langle f- \tilde{P}_y(\,\cdot\,), \psi_y^\lambda \rangle+\int \big(\tilde{P}_y(z) - \tilde{P}_z (z)\big) \psi_y^\lambda(z) \, dz \to 0,\end{equation*}

as λ → 0. Thus we find that the distribution f agrees with the continuous function $x\mapsto \tilde{P}_x(x)$, concluding the proof.

2.3. Discrete subgroups

Recall that, given a discrete subgroup $\mathfrak{G}\subset \mathbb{G}$ acting on $\mathbb{G}$ (say) on the left, then the quotient space $ S:=\mathbb{G} / \mathfrak{G}=\{\mathfrak{G} x \ : \ x\in \mathbb{G}\}$ is a smooth manifold. Furthermore, the quotient map $\pi: \mathbb{G}\to S$ is a smooth normal covering map, cf. [Reference Lee53, Thm. 21.29], and the canonical $\mathbb{G}$ right action

\begin{equation*}S\times \mathbb{G}\to S, \qquad (\mathfrak{G} x, x') \mapsto \mathfrak{G} x x' \ \end{equation*}

makes it a homogeneous space. We call $\mathfrak{G}$ a lattice if $ S= \mathbb{G} / \mathfrak{G} $ is a compact space. This is equivalent to S carrying a finite $\mathbb{G}$ invariant measure, which we shall denote by $d(\mathfrak{G} x)$, cf. [Reference Raghunathan62, Thm. 2.1]. Throughout this article we assume that every homogeneous Lie group we work with carries a lattice, the following theorem, [Reference Raghunathan62, Thm. 2.12], gives sufficient conditions for this to be the case, which all our examples satisfy.

Let us point out that there do exist nilpotent Lie groups that do not admit a basis with respect to which the structure constants are rational, see [Reference Raghunathan62, Rem. 2.14] for an example.

Denote by $\pi^\ast: C(S) \to C(\mathbb{G})$ the pull-back under π. We define a convolution map

\begin{equation*}*_S : C^\infty(S)\times C_{c}^\infty(\mathbb{G}) \to C^\infty(S),\quad (f, \phi) \mapsto f *_S \phi\end{equation*}

as the unique map, such that the following diagram commutes:

where convolution on the bottom row is convolution on the group as defined in 2.12. In order to check that this map is well defined, we use the following terminology. A function $f\in C^\infty (\mathbb{G})$ is called left $\mathfrak{G}$ periodic, if for every $x\in \mathbb{G}$ and $n\in \mathfrak{G}$ it holds that

\begin{equation*}f(x)= f(nx).\end{equation*}

Thus, we only need to check that for any $\phi \in C_{c}^\infty(\mathbb{G})$ and any left $\mathfrak{G}$ periodic function $f\in C^\infty (\mathbb{G})$, the function $f* \phi$ is left $\mathfrak{G}$ periodic. Indeed for any $n\in \mathfrak{G}$ and $x\in \mathbb{G}$, it holds that

\begin{align*} f* \phi (nx)&= \int_{\mathbb{G}} f(y) \phi(y^{-1}nx) dy \\ &= \int_{\mathbb{G}} f(y) \phi((n^{-1}y)^{-1}x) dy \\ &= \int_{\mathbb{G}} f(ny) \phi(y^{-1}x) dy \\ &= \int_{\mathbb{G}} f(y) \phi(y^{-1}x) dy. \end{align*}

By duality, one naturally extends the notion of convolution on S to pairs, $(\xi,\zeta)\in \mathcal{D}'(S)\times \mathcal{D}'_c(\mathbb{G})$, where $\mathcal{D}'_c(\mathbb{G})$ denotes distributions on $\mathbb{G}$ with compact support.

2.4. Concrete examples

In order to cement ideas we present two concrete examples of non-abelian, homogeneous Lie groups, both of which are identified with fixed global charts. These will be revisited in $\S$ 4.4 when we discuss the heat operator on the Heisenberg group and Kolmogorov-type operators.

2.4.1. The Heisenberg group

Given $n\geqslant 1$ we define the Heisenberg group $\mathbb{H}^n$ as the set $\mathbb{R}^{2n}\times \mathbb{R}$ with the group law,

\begin{equation*} (x,y,z) (x',y',z') = \left(x+x',y+y', z+z' + \sum_{i=1}^n \left(x'_iy_i - x_iy'_i\right) \right). \end{equation*}

Remark 2.26. Note that one may equally define the complex Heisenberg group on $\mathbb{C}^{n}\times \mathbb{R}$ equipped with the group law,

\begin{equation*} (u,z)(u',z') = \left(u+u',z+z' + \Im \sum_{i=1}^{n} u_i \bar{u}'_i\right), \end{equation*}

where $\Im z$ denotes the imaginary part. One sees that these definitions are equivalent after identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$.

The origin $e=(0,0,0)$ is clearly the identity, and for $(x,y,z)\in \mathbb{R}^{2n}\times \mathbb{R}$ one has $(x,y,z)^{-1}= (-x,-y,-z)$. The Lie algebra, $\mathfrak{h}$ of $\mathbb{H}^n$, is identified with $\mathbb{R}^{2n+1}$, spanned by the basis of left-invariant vector fields,

\begin{equation*} A_i(x,y,z) = \partial_{x_i} + y_i \partial_z,\quad B_i(x,y,z) = \partial_{y_i} - x_i \partial_z,\quad C(x,y,z) = \partial_z, \end{equation*}

and equipped with the Lie bracket $[A,B]= AB-BA$. We observe that for any $(x,y,z)\in \mathbb{H}^n$ and $i =1,\ldots,n$,

(2.15)\begin{equation} [A_i,B_i](x,y,z) =- 2C. \end{equation}

We say that a Lie algebra is graded if there exist vector spaces $\{W_k\}_{k \geqslant 1}$ where only finitely many W_k are non-zero, $\mathfrak{g} = \bigoplus_{k=1}^\infty W_k$ and $[W_i,W_j]\subset W_{i+j}$.

Definition 5. A homogeneous Lie group $\mathbb{G}$ is called stratified (or a Carnot group) if its Lie algebra is graded and generated by W ₁.

Due to (2.15), we see that if we equip $\mathbb{H}^n$ with the dilation map,

\begin{equation*} \lambda \cdot (x,y,z) := (\lambda x, \lambda y, \lambda^2z), \end{equation*}

then $\mathbb{H}^n$ becomes a stratified group. Refer to [Reference Bramanti12, $\S$ 3.3.6] for a discussion of generalizations of this structure.

A simple example of a lattice on the Heisenberg group is the set of integer vectors $(a,b,c)\in \mathfrak{H}^n:= \mathbb{Z}^{2n}\times \mathbb{Z}$ equipped with the group law as defined above. Note that it is not a normal subgroup, and thus the quotient space $\mathbb{H}^n/\mathfrak{H}^n$ is not a group. However, since it is a lattice the homogeneous space $\mathbb{H}^n/\mathfrak{H}^n$ is compact, cf. $\S$ 2.3.

2.4.2. Matrix exponential groups

For $n\geqslant 1$, let B be a rational n × n block matrix of the form,

\begin{equation*} B = \begin{pmatrix} 0 & B_1 & 0& \cdots &0\\ 0& 0 & B_2 &\cdots &0\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 0 & 0&\cdots & 0 & B_{k}\\ 0 & 0 &\cdots &0 &0 \end{pmatrix} \end{equation*}

with each B_i a $p_{i-1}\times p_{i}$ block matrix of rank p_i, where $n\geqslant p_0 \geqslant p_1\geqslant \cdots \geqslant p_{k}$ and $\sum_{i=0}^{k}p_i=n$. Note that this implies the zero blocks on the diagonal are all $p_i\times p_i$ square matrices. We can equip $\mathbb{R}\times \mathbb{R}^{n}$ with an associated Lie structure by defining the group law

\begin{equation*} (t,z)(s,z') := \left(t+s,z'+\exp\left(sB^\top\right)z\right). \end{equation*}

To define the dilation we begin by decomposing according to the structure of the block matrix B, writing

\begin{equation*} \mathbb{R}^n = \mathbb{R}^{p_0} \times \cdots \times \mathbb{R}^{p_{n}}. \end{equation*}

Thus the action of $B^\top$ on $z=(z_0,..., z_k)\in \mathbb{R}^{p_0} \times \cdots \times \mathbb{R}^{p_{k}}$ is written as $B^\top z=(B^{\top}_1 z_0,\,B^{\top}_2z_{1},...B^{\top}_kz_{k-1})$ etc. Then we set

\begin{equation*} \lambda\cdot (t,z_0,\ldots,z_k):= (\lambda^2 t, \lambda z_0, \ldots,\lambda^{2k+1} z_k). \end{equation*}

The origin $e=(0,0)$ is again the identity element and $(t,z)^{-1} = (-t,-\exp(-tB^\top)z)$. The Lie algebra is spanned by the translation invariant vector fields,

\begin{equation*} X_i(t,z) = \partial_{z_i}\quad \text{for } i=1,\ldots,p_0 \quad \text{and}\quad Y(t,z) = \partial_t - (B z) \cdot \nabla = \partial_t - \sum_{i,j = p_0+1}^n b_{ij}z_i \partial_{z_i}. \end{equation*}

This defines the matrix exponential group associated with B and one sees that it is not stratified.

The simplest non-trivial example is to set n = 2 and

\begin{equation*} B= \begin{pmatrix} 0 & 1\\ 0&0 \end{pmatrix}, \end{equation*}

so that using the suggestive notation $(t,v,x)\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}$, the group law becomes,

\begin{equation*} (t,v,x)(s,w,y) = (t+s,v+w,x+y+sv). \end{equation*}

The equivalent scaling as above is to set $\lambda \cdot (t,x,v) = (\lambda^2t,\lambda v, \lambda^3 x)$. We refer to [Reference Manfredini57] for more details and $\S$ 4.4 for a discussion of natural second-order hypoelliptic linear operators associated with these groups.

3.1. The polynomial regularity structure

As an important example we describe the polynomial regularity structure and canonical model which are crucial for the analysis of singular SPDEs on homogeneous Lie groups. We define the structure space $\bar{\operatorname{T}}$ to be the symmetric tensor (Hopf) algebra generated by $\{\pmb{\eta}_i\}_{i=1}^d$, which we think of as abstract lifts of the monomials $\{\eta_i\}_{i=1}^d$. For a multi-index I, we write $\pmb{\eta}^{I}:=\pmb{\eta}_1^{i_1}\cdot...\cdot\pmb{\eta}_d^{i_d}$ as well as $\mathbf{1}:=\pmb{\eta}^{0}$. Group $\bar{\operatorname{G}}$ is given by a copy of $\mathbb{G}$ acting by

\begin{equation*} g\mapsto \Big( \Gamma_{g}: \pmb{\eta}_j\mapsto \pmb{\eta}_j + \sum_{I,J\neq 0,\ d(I)+d(J)=\mathfrak{s}_j} C_j^{I,J}\eta^I(g)\pmb{\eta}^J\ +\eta_j (g)\mathbf{1}\Big) \end{equation*}

and $\Gamma_{g} \pmb{\eta}^I= \big(\Gamma_{g} \pmb{\eta}\big)^I$. The canonical polynomial model is defined by setting

\begin{equation*} \Pi_{x}\pmb{\eta}_j (z)=\eta_j (x^{-1}z)\end{equation*}

and

\begin{equation*} \Gamma_{x,y}\pmb{\eta}_j=\pmb{\eta}_j + \sum_{I,J\neq 0,\ d(I)+d(J)=\mathfrak{s}_j} C_j^{I,J}\eta^I(y^{-1} x)\pmb{\eta}^J\ +\eta_j (y^{-1} x)\mathbf{1}.\end{equation*}

These maps are extended multiplicatively to all of $\bar{\operatorname{T}}$. Using Prop.  2.5 in the third equality,

\begin{align*} \Pi_{x}\Gamma_{x,y}\pmb{\eta}_j(z) &= \Pi_{x}\Big(\pmb{\eta}_j + \sum_{I,J\neq 0,\ d(I)+d(J) =\mathfrak{s}_j} C_j^{I,J}\eta^I(y^{-1} x)\pmb{\eta}^J\ +\eta_j (y^{-1} x)\Big)(z)\\ &= \eta_j(x^{-1} z) + \sum_{I,J\neq 0,\ d(I)+d(J)=\mathfrak{s}_j} C_j^{I,J}\eta^I(y^{-1} x)\eta^J(x^{-1} z)\ +\eta_j (y^{-1} x) \\ &= \eta_j(y^{-1} x x^{-1}z)\\ &= \eta_j(y^{-1} z)\\ &= \Pi_y \pmb{\eta}_j (z). \end{align*}

3.1.1. Derivatives and abstract polynomials

Next we lift the vector fields X_i to abstract differential operators $\mathcal{X}^i$ on $\bar{\operatorname{T}}$ of degree $\mathfrak{s}_j$, cf. $\S$ 3.5. We set

\begin{equation*}\mathcal{X}^i \pmb{\eta}^j = \delta_{i,j}\mathbf{1} + \sum_{I\neq 0,\ d(I)= \mathfrak{s}_j-\mathfrak{s}_i} C_j^{I, e_i} \pmb{\eta}^I\end{equation*}

and extend this definition to the whole regularity structure by the Leibniz rule. The conditions to be an abstract differential operator are checked directly:

1. (i) Indeed $\mathcal{X}^i: \operatorname{T}_\alpha\mapsto T_{\alpha-\mathfrak{s}_i}$.

2. (ii) By the Leibinz rule the second property is checked by showing that

   (3.1)\begin{equation} \mathcal{X}^i \Gamma_{x,y}\pmb{\eta}^j= \Gamma_{x,y}\mathcal{X}^i \pmb{\eta}^j. \end{equation}

   Note that $\Pi_x \mathcal{X}^i \pmb{\eta}^j =X_i \Pi_x\pmb{\eta}^j,$ since

   \begin{align*}& \Pi_x \mathcal{X}^i \pmb{\eta}_j (z) = \\ & \Pi_x \Big(\delta_{i,j}\mathbf{1} \,+ \sum_{I\neq 0,\ d(I) = \mathfrak{s}_j-\mathfrak{s}_i} C_j^{I, e_i} \pmb{\eta}^I\Big) = \delta_{i,j}\, + \sum_{I\neq 0,\ d(I)= \mathfrak{s}_j-\mathfrak{s}_i} C_j^{I, e_i} \eta^I(x^{-1}z)\end{align*}

   and using left invariance of the vector field X_i as well as (2.4)

   \begin{align*}& X_i \Pi_x\pmb{\eta}_j (z)= (X_i \eta_{j}( x^{-1} \cdot )) (z) \\ & = (X_i \eta_{j}) (x^{-1}z)= \delta_{i,j} + \sum_{I\neq 0,\ d(I)= \mathfrak{s}_j-\mathfrak{s}_i} C_j^{I, e_i} \eta^I(x^{-1}z).\end{align*}

   Thus (3.1) follows from the injectivity of maps $\Pi_x$ and relation $\Pi_x\Gamma_{x,y}= \Pi_y$.

Remark 3.2. In addition to showing that $\mathcal{X}_i$ is an abstract differential operator, we have shown that it also realizes X_i for the polynomial model.

3.1.2. Abstract Taylor expansions

The following operator, which sends a smooth function to its abstract Taylor expansion, will be used throughout the article.

Definition 8. For $x\in \mathbb{G}$, we define the family of maps

\begin{equation*}\pmb{\operatorname{{P}}}^a_x: \mathcal{C}^a (\mathbb{G}) \to \bar{\operatorname{T}}\end{equation*}

where for a > 0 $\pmb{\operatorname{{P}}}^a_x[f]$ is characterized by the fact that $\Pi_e \pmb{\operatorname{{P}}}^a_x[f] =\operatorname{P}^a_x[f]\in \mathcal{P}_a$ and for $a\leqslant 0$ it satisfies $\pmb{\operatorname{{P}}}^a_x[f]=0$.

Remark 3.3. Note that the map $\pmb{\operatorname{{P}}}^a_x$ is well defined by Remark 2.24. We shall almost exclusively use it, when its argument is a smooth function $f\in C^\infty(\mathbb{G})\subset \mathcal{C}^a(\mathbb{G})$.

Lemma 3.4. Let $f\in C^\infty(\mathbb{G})$ and $a \geqslant 0$. Then for each multi-index I the coefficient of $\pmb{\eta}^I$ in $\pmb{\operatorname{{P}}}^a_x[f]$ is given by a linear combination (depending only on $\mathbb{G}$) of $\{X^Kf(x)\}_{d(K)\leqslant d(I)}$. Furthermore, for $b\geqslant a$ one has

(3.2)\begin{equation} \pmb{\operatorname{{P}}}_x^a[f]= \mathcal{Q}_{ \lt a} \pmb{\operatorname{{P}}}^{b}_x [f]. \end{equation}

Proof. We first observe that (3.2) holds since the polynomial $\Pi_e \mathcal{Q}_{\leqslant a} \pmb{\operatorname{{P}}}^{b}_x [f]$ satisfies the property required in Definition 3. The first part of the lemma follows by combining Proposition 2.11 with (3.2).

Lemma 3.5. In the setting of the above lemma one has

\begin{equation*}|X^I \Pi_e \pmb{\operatorname{{P}}}^a_x[f] (z)|= |X^I \operatorname{P}^a_x[f] (z)| \lesssim \sum_{d(I)\leqslant d(J) \lt a} \sup_{d(J')\leqslant d(J)} |X^{J'}f(x)||z|^{d(J)-d(I)}.\end{equation*}

Proof. Since the coefficient of η ^J in $ {\operatorname{P}}^a_x[f]$ is bounded by a universal multiple of $\sup_{d(J')\leqslant d(J)} |X^{J'}f(x)|$ by Lemma 3.4 the claim follows using the fact that $|X^I\Pi_e \pmb{\eta}^J(z)|\lesssim |z|^{{d(J)-d(I)}}$.

The next lemma will be useful when proving Schauder estimates in $\S$ 3.4, since it allows us to circumvent the issue of having non-explicit Taylor polynomials.

Lemma 3.6. Let δ > 0 and $P\in \bar{\operatorname{T}}_{ \lt \delta}$. If, for some $\varepsilon \in [0,1]$,

\begin{equation*}\left| (X^{I}\Pi_e P) (e)\right|\leqslant \varepsilon^{\delta-d(I)}\end{equation*}

for all $I\in \mathbb{N}^d$ such that $\delta \gt d(I)$, then it holds that $|P|_a\lesssim_{\delta,\mathbb{G}} \varepsilon^{\delta-a}$ for all $a\leqslant \delta$.

Proof. We observe that $\operatorname{P}_x^\delta[\Pi_e P]=\Pi_e P$. Thus, the claim follows directly from Lemma 3.4.

3.2. Modelled Distributions

Definition 9. Given a regularity structure $\mathcal{T}=(\operatorname{T},\operatorname{G})$, a model $M=(\Pi, \Gamma)$ and $\gamma\in \mathbb{R}$ we define $ \mathscr{D}_M^\gamma$ as the space of all continuous maps $f:\mathbb{G}\to \operatorname{T}_{ \lt \gamma}$, such that for all $\zeta \in A\cap (-\infty,\gamma)$ the following bounds hold for every compact set $\mathfrak{K} \subset \mathbb{G}$

\begin{equation*}\sup_{x\in \mathfrak{K}} |f(x)|_\zeta \lt +\infty, \qquad \sup_{\substack{x,y\in \mathfrak{K},\\0 \lt {|y^{-1}x|}\leqslant 1}} \frac{|f(y)-\Gamma_{x,y}f(x)|_\zeta}{{|y^{-1}x|}^{\gamma-\zeta}} \lt +\infty.\end{equation*}

We define the corresponding semi-norm $\|\,\cdot\,\|_{\gamma;\mathfrak{K}}$ on $\mathscr{D}_M^\gamma$ by setting,

\begin{equation*} \|f\|_{\gamma;\mathfrak{K}} := \sup_{x\in \mathfrak{K}} \sup_{\zeta \lt \gamma}|f(x)|_\zeta + \sup_{\zeta \lt \gamma} \sup_{\substack{x,y\in \mathfrak{K}, \\ 0 \lt {|y^{-1}x|}\leqslant 1}}\frac{|f(y)-\Gamma_{x,y}f(x)|_\zeta}{{|y^{-1}x|}^{\gamma-\zeta}}. \end{equation*}

Given two models $M = (\Pi,\Gamma),\, \bar{M}=(\bar{\Pi},\bar{\Gamma})$, two modelled distributions $f\in \mathscr{D}^\gamma_M,\, \bar{f}\in \mathscr{D}^\gamma_{\bar M}$ and a compact set $\mathfrak{K}\subset\mathbb{G}$ we define the quantity,

\begin{equation*} \begin{array}{rcl} \|f;\bar{f}\|_{\gamma;\mathfrak{K}} &:=& \sup_{x \in \mathfrak{K}}\sup_{\zeta \lt \gamma}|f(x)-\bar{f}(x)|_\zeta \\ &&+ \sup_{\substack{x,y\in \mathfrak{K}\\ 0 \lt {|y^{-1}x|} \leqslant 1}}\sup_{\zeta \lt \gamma} \frac{|f(y)-\bar{f}(y)-\Gamma_{x,y}f(y)+\bar{\Gamma}_{x,y}\bar{f}(x)|_\zeta}{{|y^{-1} x|}^{\gamma-\zeta}}. \end{array} \end{equation*}

For the set of modelled distributions taking values in a sector $V\subset \operatorname{T}$ we write $\mathscr{D}^\gamma_M(V)$ and if the regularity of the sector is $\alpha \in A$ we often use the shorthand $\mathscr{D}^\gamma_{\alpha;M}$. We will freely drop the explicit dependence on the model, image sector and its regularity when the context is clear.

3.2.1. Reconstruction

Theorem 3.7 (Reconstruction theorem)

Let $\mathcal{T}=(\operatorname{T},\operatorname{G})$ be a regularity structure with $\underline{\alpha} = \min A$. Then for every γ > 0 and $M = (\Pi,\Gamma) \in \mathcal{M}_{\mathcal{T}}$, there exists a unique, continuous linear map $\mathcal{R}_{M}:\mathscr{D}_M^\gamma \rightarrow \mathcal{C}^{\underline{\alpha}}$, called the reconstruction operator associated with M, such that for any compact $\mathfrak{K}$ and $\lambda \in (0,1]$

(3.3)\begin{equation} \sup_{\psi \in \mathfrak{B}_{r}} |\langle \mathcal{R}_M f- \Pi_x f(x),\psi^\lambda_x\rangle|\lesssim_{\mathfrak{K}} \lambda^\gamma\|f\|_{\gamma;B_{2\mu\lambda}(x)}\|\Pi\|_{\gamma;B_{2\mu\lambda}(x)}, \end{equation}

uniformly over $x\in \mathfrak{K}$, where $\mu\geqslant 1$ was defined in Remark 2.4. Furthermore the map $M \rightarrow \mathcal{R}_M$ is locally Lipschitz continuous in the sense that for a second model $\bar{M}=(\bar{\Pi},\bar{\Gamma})$ and $\bar{f}\in \mathscr{D}_{\bar{M}}^\gamma$, for every $\lambda \in (0,1],$

(3.4)\begin{align} &\sup_{\psi \in \mathfrak{B}_{r}} |\langle \mathcal{R}_M f-\mathcal{R}_{\bar{M}}\bar{f} -\Pi_x f(x)+\bar{\Pi}_x \bar{f}(x),\psi^{\lambda}_x\rangle| \nonumber\\ &\quad \lesssim_{\mathfrak{K}} \lambda^\gamma \left(\|f;\bar{f}\|_{\gamma;B_{2\mu\lambda}(x)}\|\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)}+\|f\|_{\gamma;B_{2\mu\lambda}(x)}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)} \right). \end{align}

Remark 3.8. Existence of a reconstruction operator for $\gamma \leqslant 0$ also holds; however, uniqueness does not. In the case γ = 0 the analogous bound to (3.3) contains an additional logarithmic correction on the right-hand side, cf. [Reference Caravenna and Zambotti15].

Remark 3.9. It follows from the definition of the reconstruction operator, that if f is a modelled distributions with values in a sector $V\subset \operatorname{T}$ of regularity $\alpha\geqslant \underline{\alpha}$, then $\mathcal{R}_Mf\in C^{\alpha}$.

In order to prove Theorem 3.7 we require two preliminary results. As in [Reference Friz and Hairer33, $\S$ 13.4] we make the observation that for every $N\geqslant 0$ there exists a $\rho : \mathbb{G}\rightarrow \mathbb{R}$, smooth and compactly supported in $B_1(e)$ and such that

(3.5)\begin{equation} \int \eta^I(x) \rho(x)\text{d} x = \delta_{I,0},\quad 0 \lt d(I)\leqslant N, \end{equation}

where δ here denotes the Kronecker delta applied componentwise to the multi-index. For $\mathfrak{r} \gt 1$ we define $\rho^{(n)}(x):= \mathfrak{r}^{n|\mathfrak{s}|}\rho (\mathfrak{r}^{n}\cdot x)$ as well as

\begin{equation*} \rho^{(n,m)} = \rho^{(n)}\ast \rho^{(n+1)}\ast \cdots \ast \rho^{(m)}, \end{equation*}

with the convention $\rho^{(n,n)}= \rho^{(n)}$. We then have the following result.

Lemma 3.10. If $\,\mathfrak{r} \gt \| \rho\|_{L^1} \gt 1$, then there exists a smooth, compactly supported function $\varphi^{(n)}=\lim_{m\rightarrow \infty} \rho^{(n,m)}$, where the convergence takes place in $\mathcal{D}(\mathbb{G})$ and ${\text{supp}}(\varphi^{(n)})\subset B_{C \mathfrak{r}^{-n}}$ for $C=\frac{\mathfrak{r}}{\mathfrak{r}-1}$.

Proof. First, note that since $\|\rho^{(m)}\|_{L^1}=\|\rho\|_{L^1}$ and $\rho^{(m)}$ is supported on a ball of radius $\mathfrak{r}^{-m}$, it follows from the mean value theorem on $\mathbb{G}$ (cf. (2.5) with a = 0) that

\begin{equation*} |f\ast \rho^{(m)}(x)-f(x)| = \left| \int_{\mathbb{G}} (f(y)-f(x))\rho^{(m)}(y^{-1}x)\text{d} y\right| \lesssim \max_{i=1,...,d} \|Y_i f\|_{{L^\infty}} \|\rho\|_{L^1} \mathfrak{r}^{-m}. \end{equation*}

Secondly, since

(3.6)\begin{equation} Y^I \rho^{(n,m)}= Y^I(\rho^{(n)}\ast \rho^{(n+1,m)})=(Y^I \rho^{(n)})\ast \rho^{(n+1,m),} \end{equation}

one finds by applying Young’s convolution inequality m-times, cf. $\S$ 2.2, that

\begin{align*} \|Y_i \rho^{(n,m)} \|_{L^\infty} &= \|(Y_i \rho^{(n)})\ast \rho^{(n+1,m)} \|_{L^\infty}\leqslant \| Y_i \rho^{(n)}\|_\infty \|\rho\|_{L^1}^{m-n-1} \\ &\leqslant \mathfrak{r}^{n(\mathfrak{s}_i+|\mathfrak{s}|)} \|Y_i \rho\|_{L^\infty} \|\rho\|^{m-n-1}_{L^1} \\ &\leqslant \mathfrak{r}^{2n|\mathfrak{s}|} \|Y_i \rho\|_{L^\infty} \|\rho\|^{m-n-1}_{L^1} \end{align*}

and therefore

\begin{align*} \|\rho^{(n,m)}- \rho^{(n,m-1)}\|_{L^\infty} &= \|\rho^{(n,m-1)}\ast \rho^{(m)}- \rho^{(n,m-1)}\|_{L^\infty} \\ &\lesssim \max_{i=1,...,d} \|Y_i \rho^{(n,m-1)}\|_{{L^\infty}} \|\rho\|_{L^1} \mathfrak{r}^{-m} \\ &\leqslant \mathfrak{r}^{2n|\mathfrak{s}|}\max_{i=1,...,d}\|Y_i\rho\|_{L^\infty} \|\rho\|^{m-n-2}_{L^1} \mathfrak{r}^{-m} \\ &\leqslant \mathfrak{r}^{2n|\mathfrak{s}|} \frac{\max_{i=1,...,d}\|Y_i\rho\|_{L^\infty} }{\|\rho\|^{n+2}_{L^1} } \left(\frac{\|\rho\|_{L^1}}{\mathfrak{r} }\right)^m, \end{align*}

which is summable in m since we assumed that $\mathfrak{r} \gt \|\rho\|_{L^1}$. Thus we may write

\begin{equation*} \rho^{(n,m)} =\rho^{(n)} +\sum_{k=0}^{m-n-1} \rho^{(n,m-k)} - \rho^{(n,m-1-k)}, \end{equation*}

and it follows that $\rho^{(n,m)}$ converges uniformly as $m\to +\infty$. Using (3.6) we obtain convergence in $\mathcal{D}(\mathbb{G})$. It remains to check the support of $\varphi^{(n)}$. For two functions $f_1,f_2$ such that ${\text{supp}}(f_i)\in B_{r_i}$ one has ${\text{supp}}(f_1\ast f_2)\in B_{r_1+r_2}$, and hence it follows that $ \varphi^{(n)}$ is supported in a ball of radius $\sum_{m=n}^\infty \mathfrak{r}^{-m}= \frac{\mathfrak{r}^{-n}}{1-\mathfrak{r}^{-1}} $.

It follows from the definitions that $\varphi^{(n)}= \rho^{(n)}\ast \varphi^{(n+1)}$. We set

\begin{equation*} \tilde{\rho}^{(m,n)} := \tilde{\rho}^{(m)}\ast \tilde{\rho}^{(m-1)}\ast \cdots \ast \tilde{\rho}^{(n),} \end{equation*}

and using (2.13) we note that $\tilde{\varphi}^{(m+1)}\ast \tilde{\rho}^{(m)}=\tilde{\varphi}^{(m)}$ and $ \tilde{\rho}^{(m,n)} \to \tilde{\varphi}^{(n)}$ in $\mathcal{D}(\mathbb{G})$ as $m\to \infty$.

Lemma 3.11. Let $\mathfrak{r} \gt 1$ and ρ be as in Lemma 3.10 Let α > 0 and $\xi_n :\mathbb{G}\rightarrow \mathbb{R}$ be a sequence of functions such that for every compact $ \mathfrak{K}\subset \mathbb{G}$ there exists a $C_{\mathfrak{K}}$ such that $\sup_{x \in \mathfrak{K}}|\xi_n(x)| \leqslant C_{\mathfrak{K}}\mathfrak{r}^{\alpha n}$ and such that $\xi_n = \xi_{n+1}\ast \tilde{\rho}^{(n)}$. Then the sequence ξ_n is Cauchy in $\mathcal{C}^{-\beta}(\mathbb{G})$ for every $\beta \gt \alpha$ and the limit ξ satisfies $\xi_n = \xi\ast \tilde{\varphi}^{(n)}$. If, furthermore, for some $x\in \mathbb{G}$ and $\gamma \gt -\alpha$ one has the bound

\begin{equation*}|\xi_n(y)| \leqslant \mathfrak{r}^{\alpha n}\left(|x^{-1} y|^{\gamma+\alpha}+\mathfrak{r}^{-(\gamma+\alpha)n}\right)\end{equation*}

uniformly over $n\geqslant 0$ and $y\in \mathbb{G}$ such that $|x^{-1} y|\leqslant 1$, then $|\langle \xi,\psi^\lambda_x\rangle|\lesssim \lambda^\gamma$ for all $\lambda \leqslant 1$ and $\phi \in \mathfrak{B}_{r}$, where $r= -[-\alpha]$.

Proof. The proof follows along the same steps as that of [Reference Friz and Hairer33, Lem. 13.24], but one has to be careful since convolution is non-commutative in our setting. Let $\lambda \in (0,1]$ and $\psi \in \mathfrak{B}_{r}$; we first establish the bound

(3.7)\begin{equation} |\langle \xi_{n}-\xi_{n+1},\psi^\lambda_x\rangle| \lesssim \lambda^{-\beta}\mathfrak{r}^{(\alpha-\beta)n} \end{equation}

uniformly over $\psi \in \mathfrak{B}_{r}$, $\lambda \in (0,1]$ and locally uniformly over $x \in \mathbb{G}$. First observe the trivial bound,

\begin{equation*} |\langle \xi_{n}-\xi_{n+1},\psi^\lambda_x\rangle| \leqslant \sup_{x \in \bar{\mathfrak{K}}}\left(|\xi_n(x)| + |\xi_{n+1}(x)|\right)\|\psi^\lambda_x\|_{L^1} \leqslant (1+\mathfrak{r}^{\alpha})C_{\bar{\mathfrak{K}}}\bar{C} \mathfrak{r}^{\alpha n}, \end{equation*}

where $\bar{C}:= \sup \{\int |\psi^\lambda(x)|\text{d} x \ : \ \psi \in \mathfrak{B}_{r}\} \lt +\infty$. Hence, when $\lambda\leqslant \mathfrak{r}^{-n}$ the bound (3.7) holds directly.

In the case $\mathfrak{r}^{-n}\leqslant \lambda$, using (2.12) we rewrite

\begin{equation*}|\langle \xi_{n}-\xi_{n+1},\psi^\lambda_x\rangle| =\langle \xi_{n+1}\ast \tilde{\rho}^{(n)}-\xi_{n+1},\psi^\lambda_x\rangle| = |\langle \xi_{n+1}, \psi_x^\lambda*\rho^{(n)} -\psi_x^\lambda\rangle|.\end{equation*}

By Taylor’s theorem and in particular Remark 2.16

(3.8)\begin{equation} |\psi_x^\lambda(z) - \tilde{\operatorname{P}}^r_y[\psi_x^\lambda](z)| \, \lesssim_{r} \sum_{\delta \gt 0}\lambda^{-(r+\delta)-|\mathfrak{s}|}|y^{-1} z|^{r+\delta}, \end{equation}

where the sum runs over a finite set. It follows from (2.11), Remark 2.8, Remark 2.16 and (3.5) that we have

\begin{equation*} \tilde{\operatorname{P}}^r_y[\psi^\lambda_x]*\rho^{(n)} (z){=} \int \tilde{\operatorname{P}}^r_y[\psi^\lambda_x](z w^{-1}) \rho^{(n)}(w) dw{=}\int \tilde{\operatorname{P}}^0_y[\psi^\lambda_x](z w^{-1}) \rho^{(n)}(w) dw= \psi_x^{\lambda}(y), \end{equation*}

and thus

\begin{equation*} \psi_x^\lambda*\rho^{(n)}(y) -\psi_x^\lambda(y) = (\psi_x^\lambda-\tilde{\operatorname{P}}^r_y[\psi^\lambda_x])\ast \rho^{(n)}(y), \end{equation*}

which by (3.8) is bounded uniformly by a multiple of $\sum_{\delta \gt 0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{-n(r+\delta)}\ $ and supported on a ball of radius $\lambda + \mathfrak{r}^{-n}\leqslant 2\lambda$. Note that we were able to keep the convolution with $\rho^{(n)}$ on the right in both expressions by using the rebased Taylor polynomial of Rem. 2.16. Using the bound $|\xi_{n+1}|\lesssim_\alpha \mathfrak{r}^{\alpha n}$ we conclude that

\begin{equation*} |\langle \xi_{n+1}, \psi_x^\lambda*\rho^{(n)} -\psi_x^\lambda\rangle| \lesssim \sum_{\delta \gt 0}\lambda^{-(r+\delta)}\mathfrak{r}^{-n(r+\delta)} \mathfrak{r}^{\alpha n} \lesssim \lambda^{-\beta}\mathfrak{r}^{(\alpha-\beta)n,} \end{equation*}

where we used $\mathfrak{r}^{-n}\leqslant \lambda$ and without loss of generality assumed that $r\geqslant \beta$ in the last line. Hence $\{\xi_n\}_{n\geqslant 1}$ is Cauchy in $\mathcal{C}^{-\beta}(\mathbb{G})$ and for any test function,

\begin{equation*} \langle \xi_n,\psi\rangle = \langle\xi_{n+1}, \psi* \rho^{(n)}\rangle = \langle \xi_{m+1}, \psi*\rho^{(n,m)}\rangle = \langle \xi, \psi*\varphi^{(n)}\rangle, \end{equation*}

showing that $\xi_n= \xi*\tilde{\varphi}^{(n)}$.

To prove the second claim, for any test function $\psi\in \mathfrak{B}_r$, λ > 0 and $x\in \mathbb{G}$ we write

\begin{equation*} \langle \xi,\psi^\lambda_x\rangle = \langle \xi_n, \psi^\lambda_x\rangle + \sum_{k\geqslant n} \langle \xi_{k+1}-\xi_k,\psi^\lambda_x\rangle, \end{equation*}

where n is chosen so that $\lambda \in [\mathfrak{r}^{-(n+1)},\mathfrak{r}^{-n}]$ and as a consequence

\begin{equation*} |\langle \xi_n,\psi^\lambda_x\rangle| \leqslant \lambda^{-|\mathfrak{s}|} \mathfrak{r}^{\alpha n} \int_{B_\lambda(x)} \left(|x^{-1} y|^{\gamma+\alpha}+ \mathfrak{r}^{-(\gamma+\alpha)n}\right)\text{d} y \, \lesssim\, \lambda^{\gamma+\alpha}\mathfrak{r}^{\alpha n} + \mathfrak{r}^{-\gamma n} \lesssim \lambda^\gamma. \end{equation*}

To bound the summands $\langle \xi_{k+1} -\xi_k,\varphi^\lambda_x\rangle$ we proceed as in the proof of the first claim to find that

\begin{align*} |\langle \xi_{k} -\xi_{k+1},\psi^\lambda_x\rangle| & = |\langle \xi_{k+1}, \psi^\lambda_x*\rho^{(k)}-\psi^\lambda_x\rangle| \\ &= |\langle \xi_{k+1}, (\psi^\lambda_x-\tilde{\operatorname{P}}^r_x[\psi^\lambda_x] ))*\rho^{(k)}\rangle| \\ &\lesssim \sum_{\delta \gt 0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{-k(r+\delta)} \int_{B_{2\lambda}(x)} |\xi_{k+1}(y)|\text{d} y\\ &\lesssim \sum_{\delta \gt 0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{k(\alpha-r-\delta)} \\ &\quad \times \left(\int_{B_{2\lambda}(x)} |x^{-1} y|^{\gamma+\alpha}\text{d} y+\mathfrak{r}^{-(\gamma+\alpha)(k+1)}\lambda^{|\mathfrak{s}|}\right) \\ &\lesssim \sum_{\delta \gt 0}\left(\lambda^{\gamma+\alpha-r-\delta}\mathfrak{r}^{k(\alpha-r-\delta)} + \lambda^{-(r+\delta)}\mathfrak{r}^{-k(\gamma+r+\delta)}\right), \end{align*}

where the sum in δ is again over a finite set. Since $r+\delta \gt \alpha$ the quantity on the left is summable over $k\geqslant n$ and is of order λ ^γ, concluding the proof.

We are now ready to proof Theorem 3.7.

Proof of Theorem 3.7

For m > 0 we first define the operators $\mathcal{R}^{(m,m)}:\mathscr{D}^\gamma \rightarrow C(\mathbb{G})$ by setting

\begin{equation*} \left(\mathcal{R}^{(m,m)}f\right)(y) := \left( \Pi_yf(y)\ast \tilde{\varphi}^{(m)}\right)(y) = \langle\Pi_y f(y),{\varphi}^{(m)}_{y} \rangle. \end{equation*}

We then set, for n < m,

\begin{equation*} \mathcal{R}^{(m,n)} f = \mathcal{R}^{(m,m)}f \ast \tilde{\rho}^{(m-1,n),} \end{equation*}

and recalling that $\tilde{\varphi}^{(m+1)}\ast \tilde{\rho}^{(m)}=\tilde{\varphi}^{(m)}$ we find

\begin{align*} \mathcal{R}^{(m,n)}f - \mathcal{R}^{(m+1,n)}f &= \mathcal{R}^{(m,m)}f \ast \tilde{\rho}^{(m-1,n)}- \mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m,n)}\\ &= \big(\mathcal{R}^{(m,m)}f - \mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m)}\big)\ast \tilde{\rho}^{(m-1,n)}. \end{align*}

Using the identity

\begin{equation*}(F\ast \tilde{\phi})(x)= \langle F, \phi_x\rangle= \int F(y)\phi_x(y)dy,\end{equation*}

it follows by a straightforward computation that

\begin{equation*} \begin{array}{ll} &\big(\mathcal{R}^{(m,n)}f - \mathcal{R}^{(m+1,n)}f\big)(x) \\ &\quad = \int \big(\mathcal{R}^{(m,m)}f - \mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m)}\big)(y) {\rho}_x^{(m-1,n)}(y) dy\\ &\quad = \int\int \langle\Pi_yf(y) - \Pi_zf(z), {\varphi}_z^{(m+1)}\rangle {\rho}_y^{(m)}(z)dz {\rho}_x^{(m-1,n)}(y) dy. \end{array} \end{equation*}

Therefore, using the fact that $\Pi_z = \Pi_y \Gamma_{yz}$, we have

\begin{align*} &\big(\mathcal{R}^{(m,n)}f - \mathcal{R}^{(m+1,n)}f\big)(x) \\ &\quad = \int\int \langle\Pi_y(f(y) - \Gamma_{yz} f(z)), {\varphi}_z^{(m+1)}\rangle {\rho}_y^{(m)}(z)dz {\rho}_x^{(m-1,n)}(y) dy. \end{align*}

Then successively applying the facts that

• • $\sup_{y,\,z \in B_{2\mu\mathfrak{r}^{-m}}(0)}|\langle \Pi_y \tau,\varphi^{(m+1)}_z|\lesssim \| \Pi\|_{\gamma; B_{2\mu\mathfrak{r}^{-m}}(0)} \mathfrak{r}^{-\alpha m}|\tau|_\alpha$ for $\tau\in T_\alpha$,

• • $\|f(y)-\Gamma_{yz}f(z)\|_\alpha \lesssim\|f\|_{\gamma;B_{\mathfrak{r}^{-m}}(y)} \mathfrak{r}^{(\alpha-\gamma)m}$ uniformly over $|y^{-1}z|\lesssim \mathfrak{r}^{-m},$

• • $\|\rho_x^{(n,m-1)}\|_{L^1}\lesssim 1$ uniformly over $m \gt n\geqslant 0$,

we establish the bound

\begin{equation*} \|\mathcal{R}^{(m,n)}f - \mathcal{R}^{(m+1,n)}f\|_{L^\infty(\mathfrak{K})}\lesssim \| \Pi\|_{\gamma, \bar{\mathfrak{K}}} \|f\|_{\gamma;\bar{\mathfrak{K}}} \mathfrak{r}^{-\gamma m}, \end{equation*}

uniformly over $m\geqslant n\geqslant 0$, where $\bar{\mathfrak{K}}$ denotes the two flattening of the set $\mathfrak{K}$. It follows directly from the definition and properties of a model that we also have the bound

(3.9)\begin{equation} \|\mathcal{R}^{(n,n)} f\|_{L^\infty(\mathfrak{K})} \lesssim \| \Pi\|_{\gamma, \bar{\mathfrak{K}}} \|f\|_{\gamma;\bar{\mathfrak{K}}} \mathfrak{r}^{-\underline{\alpha}n}, \end{equation}

where $\underline{\alpha}= \min A$. It follows that $\mathcal{R}^{(m,n)}f$ converges uniformly on compacts as $m\rightarrow \infty$ to $\mathcal{R}^{(n)}f$ which also satisfies the bound (3.9). Since it also holds that for every $m\geqslant n+1$,

\begin{equation*} \mathcal{R}^{(m,n)}f = \mathcal{R}^{(m,m)}f \ast \tilde{\rho}^{(m-1,n)} = \mathcal{R}^{(m,m)} f \ast \tilde{\rho}^{(m-1,n+1)}\ast \tilde{\rho}^{(n)} = \mathcal{R}^{(m,n+1)}f \ast\tilde{\rho}^{(n)}, \end{equation*}

we find that

\begin{equation*}\mathcal{R}^{(n)}f= \mathcal{R}^{(n+1)}f \ast\tilde{\rho}^{(n)}.\end{equation*}

Therefore we may apply Lemma 3.11 to see that there exists a limit $\mathcal{R} f:=\lim_{n\rightarrow \infty}\mathcal{R}^{(n)}f $.

With the validity of the limit established we now turn to show the bounds (3.3) and (3.4); this requires us to keep more careful track of the underlying sets in the proof. We begin with (3.3), first noting that if we define $f_x(y) := \Gamma_{y,x}f(x)$ then one has $\mathcal{R}^{(n,n)}f_x = \Pi_x f(x)\ast \tilde{\varphi}^{(n)}$ so that (3.3) can be written as the claim that for all $\lambda \in (0,1]$,

(3.10)\begin{equation} \sup_{\psi \in \mathfrak{B}_{r}} | \langle \mathcal{R}(f-f_x),\psi^\lambda_x\rangle|\lesssim_{\mathfrak{K}} \lambda^{\gamma}\|f\|_{\gamma;B_{2\lambda}(x)}\|\Pi\|_{\gamma;B_{2\lambda}(x)}. \end{equation}

Using that $|(f-f_x)(z)|_\alpha\lesssim\|f\|_{\gamma;B_{{|z^{-1}x|}}(x),} {|z^{-1}x|}^{\gamma-\alpha}$ for $x,\,z \in \mathfrak{K}$, it follows that for all $y\in B_\lambda(x)$ one has

(3.11)\begin{align} &|(\mathcal{R}^{(n,n)}(f-f_x)(y)| = |\langle \Pi_y(f(y)-f_x(y)),\varphi^{(n)}_y\rangle| \nonumber\\ &\quad = |\langle \Pi_y(f(y)-\Gamma_{y,x}f(x)),\varphi^{(n)}_y\rangle| \nonumber\\ &\quad \lesssim \|\Pi\|_{\gamma;B_\lambda(y)} \|f\|_{\gamma;B_{C\mathfrak{r}^{-n}}(y)} \sum_{\underline{\alpha}\leqslant\alpha\leqslant \gamma} \mathfrak{r}^{-\alpha n}|y^{-1}x|^{\gamma-\alpha} \nonumber\\ &\quad \lesssim \|\Pi\|_{\gamma;B_\lambda(y)} \|f\|_{\gamma;B_{C\mathfrak{r}^{-n}}(y)} \mathfrak{r}^{-\underline{\alpha}n}({|y^{-1}x|}^{\gamma-\underline{\alpha}}+ \mathfrak{r}^{(\underline{\alpha}-\gamma)n}), \end{align}

where $C(\mathfrak{r}):= \frac{\mathfrak{r}}{\mathfrak{r}-1}$ is as in Lemma 3.10. Given $n \gt n_0(\lambda)$ sufficiently larger, we have a uniform bound $C\mathfrak{r}^{-n}\leqslant \lambda$. By the convergence of $\mathcal{R}^{(n,n)}$ in the first exponent, it follows that $\mathcal{R}^{(n)}$ satisfies the same bound and so inspecting the proof of the second half of Lemma 3.11, in particular noticing that we integrate the above estimate over $y\in B_\lambda(x)$, we conclude that (3.10) holds for the limit $\mathcal{R}$.

Using the obvious notation we can also rewrite (3.4) as

\begin{equation*} \begin{array}{ll} &\sup_{\psi \in \mathfrak{B}_{r}} | \langle \mathcal{R}(f-f_x)- \bar{\mathcal{R}}(\bar{f}-\bar{f}_x),\psi^\lambda_x\rangle|\lesssim \lambda^\gamma \big(\|f;\bar{f}\|_{\gamma;B_{2\lambda}(x)}\|\bar{\Pi}\|_{\gamma;B_{2\lambda}(x)} \\ &\quad +\|f\|_{\gamma;B_{2\lambda}(x)}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\lambda}(x)} \big), \end{array} \end{equation*}

uniformly over $\lambda \in (0,1]$. This is seen very similar to (3.10), but using this time that for n large enough,

\begin{equation*} \begin{array}{l} |\mathcal{R}^{(n,n)}(f-f_x)- \bar{\mathcal{R}}^{(n,n)}(\bar{f}-\bar{f}_x)(y)| \\ = |\langle \Pi_y(f(y)-\Gamma_{x,y}f(x))-\bar{\Pi}_y(\bar{f}(y)-\bar{\Gamma}_{y,x}\bar{f}(x)),\varphi^{(n)}_y\rangle|\\ = |\langle \Pi_y(f(y)-\Gamma_{x,y}f(x)-\bar{f}(y)+\bar{\Gamma}_{y,x}\bar{f}(x)) + (\Pi_y-\bar{\Pi}_y)(\bar{f}(y)-\bar{\Gamma}_{y,x}\bar{f}(x)),\varphi^{(n)}_y\rangle|\\ \lesssim \big(\|\Pi\|_{\gamma;B_{\lambda}(y)} \|f;\bar f\|_{\gamma;B_{\lambda}(y)}+\|\Pi-\bar{\Pi}\|_{\gamma;B_{\lambda}(y)} \|\bar f\|_{\gamma;B_{\lambda}(y)}\big)\sum_{\underline{\alpha}\leqslant\alpha\leqslant \gamma} \mathfrak{r}^{-\alpha n}{|y^{-1}x|}^{\gamma-\alpha}. \end{array} \end{equation*}

It remains to show that the reconstruction map is unique for γ > 0 and that $\mathcal{R}f$ is indeed an element of $C^{\underline{\alpha}}$. This is done exactly as in [Reference Hairer41, $\S$ 3].

3.2.2. Functions as modelled distributions

Let $\bar{\mathcal{T}}=(\bar{\operatorname{T}},\bar{\operatorname{G}})$ be the polynomial regularity structure equipped with the polynomial model. We show that in this setting the reconstruction theorem and its inverse map Taylor polynomials to Hölder functions and vice versa.

Theorem 3.12. For γ > 0 the reconstruction operator is an isomorphism between $\mathscr{D}^\gamma (\mathbb{G})$ and $\mathcal{C}^\gamma(\mathbb{G})$. In particular the inverse of the reconstruction map is given by

\begin{equation*} \mathcal{C}^{\gamma}(\mathbb{G}) \ni f(\cdot) \mapsto \pmb{\operatorname{{P}}}^{\gamma}_{(\,\cdot\,)}[f] \in \mathscr{D}^\gamma(\mathbb{G}), \end{equation*}

where $\pmb{\operatorname{{P}}}^a$ is defined in Definition 8

Proof. Given a modelled distribution in $\mathscr{D}^\gamma$ and since we are working with the polynomial regularity structure equipped with its canonical model, it follows directly from the bound satisfied by the image of the reconstruction operator, i.e. Equation (3.3), and the definition of $\mathcal{C}^\gamma$ in (2.15) that the reconstruction operator is a map $\mathscr{D}^\gamma (\mathbb{G}) \to \mathcal{C}^{\gamma}(\mathbb{G})$. Continuity also follows from (3.3) and by linearity.

To see the other direction recall that given a Hölder continuous distribution $f\in \mathcal{C}^\gamma(\mathbb{G})$ by Proposition 2.23 $\{X^I f\}_{d(I) \lt \gamma}$ are actually functions and, in particular, that $\tilde{P}_x= \Pi_x\pmb{\operatorname{{P}}}^{[\gamma]}_{x}[f]$. Therefore, for $\lambda=|x^{-1}y|_{\mathbb{G}}$

(3.12)\begin{equation} \left| \Pi_x\big(\pmb{\operatorname{{P}}}^{[\gamma]}_{x}[f] -\Gamma_{x,y}\pmb{\operatorname{{P}}}^{[\gamma]}_{y}[f] \big) (\psi^\lambda_x) \right| = |\langle \tilde{P}_x -f,\psi^\lambda_x\rangle| +|\langle f-\tilde{P}_y, \psi^\lambda_x\rangle| \lesssim \lambda^\gamma \ \end{equation}

uniformly in $\psi\in \mathfrak{B}_{r}$. On the other hand, writing $\pmb{\operatorname{{P}}}^{[\gamma]}_{x}[f] -\Gamma_{x,y}\pmb{\operatorname{{P}}}^{[\gamma]}_{y}[f] = \sum_{d(I) \lt \gamma} c^I_{x,y} \pmb{\eta}^I$, we find that

\begin{equation*}\Pi_x\big(\pmb{\operatorname{{P}}}^{[\gamma]}_{x}[f] -\Gamma_{x,y}\pmb{\operatorname{{P}}}^{[\gamma]}_{y}[f] \big) (\psi^\lambda_x)= \sum_{d(I) \lt \gamma} c^I_{x,y} \Pi_e\pmb{\eta}^I (\psi^\lambda)= \sum_{d(I) \lt \gamma} c^I_{x,y}\lambda^{d(I)} \Pi_e\pmb{\eta}^I (\psi).\end{equation*}

Using that $\mathcal{P}_{\gamma}$ is a finite dimensional vector space and the surjectivity of the linear map

\begin{equation*}C_c^\infty(B_1(e)) \to \mathbb{R}^{\dim \mathcal{P}_{\gamma}}, \quad \psi \mapsto \{\Pi_e\pmb{\eta}^I (\psi) \}_{d(I) \lt \gamma},\end{equation*}

we find that

(3.13)\begin{equation} \sum_{d(I) \lt \gamma} |c^I_{x,y}|\lambda^{d(I)} \lesssim_{\gamma} \sup_{\psi\in \mathfrak{B}_{r}}\left|\sum_{d(I) \lt \gamma} c^I_{x,y}\lambda^{d(I)} \Pi_e\pmb{\eta}^I (\psi)\right|. \end{equation}

Together, (3.12) and (3.13) imply that $|c^I_{x,y}|\lesssim \lambda^{\gamma- d(I)}$; we may then apply Lemma 3.6 to see that $\pmb{\operatorname{{P}}}^{[\gamma]}_{(\,\cdot\,)}[f] \in \mathscr{D}^\gamma $.

3.2.3. Local reconstruction

We will require the following further refinement of the reconstruction theorem which is an analogue in our case of [Reference Hairer41, Prop.  7.2].

proposition 3.13.

In the setting of Theorem 3.7 one has the improved bound,

(3.14)\begin{equation} \sup_{\psi\in \mathfrak{B}_m}|(\mathcal{R} f - \Pi_x f(x))(\psi^\lambda_x)| \lesssim \lambda^{\gamma} \|\Pi\|_{\gamma;B_{2\mu\lambda}(x)} \sup_{y,\,z \in {\text{supp}} (\psi^\lambda_x)} \sup_{\ell \lt \gamma} \frac{|f(z)-\Gamma_{zy}f(y)|_{\ell}}{{|y^{-1}z|}^{\gamma-\ell}}, \end{equation}

as well as, given a second model $\bar{M}(\bar{\Pi},\bar{\Gamma})$ and a modelled distribution $\bar{f}\in \mathscr{D}^\gamma_{\bar{M}}$, the analogous bound that for every $\lambda \in (0,1]$,

(3.15)\begin{align} &\sup_{\psi \in \mathfrak{B}_{r}} |\langle \mathcal{R}_M f-\mathcal{R}_{\bar{M}}\bar{f} -\Pi_x f(x)+\bar{\Pi}_x \bar{f}(x),\psi^{\lambda}_x\rangle| \nonumber\\ &\quad \lesssim \lambda^\gamma \bigg(\|f;\bar{f}\|_{\gamma;{\text{supp}} (\psi^\lambda_x)}\|\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)} +\|f\|_{\gamma;{\text{supp}} (\psi^\lambda_x)}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)} \bigg), \end{align}

for any $x\in \mathbb{G}$ and any $\lambda \in (0,1]$.

Proof. Since the right-hand side of (3.14) is linear in f, as in [Reference Hairer41] we may assume it to be equal to 1. We use the functions $\rho^{(n)}$ and $\varphi^{(n)}$ from $\S$ 3.2.1 and recall that they satisfy

\begin{equation*} \varphi^{(n-1)}(x) = \int_{B_{\mathfrak{r}^{-n+1}}} \rho^{(n-1)}(y) \varphi_y^{(n)}(x) dy, \quad \varphi^{(n-1)}_y= \int_{B_{\mathfrak{r}^{-n+1}}} \rho^{(n-1)}(w) \varphi_{yw}^{(n)}(x) dw; \end{equation*}

in particular since $\int \rho^{(n)}(x)\text{d} x = 1$ we have $\int \varphi_y^{(n)} (\,\cdot\,)\text{d} y =1$ for all $n\geqslant 0$. Define, for fixed $\psi^\lambda_x$, the set

\begin{equation*} \Lambda_n= \left\{y\in \mathbb{G} \ : \ {\text{supp}} (\varphi^{(n)}_y)\cap {\text{supp}} (\psi^\lambda_x) \neq \emptyset \right\}\subset \mathbb{G} \end{equation*}

which by definition is contained in $B_{\lambda+ C\mathfrak{r}^{-n}}(x)$, with $C=\frac{\mathfrak{r}}{\mathfrak{r}{-1}}$ as in Lemma 3.10, as well as a (measurable) function

\begin{equation*} \pi_n: \Lambda_n\to {\text{supp}} (\varphi^{(n)}_y)\cap {\text{supp}} (\psi^\lambda_x). \end{equation*}

Next, let

\begin{align*} R_n &= \int_{\Lambda_n} \big( \mathcal{R}f- \Pi_{\pi_n(y)}f(\pi_n(y))\big) (\psi_x^\lambda \varphi^n_y) dy\\ &= \langle\mathcal{R} f,\psi_x^\lambda\rangle - \int_{\Lambda_n}\Pi_{\pi_n(y)}f(\pi_n(y)) (\psi_x^\lambda \varphi^n_y) dy. \end{align*}

It follows that for $n_0= \min \{n\in \mathbb{N} \ : \ \mathfrak{r}^{-n}\leqslant \lambda \}$, one has

(3.16)\begin{equation} \left|(\mathcal{R} f - \Pi_x f(x))(\psi^\lambda_x)- R_n\right| = \left| \int_{\Lambda_{n}}\big(\Pi_x f(x)- \Pi_{\pi_n(y)} f (\pi_n(y))\big) (\psi_x^\lambda \varphi^n_y) dy\right| \lesssim \lambda^\gamma \end{equation}

as well as for $n \gt n_0$

\begin{align*} R_{n-1}- R_{n} &= \int_{\Lambda_{n-1}}\Pi_{\pi_{n-1}(y)}f(\pi_{n-1}(y)) (\psi_x^\lambda \varphi^{n-1}_y) dy \\ &\quad - \int_{\Lambda_n}\Pi_{\pi_n(z)}f(\pi_n(z)) (\psi_x^\lambda \varphi^n_z) dz\\ &= \int_{B_{\mathfrak{r}^{-n+1}}} \rho^{(n-1)}(w)\int_{\Lambda_{n}}\Big( \Pi_{\pi_{n-1}(zw^{-1})}f(\pi_{n-1}(zw^{-1})) \\ &\quad -\Pi_{\pi_n(z)}f(\pi_n(z))\Big) (\psi_x^\lambda \varphi^n_z) dzdw. \end{align*}

Since $|\pi_n(z)^{-1}\pi_{n-1}(zw^{-1})|\leqslant \bar{C}\mathfrak{r}^{-n}$ and for $\tau\in \operatorname{T}_{\alpha}$ such that $|\tau| \leqslant 1$

\begin{equation*} |\Pi_{\pi_n(z)} \tau (\psi_x^\lambda \varphi^n_z)| \lesssim \lambda^{|\mathfrak{s}|} \mathfrak{r}^{-\alpha n}, \end{equation*}

we find that

\begin{equation*} \int_{\Lambda_{n}} \Big| \big(\Pi_{\pi_{n-1}(zw^{-1})}f(\pi_{n-1}(zw^{-1})) -\Pi_{\pi_n(z)}f(\pi_n(z))\big) (\psi_x^\lambda \varphi^n_z) \Big| dz\lesssim \mathfrak{r}^{-\gamma n} \end{equation*}

and thus conclude

\begin{equation*} |R_n-R_{n-1}| \lesssim \mathfrak{r}^{-\gamma n}. \end{equation*}

A similar argument gives $|R_n|\to 0$ as $n \to \infty$ which combined with (3.16) concludes the proof of (3.14). The proof of the analogous bound (3.15) follows in a similar manner.

3.3. Singular modelled distributions

As in [Reference Hairer41] we will eventually be concerned with solutions to SPDEs that take values in spaces of modelled distributions with permissible singularities in some regions of the domain. Our main example will be modelled distributions on space-time domains that are allowed to be discontinuous at $\{t=0\}$, see $\S$ 4. However, as in [Reference Hairer41] we build the notion of singular modelled distributions allowing for singularities on more general sets, generalizations of which have been used in [Reference Gerencsér and Hairer35, Reference Gerencsér and Hairer37] to study singular equations with boundary conditions.

We fix a homogeneous sub-Lie group $P\subset \mathbb{G}$ with associated Lie algebra for which we write $\mathfrak{p} \subset \mathfrak{g}$. The assumption that P be a homogeneous sub-Lie group means that the scaling map $\mathfrak{s}$ restricts to a map $\bar{\mathfrak{s}}:=\mathfrak{s}|_{\mathfrak{p}}:\mathfrak{p} \rightarrow \mathfrak{p}$ which is diagonalizable. We fix a decomposition $\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{p}^c$ such that $\mathfrak{p}^c$ is also invariant under $\mathfrak{s}$ and define the homogeneous dimension of P and its complement $P^c := \exp(\mathfrak{p}^c)$ as

(3.17)\begin{equation} |\bar{\mathfrak{s}}|:= \text{trace} ( \mathfrak{s}|_{\mathfrak{p}}) \quad\text{and}\quad |\mathfrak{m}| = \text{trace}(\mathfrak{s}|_{\mathfrak{p}^c}). \end{equation}

Furthermore, we set

\begin{equation*} |x|_P := 1 \wedge d_{\mathbb{G}}(x,P) = 1\wedge \inf\left\{z \in P\,:\, {|x^{-1}z|} \right\}, \quad |x,y|_P := |x|_P \wedge |y|_P. \end{equation*}

Note that since P is closed (being the image under exp of a linear subspace of $\mathfrak{g}$), one sees easily that the infimum above is actually a minimum. Given $\mathfrak{K} \subset \mathbb{G}$ we define the set

\begin{equation*} \mathfrak{K}_P := \left\{(x,y) \in (\mathfrak{K}\setminus P)^2\,:\, x\neq y \, \text{and }\, {|x^{-1}y|} \leqslant |x,y|_{P}\right\}. \end{equation*}

That is $\mathfrak{K}_P$ contains all the points in $\mathfrak{K}$ that are closer to each other than they are to P.

Definition 10. (Singular modelled distributions)

Given a regularity structure $\mathcal{T}$ and a subgroup P as above, for any γ > 0, $\eta \in \mathbb{R}$ and maps $f:\mathbb{G}\setminus P \rightarrow \mathcal{T}$, we set

\begin{equation*} \|f\|_{\gamma,\eta;\mathfrak{K}} := \sup_{x \in \mathfrak{K}\setminus P} \sup_{\zeta \lt \gamma} \frac{|f(x)|_{\zeta}}{|x|_P^{(\eta-\zeta)\wedge 0}},\quad \unicode{x27E6} f\unicode{x27E7}_{\gamma,\eta;\mathfrak{K}} := \sup_{x \in \mathfrak{K}\setminus P} \sup_{\zeta \lt \gamma} \frac{|f(x)|_{\zeta}}{|x|^{\eta-\zeta}_P}. \end{equation*}

Then given a model $M=(\Pi,\Gamma)$ as well as a sector V, the space $\mathscr{D}^{\gamma,\eta}_{P,M}(V)$ consists of all functions $f: \mathbb{G}\setminus P \rightarrow V_{ \lt \gamma}$ such that for every compact set $\mathfrak{K} \subset \mathbb{G}$,

\begin{equation*} {\left\vert\left\vert\left\vert{f}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}} := \|f\|_{\gamma,\eta;\mathfrak{K}} + \sup_{(x,y)\in \mathfrak{K}_P} \sup_{\zeta \lt \gamma} \frac{|f(x)-\Gamma_{x,y}f(y)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta} |x,y|^{\eta-\gamma}_P} \lt +\infty. \end{equation*}

For two models $M=(\Pi,\Gamma),\, \bar{M}=(\bar{\Pi},\bar{\Gamma})$ and two modelled distributions $f \in \mathscr{D}^{\gamma,\eta}_{P,M},\,\bar{f} \in \mathscr{D}^{\gamma,\eta}_{P,\bar{M}}$ we also set

\begin{equation*} {\left\vert\left\vert\left\vert{f;\bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}} := \|f-\bar{f}\|_{\gamma,\eta;\mathfrak{K}} + \sup_{(x,y) \in \mathfrak{K}_P} \sup_{\zeta \lt \gamma} \frac{|f(x)-\bar{f}(x) - \Gamma_{x,y} f(y) + \bar{\Gamma}_{xy}\bar{f}(y)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}|x,y|^{\eta-\gamma}_{P}}. \end{equation*}

If V is a sector of regularity $\alpha \in A$, where appropriate we will use the shorthand $\mathscr{D}^{\gamma,\eta}_{\alpha;P,M} = \mathscr{D}^{\gamma,\eta}_{P,M}(V)$ and we will drop the dependence on the model when the context is clear.

Lemma 3.16. Let $\mathfrak{K}\subset \mathbb{G}$ be a compact domain such that for every $x\in \mathfrak{K}$ and $\bar{x}:= \textrm{arg\,min}_{y\in P}|x^{-1}y|$ one has that the points $\bar{x}\big( \lambda\cdot (\bar{x}^{-1} x)\big) \in \mathfrak{K}$ for every $\lambda\in [0,1]$. Also let $f \in \mathscr{D}^{\gamma,\eta}_P$ for some γ > 0 and assume that for every $\zeta \lt \eta$ the map $x\mapsto \mathcal{Q}_{\zeta} f(x)$ extends continuously in such a way that for $x\ \in P$ one has $\mathcal{Q}_{\zeta} f(x) =0$. Then, one has the bound

\begin{equation*} \unicode{x27E6} f\unicode{x27E7}_{\gamma,\eta;\mathfrak{K}} \lesssim {\left\vert\left\vert\left\vert{f}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}}, \end{equation*}

where the implied constant depends affinely on $\|\Gamma\|_{\gamma;\mathfrak{K}}$ but is otherwise independent of $\mathfrak{K}$. Similarly, if $\bar{f}\in \mathscr{D}^{\gamma,\eta}_{P,\bar{M}}$ with respect to a different model $\bar{M}=(\bar{\Pi},\bar{\Gamma})$ and is such that

\begin{equation*}\lim_{x\to P}\mathcal{Q}_\zeta(f(x)-\bar{f}(x)) =0\end{equation*}

for every $\zeta \lt \eta$, then one has the bound

\begin{equation*} \unicode{x27E6} f -\bar{f} \unicode{x27E7}_{\gamma,\eta;\mathfrak{K}} \lesssim {\left\vert\left\vert\left\vert{f;\bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}} + \|\Gamma-\bar{\Gamma}\|_{\gamma;\mathfrak{K}}\left({\left\vert\left\vert\left\vert{f}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}}+ {\left\vert\left\vert\left\vert{\bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}}\right), \end{equation*}

with proportionality constant also depending affinely on $\|\Gamma\|_{\gamma;\mathfrak{K}}$ and $\|\bar{\Gamma}\|_{\gamma;\mathfrak{K}}$.

Lemma 3.17. Let γ > 0, $\kappa \in (0,1)$ and assume that $f,\,\bar{f}$ satisfy the assumptions of Lemma 3.16. Then, for every compact $\mathfrak{K} \subset \mathbb{G}$, one has

\begin{equation*} {\left\vert\left\vert\left\vert{f;\bar{f}}\right\vert\right\vert\right\vert}_{(1-\kappa)\gamma,\eta;\mathfrak{K}} \lesssim \unicode{x27E6} f-\bar{f}\unicode{x27E7}^\kappa_{\gamma,\eta;\mathfrak{K}}\left({\left\vert\left\vert\left\vert{f}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}} + {\left\vert\left\vert\left\vert{\bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\mathfrak{K}}\right)^{1-\kappa}. \end{equation*}

3.3.1. Reconstruction theorem for singular modelled distributions

Since the reconstruction is purely local, it follows from our earlier proof that for any singular modelled distribution, $f\in \mathscr{D}^{\gamma,\eta}_P$, there exists a unique element $\tilde{\mathcal{R}}f\in \mathcal{S}'(\mathbb{G}\setminus P)$, i.e. in the dual of smooth functions compactly supported away from P, such that

\begin{equation*} \langle \tilde{\mathcal{R}}f - \Pi_x f(x), \psi^\lambda_x\rangle \lesssim \lambda^{\gamma}, \end{equation*}

for all $x\notin P$ and $\lambda \ll d_{\mathbb{G}}(x,P)$. However, we show below that under appropriate assumptions there exists a natural extension of $\tilde{\mathcal{R}}f$ to an actual distribution on $\mathbb{G}$ with regularity $\mathcal{C}^{\underline{\alpha}}$.

Proposition 3.18 (Singular reconstruction)

Let $f\in \mathscr{D}^{\gamma,\eta}_P(V)$, γ > 0 and $\eta\leqslant \gamma$. Then, provided $\alpha \wedge \eta \gt -|\mathfrak{m}|$, where $|\mathfrak{m}|$ is the scaled dimension of the complement of the singular hyperplane defined by (3.17), there exists a unique distribution $\mathcal{R} f \in \mathcal{C}^{{\alpha} \wedge \eta}_\mathfrak{s}$ such that $(\mathcal{R} f)(\varphi) = (\tilde{\mathcal{R}}f)(\varphi)$ for any smooth test function compactly supported away from P. If f and $\bar{f}$ are given with respect to two models M and $\bar{M}$ then, for any compact $\mathfrak{K}$, it holds that

\begin{equation*} \|\mathcal{R}_M f- \mathcal{R}_{\bar{M}} \bar{f}\|_{\mathcal{C}^{{\alpha} \wedge \eta}(\mathfrak{K})} \lesssim {\left\vert\left\vert\left\vert{f;\bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\bar{\mathfrak{K}}} + {\left\vert\left\vert\left\vert{M;\bar{M}}\right\vert\right\vert\right\vert}_{\gamma;\bar{\mathfrak{K}}}, \end{equation*}

where the constant depends on semi-norms of $f,\,\bar{f}$ and $M,\,\bar{M}$ on $\bar{\mathfrak{K}}$.

We provide the proof of Proposition 3.18 at the end of this section, let us first make some preparatory observations. Recalling the decomposition $\mathfrak{g}=\mathfrak{p} \oplus \mathfrak{p}^c$ defined at the start of the subsection, we define the projections $\pi_c: \mathfrak{g}\to \mathfrak{p}^c$ and $\pi_{\mathfrak{p}}: \mathfrak{g}\to \mathfrak{p}$. Then using the decomposition $X = X^{\mathfrak{p}}+ X^c \in \mathfrak{p} \oplus \mathfrak{p}^c$, we define the map

(3.18)\begin{equation} \tilde{\Phi}:\mathfrak{g} \rightarrow \mathbb{G}, \quad \tilde{\Phi}(X) = \exp(X^{\mathfrak{p}})\exp(X^c). \end{equation}

Similar to Remark 2.19 one sees that $\tilde{\Phi}$ is a global diffeomorphism. We then define the map

(3.19)\begin{equation} N_P: \mathbb{G}\to \mathbb{R}_+, \qquad x\mapsto \left|\exp\left(\pi_c\circ \tilde{\Phi}^{-1} (x)\right)\right| \end{equation}

and observe the following properties:

• • For $x\in \mathbb{G}$ one has that $N_P(x)=0$ if and only if $x\in P.$ This follows from the fact that $P= \exp(\mathfrak{p})$.

• • For $x\in \mathbb{G}$ and δ > 0 one has the identity

  (3.20)\begin{equation} N_P (\delta \cdot x)= \delta N_P(x). \end{equation}

  Indeed, writing $x= \tilde{\Phi} (\tilde{X})$, we have

  \begin{equation*}N_P (\delta \cdot x)=|\exp( \pi_c(\operatorname{D}_\delta \tilde{X}))|=|\exp(\operatorname{D}_\delta \pi _c( \tilde{X}))|= |\delta\,\cdot\,\exp( \pi_c( \tilde{X}))|=\delta N_P(x).\end{equation*}

• • For any $x\in \mathbb{G}$ and $y\in P$ one has

  (3.21)\begin{equation} N_P (y x)= N_ P(x). \end{equation}

  This follows from the observation that writing $x= \tilde{\Phi}(\tilde{X})$ and $y=\tilde{\Phi}(Y)= \exp(Y)$ one finds that $yx=\exp(\operatorname{H} (Y,\tilde{X}^\mathfrak{p}))\exp(\tilde{X}^{c})$, where $\operatorname{H}$ was defined in (2.1). Identity (3.21) then follows from the fact that $\operatorname{H} (Y,\tilde{X}^\mathfrak{p}) \in \mathfrak{p}$.

• • The map N_P is Lipschitz continuous on $\mathbb{G}$, and it follows from Remark 2.4 that N_P is smooth on ${\mathbb{G}\setminus P}$.

• • There exists a constant C > 0 such that for all $x\in \mathbb{G}$

  (3.22)\begin{equation} C N_P(x)\leqslant d_{\mathbb{G}}(x,P). \end{equation}

  In a neighbourhood of the origin this follows directly from the fact that N_P is Lipschitz continuous. Homogeneity of N_p (given by (3.20) above) and of the distance function $x\mapsto d_{\mathbb{G}}(x,P)$ then shows that this constant is in fact uniform on all of $\mathbb{G}$.

• • For all $x\in \mathbb{G}$

  (3.23)\begin{equation} d_{\mathbb{G}}(x,P)\leqslant N_P(x). \end{equation}

  Indeed, write $x= \tilde{\Phi}({X})= \exp(X^{\mathfrak{p}})\exp(X^c)$ and note that

  \begin{equation*}d(x,P)\leqslant |\exp(X^{\mathfrak{p}})^{-1} x|= |\exp(X^c)|= N_P(x).\end{equation*}

Proof of Proposition 3.18

The proof is analogous to that of [Reference Hairer41, Lem.6.9], the only element that does not adapt ad verbatim is the construction of the partition of unity $\varphi_{x,n}$. We therefore present an alternative construction of a partition of unity on $\mathbb{G}$, which satisfies all the required conditions.

First let $\varphi:\mathbb{R}_+\rightarrow [0,1]$ be a smooth compactly, supported function such that ${\text{supp}}(\varphi)=[1/2,2]$ and with the property that for all $r\in \mathbb{R}_+$,

\begin{equation*} \sum_{n\in \mathbb{Z}} \varphi(2^nr) =1. \end{equation*}

Secondly, let $\mathfrak{Z}\subset P$ be a lattice (see $\S$ 2.3) and let $\tilde{\varphi}$ be smooth, compactly supported, such that

(3.24)\begin{equation} \sum_{y\in\mathfrak{Z}} \tilde{\varphi}_y(x) =1 \end{equation}

for all x in the $\frac{2}{C}$-fattening of $P\subset \mathbb{G}$, where C is the constant in (3.22) and (3.23). For $y\in \mathfrak{Z}$ we then set

\begin{equation*}\phi_{y} (x) := \varphi(C N_P(x)) \tilde{\varphi}_y(x), \end{equation*}

where the constant C is as in (3.22). To conclude, we then set for every $n \geqslant 0$, $y\in \mathfrak{Z}$ and $x \in \mathbb{G}$,

\begin{equation*}\phi_{n,y} (x) :=\phi_{y} (2^n \cdot x).\end{equation*}

By using (3.20) and (3.21), it directly follows that $\phi_{n,y}(x) =(\phi_{0,e}) \left( y^{-1} (2^n\cdot x) \right)$. Since $\phi_{1,e}$ has compact support and is such that for all $x\in {\text{supp}}(\phi_{1,e})$, one has $ d_{\mathbb{G}}(x,P)\geqslant C N_P(x)\geqslant 1/2$, it only remains to check that $\{\phi_{n,y}\}_{n \in \mathbb{Z}, y\in \mathfrak{Z}}$ is in fact a partition of unity. Indeed let $x\in \mathbb{G}$, then

\begin{align*}&\sum_{n\in \mathbb{Z}, y\in \mathfrak{Z}} \phi_{n,y} (x) = \sum_{n \in \mathbb{Z}, y\in \mathfrak{Z}} \varphi(CN_P(2^n\cdot x)) \tilde{\varphi}_y(2^n\,\cdot\,x) \\& \qquad\qquad\qquad\quad = \sum_{n\in \mathbb{Z}} \varphi(2^n C N_P(x)) {\sum_{y\in \mathfrak{Z}}{\tilde{\varphi}_y(2^n\,\cdot\,x)}}= 1,\end{align*}

where in the second last equality we used that $ \sum_{y\in \mathfrak{Z}}\tilde{\varphi}_y(2^n \cdot x)=1 $ whenever $d(x, P)\leqslant \frac{1}{C}2^{-n+1}$ and in the last equality we used (3.24). The remainder of the proof of [Reference Hairer41, Lem. 6.9] then adapts ad verbatim by also making use of Proposition 3.13.

Remark 3.19. If the model M is smooth, i.e. $\Pi_x\tau \in C^\infty(\mathbb{G})$ for every $\tau \in \operatorname{T}$, one finds exactly as in [Reference Hairer41, Rem. 3.15] that for any modelled distribution $f\in \mathscr{D}^{\gamma}$ with γ > 0 one has the identity $\mathcal{R}_Mf(x) = \left(\Pi_x f(x)\right)(x)$ (and in particular $\mathcal{R}_Mf$ is a continuous function).

3.4. Convolution with singular kernels

In this section we describe how to lift the action of singular kernels onto the regularity structure. While most arguments adapt from [Reference Hairer41] some care has to be taken due to the fact that convolutions are not commutative and we do not have an explicit formula for Taylor expansions. This latter issue is circumvented using Lemma 3.6, which allows us to reduce our analysis to similar expressions as shown in [Reference Hairer41]. The examples we have in mind are the singular part of Greens functions of left invariant differential operators satisfying the following assumption.

Assumption 3.20. For $\beta \in (0, |\mathfrak{s}|)$, the kernel $K:\mathbb{G}\setminus \{e\}\to \mathbb{R}$ can be decomposed as

(3.25)\begin{equation} K(x)=\sum_{n\in\mathbb{N}} K_n(x), \end{equation}

where the smooth functions K_n are supported on $B_{2^{-n}}$ and

• • for each $I\in \mathbb{N}^d$ there exists a constant $C(I) \gt 0$, uniform in $n\in \mathbb{N}$ such that

  \begin{equation*} \sup_{x \in \mathbb{G}} |X^I K_n(x)|\leqslant C(I) 2^{(|\mathfrak{s}|-\beta+d(I))n}, \end{equation*}

• • for any multi-indices $I,\,J\in \mathbb{N}^d$ there exists a constant $C(I,J) \gt 0$ uniform in $n\in \mathbb{N}$ such that

  \begin{equation*} \left|\,\int_{\mathbb{G}} \eta^I(x) X^J K(x) dx\,\right|\leqslant C(I,J) 2^{-\beta n}, \end{equation*}

• • there exists an integer r, such that

  \begin{equation*} \int_{\mathbb{G}} \eta^I(x) K(x) \, dx=0 \end{equation*}

  for all multi-indices $I\in \mathbb{N}^d$ with scaled degree $d(I)\leqslant r$.

From now on we shall exclusively work with regularity structures $\mathcal{T}$ models M satisfying the following assumption.

We point out that the assumption K annihilates polynomials causing no real restriction on the type of kernels since the result [Reference Hairer41, Lem. 5.5] adapts in a straightforward manner to our setting, see also Lemma 4.4. We point out that this assumption is convenient but not crucial for the theory, cf. [Reference Hairer and Singh46]. In the remainder of this subsection we show how the action of kernels of this type is lifted onto the regularity structure and acts on modelled distributions. Given a $\gamma \in \mathbb{R}\setminus \triangle$ we write $\mathcal{K}_\gamma$ for this lift; it corresponds to K in the sense that for $f\in\mathscr{D}^\gamma$

(3.26)\begin{equation} \mathcal{R}\mathcal{K}_\gamma f=K(\mathcal{R}f) \end{equation}

and in that it satisfies an appropriate version of the classical Schauder estimates.

Assume that the kernel K satisfies Assumption 3.20 for some β > 0. We associate with K the map $\mathcal{J}:\mathbb{R}^d\to L(\operatorname{T},\bar{\operatorname{T}})$ which for every $\tau\in \operatorname{T}_\alpha$ is given by

(3.27)\begin{equation} \mathcal{J}(x)\tau:= \sum_n \pmb{\operatorname{{P}}}^{\alpha+\beta}_x (\Pi_x \tau\ast K_n), \end{equation}

where the last sum is seen to converge absolutely by first observing that by Assumption 3.20 for any $\tau\in \operatorname{T}_\alpha$

\begin{equation*} |X^I (\Pi_x \tau\ast K_n)(x)|= | \Pi_x \tau\ast X^I K_n(x)|= |\Pi_x \tau (X^I_1K_n(x,\cdot))|\lesssim 2^{-n(\alpha+\beta- d(I))} \end{equation*}

and then using Lemma 3.4.

Definition 12. Given a regularity structure $\mathcal{T}$ equipped with an integration map $\mathcal{I}$, a kernel K and a model $M=(\Pi,\Gamma)$, we say the model M realizes K for $\mathcal{I}$ if for each $\tau\in \operatorname{T}_\alpha$ and each $x\in \mathbb{G}$,

\begin{equation*} \Pi_x\mathcal{I}\tau= K(\Pi_x\tau)- \Pi_x \mathcal{J}(x)\tau. \end{equation*}

Now we can define the lift of the kernel K, namely, for $f\in \mathscr{D}^\gamma (V)$ we set

\begin{equation*} \mathcal{K}_\gamma f(x):=\mathcal{I} f(x)+\mathcal{J}(x)f(x)+\mathcal{N}_\gamma f(x), \end{equation*}

where

\begin{equation*}(\mathcal{N}_\gamma f)(x)=\sum_n \pmb{\operatorname{{P}}}^{\gamma+\beta}_x \big( (\mathcal{R}f-\Pi_xf(x)) \ast K_n\big),\end{equation*}

where the last sum converges by the same argument as for $\mathcal{J} (x)\tau$.

The next theorem, which is an analogue of [Reference Hairer41, Thm. 5.12], confirms that $\mathcal{K}_\gamma$ does indeed correspond to K in the sense of (3.26) and satisfies the desired Schauder estimates.

Theorem 3.25. Let $\gamma\in \mathbb{R}\setminus \triangle$ and β > 0 be such that $\gamma+\beta\not\in \triangle$ and let $K:\mathbb{G}\setminus\{e\}\rightarrow \mathbb{R}$ be a kernel satisfying Assumption 3.20 for $r\geqslant\gamma+\beta$, let $\mathcal{T}=(\operatorname{T},\operatorname{G})$ be a regularity structure and $M=(\Pi,\Gamma)$ be a model satisfying Assumption 3.23. Furthermore assume that $\mathcal{T}$ is equipped with an abstract integration operator and M realizes K for $\mathcal{I}$. Then for any sector V of regularity $\alpha \in A$ the operator $\mathcal{K}_\gamma$ is a continuous linear map from $\mathscr{D}_M^\gamma(V)$ to $\mathscr{D}_{(\alpha+\beta)\wedge 0}^{\gamma+\beta}$ satisfying the identity

\begin{equation*} \mathcal{R}\mathcal{K}_\gamma f=K(\mathcal{R}f), \end{equation*}

for all $f\in \mathscr{D}_M^\gamma(V)$. Furthermore, if we denote by $M=(\Pi,\Gamma),\ \bar{M}=(\bar{\Pi},\bar{\Gamma})$ two models as above and by $\mathcal{K}_\gamma$, respectively $\bar{\mathcal{K}}_{\gamma}$ the associated operators, one has the bound

\begin{equation*} \|\mathcal{K}_\gamma f; \bar{\mathcal{K}}_{\gamma}\bar{f} \|_{\gamma+\beta;\mathfrak{K}} \lesssim_C {\left\vert\left\vert\left\vert{f,\bar{f}}\right\vert\right\vert\right\vert}_{\gamma;\bar{\mathfrak{K}}} \\ +\|\Pi-\bar{\Pi}\|_{\gamma;\bar{\mathfrak{K}}} +\|\Gamma-\bar{\Gamma}\|_{\gamma;\bar{\mathfrak{K}}}, \end{equation*}

where the implicit constant depends on the norms of $M, \bar M$ and $f\in\mathscr{D}_M^\gamma(V)$ and $\bar{f}\in \mathscr{D}^\gamma_{\bar M}(V)$.

The fact that the next lemma ([Reference Hairer41, Lem. 5.16]) still holds in our setting is the underlying reason that all proofs extend in a rather straightforward manner from [Reference Hairer41] and one does not require a more involved notion of abstract integration map, which is, for example, the case on general Riemannian manifolds cf.[Reference Hairer and Singh46].

Lemma 3.26. In the setting of Theorem 3.25 one has the identity

\begin{equation*} \Gamma_{x,y}(\mathcal{I}+\mathcal{J}(y))= (\mathcal{I}+\mathcal{J}(x))\Gamma_{x,y} \end{equation*}

for all $x,y\in \mathbb{G}$.

We introduce the following quantity; for $I\in \mathbb{N}^d, \alpha \gt 0, n\in \mathbb{N}$ and $x,y,z\in \mathbb{G}$, set

(3.28)\begin{align} K^{I,\alpha}_{n,xy}(z)&:= X^I_1 K_n(y,z)-\tilde{\operatorname{P}}^{\alpha+\beta-d(I)}_x[X^I_1 K_n(\cdot,z)] (y)\nonumber \\ &=X^I_1 \big(K_n(\,\cdot\,,z)-\tilde{\operatorname{P}}^{\alpha+\beta}_x[K_n(\cdot,z)]\big)(y), \end{align}

where the second equality follows from Remark 2.12. Here we reiterate that we are using the notation $K(x,y):=K(y^{-1}x)$ where K satisfies Assumption 3.20. Taylor’s theorem and Remark 2.16 then yield that

(3.29)\begin{equation} K^{I,\alpha}_{n,xy}(z) =\sum_{|J|\leqslant [\alpha+\beta-d(I)]+1, d(J)\geqslant \alpha+\beta-d(I)} \int_{\mathbb{G}} X_1^{J}(X_1^I K_n)(x\tilde{z},z) Q^J(x^{-1}y, d\tilde{z}). \end{equation}

As in [Reference Hairer41] the motivation for defining these quantities comes from the identity

\begin{align*} \Pi_x (\mathcal{I}\tau)(\phi)&= K(\Pi_x\tau)(\phi) - \Pi_xJ(x)\tau(\phi) \\ &\quad = \sum_{n} \int \big(\Pi_x\tau (K_n(y,\cdot)) - \tilde{\operatorname{P}}_x^{\alpha+\beta}[K_n(\Pi_x\tau)](y)\big)\phi(y) \, dy \\ &\quad = \sum_{n} \int \Pi_x\tau\big(K_n(y,\cdot) - \tilde{\operatorname{P}}_{x,1}^{\alpha+\beta}[K_n(\cdot,\cdot) ](y)\big) \phi(y) \, dy \\ &\quad = \sum_{n} \int \Pi_x\tau(K^{0,\alpha}_{n,xy}) \phi(y) \, dy, \end{align*}

where we use the subscript for $\tilde{\operatorname{P}}_{x,1}[K(\cdot,\cdot)](y)$ to clarify that one expands in the first coordinate. The next Lemma collects the results of [Reference Hairer41, Lem. 5.18, Lemma 5.19], the proofs of which adapt ad verbatim.

Lemma 3.27. Let $\alpha\in A$ and $\tau\in \operatorname{T}_\alpha$ and assume $\alpha+\beta\notin \triangle$, then the following bound holds:

(3.30)\begin{equation} |(\Pi_y\tau) (K^{I,\alpha}_{n,xy})|\lesssim \|\Pi\|_{\alpha,B_2(x)}(1+\|\Gamma\|_{\alpha,B_2(x)})\sum_{\delta \gt 0} 2^{\delta n}|y^{-1}x|^{\delta+\alpha+\beta -d(I)}, \end{equation}

where the sum over δ runs over a finite set of positive real numbers. The same bound holds for $|(\Pi_x\tau) (K^{I,\alpha}_{n,xy})|$, as well as the analogue bound on the difference of two models.

Furthermore one has the bound

(3.31)\begin{equation} \sum_n \left|\int_{\mathbb{G}} (\Pi_x\tau) (K^{I,\alpha}_{n,xy})\phi_x^\lambda(y) \text{d} y\,\right|\lesssim \lambda^{\alpha+\beta} \|\Pi\|_{\alpha;B_2(x)}(1+\|\Gamma\|_{\alpha;B_2(x)}), \end{equation}

as well as the analogous bound for the difference of two models, where all these bounds hold uniformly over $x\in \mathbb{G}$, $\lambda \in (0,1]$ and $\phi\in \mathfrak{B}_r$

As in [Reference Hairer41] we introduce for $x,y\in \mathbb{G}$ the following operator:

(3.32)\begin{equation} \mathcal{J}_{x,y}:=\mathcal{J}(x)\Gamma_{x,y}-\Gamma_{x,y} \mathcal{J}(y). \end{equation}

Lemma 3.28. Under the assumptions of Theorem 3.25 one has for each $a\in \Delta$, $\tau\in \operatorname{T}_\alpha$

\begin{equation*} |\mathcal{J}_{x,y}\tau|_a\lesssim \|\Pi\|_{\alpha, B_2(x)}(1+\|\Gamma\|_{\alpha, B_2(x)})|y^{-1}x|^{\alpha+\beta-a}|\tau| \end{equation*}

uniformly in $x,y\in \mathbb{G}$ satisfying $x\in B_1(y)$. Again, the analogous bound for the difference of two models holds as well.

Proof. First we write $\mathcal{J}_{x,y}=\sum_n \mathcal{J}^n_{x,y}$ where each $\mathcal{J}^n_{x,y}$ is defined by replacing K by only one summand K_n in the definition of $\mathcal{J}_{x,y}$. Observe that since $\mathcal{J}^n_{x,y}\tau\in \bar{\operatorname{T}}_{ \lt \alpha+\beta}$ it suffices by Lemma 3.6 to show that

\begin{equation*}p^n_\tau(w):=\Pi_e(\mathcal{J}^n_{x,y}\tau)(w)\in \mathcal{P}_{\alpha+\beta}\end{equation*}

satisfies

(3.33)\begin{equation} \sum_{n} |(X^I p^n_\tau)(e)|\lesssim \|\Pi\|_{\alpha, B_2(x)}(1+\|\Gamma\|_{\alpha, B_2(x)})|y^{-1}x|^{\alpha+\beta-d(I)}|\tau|. \end{equation}

We treat the cases $|y^{-1}x|\leqslant 2^{-n}$ and $|y^{-1}x| \gt 2^{-n}$ separately. In the case $|y^{-1}x|\leqslant 2^{-n}$, using the definitions of the polynomial regularity structure, we find

\begin{equation*} p^n_\tau(x^{-1} z)= \Pi_e(\mathcal{J}^n_{x,y}\tau)(x^{-1}z)= \Pi_x(\mathcal{J}^n_{x,y}\tau)(z) \end{equation*}

and thus

\begin{equation*} X^Ip^n_\tau(e)= X^I (\Pi_x\mathcal{J}^n_{x,y}\tau)(x). \end{equation*}

Using the analogous notation $\mathcal{J}^n(x)$ for the operator consisting only of one summand in (3.27), we find

\begin{align*} \Pi_x\mathcal{J}^n(x)\Gamma_{x,y}\tau (z)& = \sum_{\gamma\leqslant\alpha} \Pi_x\left(\mathcal{J}^n(x)\mathcal{Q}_\gamma\Gamma_{x,y}\tau\right) (z) \\ &\quad =\sum_{\gamma\leqslant\alpha} \Pi_x \mathbf{P}^{\gamma+\beta}_x [\big(\Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau\big)_2 K_n(\cdot,\cdot)] (z)\\ &\quad=(\Pi_y \tau) \left(\tilde{\operatorname{P}}^{\alpha+\beta}_{x,1}[K_n(\cdot,\cdot)](z)\right)\\ &\qquad -\sum_{\gamma\leqslant\alpha} \Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau\big( \tilde{\operatorname{P}}^{\alpha+\beta}_{x,1}[K_n(\cdot,\cdot)]-\tilde{\operatorname{P}}^{\gamma+\beta}_{x,1}[K_n(\cdot,\cdot)]\big)(z)\\ &\quad= \tilde{\operatorname{P}}^{\alpha+\beta}_{x}[(\Pi_y \tau)_2K_n(\cdot,\cdot)](z)\\ &\qquad -\sum_{\gamma \lt \alpha} \Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau\big( \tilde{\operatorname{P}}^{\alpha+\beta}_{x,1}[K_n(\cdot,\cdot)]-\tilde{\operatorname{P}}^{\gamma+\beta}_{x,1}[K_n(\cdot,\cdot)]\big)(z) \end{align*}

and

\begin{align*} \Pi_x\Gamma_{x,y}\mathcal{J}^n(y)\tau (z) &= \Pi_y\mathcal{J}^n(y)\tau (z) = \tilde{\operatorname{P}}_y^{\alpha+\beta}[(\Pi_y\tau)_2 K_n(\cdot,\cdot)] (z) \\ &=\tilde{\operatorname{P}}^{\alpha+\beta}_{x}\left[ \left(\Pi_y\tau\right)_2\big( \tilde{\operatorname{P}}_{y,1}^{\alpha+\beta}[K_n(\cdot,\cdot)]\big) \right](z), \end{align*}

where in the third equality we used the fact that $\tilde{\operatorname{P}}^{\alpha+\beta}_{x}$ acts as the identity map on polynomial functions of degree less then $\alpha+\beta$. Therefore,

\begin{align*} p_\tau^n(x^{-1}z) &= \underbrace{\tilde{\operatorname{P}}^{\alpha+\beta}_{x}\Big[(\Pi_y \tau)_2\left(K_n(\cdot,\cdot)- \tilde{\operatorname{P}}_{y,1}^{\alpha+\beta}[K_n(\cdot,\cdot)]\right) \Big](z)}_{:=p^n_{\tau,1}(x^{-1}z)} \\ &\quad-\sum_{\gamma \lt \alpha} \underbrace{\Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau\big( \tilde{\operatorname{P}}^{\alpha+\beta}_{x,1}[K_n(\cdot,\cdot)]-\tilde{\operatorname{P}}^{\gamma+\beta}_{x,1}[K_n(\cdot,\cdot)]\big)(z)}_{:=p^{n,\gamma}_{\tau, 2}(x^{-1}z)}. \end{align*}

Using the general formula $X^I(\tilde{\operatorname{P}}_x^a[f])(x)=X^I({\operatorname{P}}_x^a[f])(e)= X^If(x)$ for $d(I) \lt a$ we find that

\begin{equation*} X^I p^n_{\tau,1} (e)=X^I \Big[(\Pi_y \tau)_2\big(K_n(\cdot,\cdot)- \tilde{\operatorname{P}}_{y,1}^{\alpha+\beta}[K_n(\cdot,\cdot)]\big) \Big](x)= (\Pi_y \tau)(K^{I,\alpha+\beta}_{n,x,y}) \end{equation*}

and so by Lemma 3.27 we find that

\begin{equation*}\sum_{n\, :\, |y^{-1}x|\leqslant 2^{-n}} |X^I p^n_{\tau,1} (e)| \lesssim |y^{-1}x|^{\alpha+\beta-d(I)}|\tau|. \end{equation*}

Concerning $p^{n,\gamma}_{\tau, 2}$, since one has the identity

\begin{equation*} p^{n,\gamma}_{\tau, 2}(x^{-1}z)=-\tilde{\operatorname{P}}^{\gamma+\beta}_{x}[(\Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau)_2K_n(\cdot,\cdot)](z)+ \tilde{\operatorname{P}}^{\alpha+\beta}_{x}[(\Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau)_2K_n(\cdot,\cdot)](z), \end{equation*}

we find that $X^I p^{n,\gamma}_{\tau, 2}(e)= 0$ unless $d(I)\in [\gamma+\beta,\alpha+\beta)$, in which case

\begin{equation*} X^I p^{n,\gamma}_{\tau, 2}(e)= \Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau\big( X^I_1 K_n(x,\cdot)\big). \end{equation*}

Thus,

\begin{align*} \sum_{n\, :\, |y^{-1}x|\leqslant 2^{-n}} |X^I p^{n,\gamma}_{\tau, 2}(e)| &\lesssim \sum_{n\, :\, |y^{-1}x|\leqslant 2^{-n}} |y^{-1}x|^{\alpha-\gamma} 2^{-n(\gamma +\beta -d(I))}|\tau|_a \\ &\lesssim |y^{-1}x|^{\alpha+\beta -d(I)}|\tau|, \end{align*}

where we used that the case $d(I)= \gamma+\beta$ does not arise as by Assumption 3.23.

To treat the case $|y^{-1}x| \gt 2^{-n}$, one writes

\begin{equation*} p_\tau^n(z) = \sum_{\gamma\leq \alpha} \underbrace{\Pi_x\mathcal{J}^n(x)(\mathcal{Q}_\gamma\Gamma_{x,y}\tau)(xz)}_{=:q_{\tau, 1}^{n,\gamma}(z) } - \underbrace{\Pi_x\Gamma_{x,y}\mathcal{J}^n(y)\tau(xz)}_{=:q_{\tau, 2}^n(z)} \end{equation*}

and using the fact that $\gamma+ \beta\notin \triangle$ one finds

\begin{equation*} |X^I q_{\tau, 1}^{n,\gamma}(e)|= \begin{cases} \left| \Pi_x \mathcal{Q}_\gamma\Gamma_{x,y}\tau \left( X^I_1 K_n(x,\cdot) \right) \right|, & \text{if }d(I) \lt \gamma+ \beta,\\ 0,&\text{otherwise}, \end{cases} \end{equation*}

which is bounded uniformly over $|y^{-1}x| \lt 1$ by a constant multiple of $|y^{-1}x|^{\alpha-\gamma} 2^{-n(\gamma+\beta- d(I))}$. Concerning $q_{\tau, 2}^n$, we find that

\begin{equation*} q_{\tau, 2}^n(z)= \Pi_x\Gamma_{x,y}\mathcal{J}^n(y)\tau(xz)=\Pi_y \mathcal{J}^n(y)\tau(xz) = {\operatorname{P}}_{y,1}^{\alpha+\beta}[(\Pi_y\tau)_2 K_n(\cdot,\cdot)] (y^{-1}xz)), \end{equation*}

and therefore by Lemma 3.5

\begin{align*} |X^I q_{2}^n(e)| &\lesssim \sum_{d(I) \lt d(J)\leqslant \alpha+\beta} \sup_{d(J')\leqslant d(J)} \left| \Pi_y\tau \left( X^{J'}_1K_n(y,\cdot)\right)\right| \, |y^{-1}x|^{d(J)-d(I)}|\tau|\\ & \lesssim \sum_{d(I)\leqslant d(J) \lt \alpha+\beta} \sup_{d(J')\leqslant d(J)} 2^{-n(\alpha+\beta- d(J'))}|y^{-1}x|^{d(J)-d(I)}|\tau|\\ &\lesssim\sum_{d(I)\leqslant d(J) \lt \alpha+\beta} 2^{-n(\alpha+\beta- d(J))}|y^{-1}x|^{d(J)-d(I)}|\tau|. \end{align*}

Since $p^n_{\tau}= \sum_{\gamma\leqslant \alpha} q_{\tau, 1}^{n,\gamma} + q_{\tau,2}^n$ one obtains (3.33) by summing over $\left\{n\in \mathbb{N} \ : \ |y^{-1}x| \gt 2^{-n}\right\}$.

Proof of Theorem 3.25

First, we need to check that for $a \in A$ it holds that

(3.34)\begin{equation} | \mathcal{K}_\gamma f(x)-\Gamma_{x,y}\mathcal{K}_\gamma f(y)|_a\lesssim |y^{-1}x|^{\gamma+\beta-a}. \end{equation}

For $a\notin \triangle$ this bound follows from Lemma 3.26, continuity of the operator of $\mathcal{I}$ and properties of modelled distributions by an ad verbatim adaptation of the same argument in the proof of [Reference Hairer41, Thm.  5.12]. The only place where the proof [Reference Hairer41, Thm.  5.12] does not adapt directly is when showing the bound for $a \in \triangle$. As in the proof of Lemma 3.28 we use Lemma 3.6 to circumvent the difficulty of not having explicit Taylor expansions in our setting. The rest of the proof adapts almost ad verbatim.

We first note that the polynomial part of $\mathcal{K}_\gamma f(x)-\Gamma_{x,y}\mathcal{K}_\gamma f(y)$ is given by

(3.35)\begin{equation} P:=\underbrace{\Gamma_{x,y}\mathcal{N}_\gamma f(y)}_{=:P_1} - \underbrace{\mathcal{N}_\gamma f(x)}_{=:P_2} + \underbrace{\mathcal{J}(x)(\Gamma_{x,y}f(y)-f(x) )}_{=:P_3}, \end{equation}

and thus, in order to prove (3.34) for the polynomial part it suffices by Lemma 3.6 to show that for any $d(I) \lt \gamma+\beta$, $p:=\Pi_e P$ satisfies

(3.36)\begin{equation} |X^I p(e)| \lesssim |y^{-1}x|^{\gamma+\beta-d(I)}. \end{equation}

Using (3.25) we define the analogous decompositions $\mathcal{J}=\sum_n \mathcal{J}^n$ and $\mathcal{N}_\gamma=\sum_n \mathcal{N}^n_\gamma$ and similarly write $P=\sum_{n=0} P^n$, etc.

As usual we use different strategies for small and large scales, starting with the case $2^{-n}\leqslant |y^{-1}x|$. In this case we separately estimate $p_i: =\Pi_e P_i$ for $i\in \{1,2,3\}$.

• • Noting that

  \begin{equation*}p^n_{1}(z)= \Pi_x \Gamma_{x,y}\mathcal{N}_\gamma f(y) (xz)= {P}_{y,1}^{\gamma+\beta}\left[(\mathcal{R}f-\Pi_yf(y))_2 \big( K_n(\cdot,\cdot)\big)\right] (y^{-1}xz),\end{equation*}

  we find by Lemma 3.5

  \begin{align*} |X^I p^n_{1}(e)| &\lesssim \sum_{d(I)\leqslant d(J) \lt \gamma +\beta} \sup_{d(J')\leqslant d(J)} \left|\big(\mathcal{R}f-\Pi_yf(y)\big) \big( X^{J'}_1K_n(y,\cdot)\big)\right| \\ &\quad \times |y^{-1}x|^{d(J)-d(I)}\\ & \lesssim \sum_{d(I)\leqslant d(J) \lt \gamma+\beta} \sup_{d(J')\leqslant d(J)} 2^{-n(\gamma+\beta- d(J'))}|y^{-1}x|^{d(J)-d(I)}\\ &\lesssim\sum_{d(I)\leqslant d(J) \lt \gamma+\beta} 2^{-n(\gamma+\beta- d(J))}|y^{-1}x|^{d(J)-d(I)}. \end{align*}

• • Regarding p ⁿ₂ we find that $X^I p^n_2(e)= \left( \mathcal{R}f-\Pi_xf(x)\right) \left( X^{I}_1K_n(x,\cdot)\right)$ and thus by Reconstruction, see Theorem 3.7,

  \begin{equation*} | X^I p^n_2(e)| \lesssim 2^{-n(\gamma+\beta- d(I))}. \end{equation*}

• • Lastly, for p ⁿ₃, using the definition of $\mathcal{J} (x)$ and properties of models and modelled distributions we find

  \begin{equation*} \begin{array}{ll} |X^I p^n_3(e)| &\leqslant \sum\limits_{\delta\in A :\ d(I)-\beta \lt \delta \lt \gamma} \left|\big(\Pi_x \mathcal{Q}_{\delta}(\Gamma_{x,y}f(y)-f(x) \big)(X^I_1K(x,\cdot))\right| \\ &\lesssim \sum\limits_{\delta\in A :\ d(I)-\beta \lt \delta \lt \gamma} |y^{-1}x|^{\gamma-\delta} 2^{-n(\beta+\delta-d(I))}. \end{array} \end{equation*}

Therefore summing over $2^{-n}\leqslant |y^{-1}x|$, since $\beta+\delta\notin \triangle$ for any $\delta\in A$ and $\beta+\gamma\notin \triangle$, we find

\begin{equation*} \sum_{n \,:\, 2^{-n}\leqslant |y^{-1}x|} |X^I p^n(e)|\leqslant \sum_{n \,:\,\ 2^{-n}\leqslant |y^{-1}x|}\sum_{i=1}^3 |X^I p_i^n(e)| \lesssim |y^{-1}x|^{\gamma+\beta-d(I)}. \end{equation*}

Next we turn to the case $2^{-n} \gt |y^{-1}x|$; here a computation analogous to the one preceding [Reference Hairer41, Eq. (5.48)] gives

(3.37)\begin{equation} X^I p^n(e)= (\Pi_yf(y)-\mathcal{R}f)(K^{I,\gamma}_{n,x,y}) -\sum_{\zeta\leqslant d(I)-\beta} \big( \Pi_x\mathcal{Q}_\zeta(\Gamma_{x,y}f(y)-f(x)\big) \big(X^I_1K_n(x,\cdot)\big), \end{equation}

which can be bounded exactly the way it is done there.

Finally, one may follow the concluding steps of the proof of [Reference Hairer41, Thm. 5.12], finding that for any $\phi \in \mathfrak{B}$,

\begin{equation*} \big(\Pi_x(\mathcal{K}_\gamma f(x))-K(\mathcal{R}f)\big) (\phi^\lambda_x)= \sum_n \int \big( \Pi_x f(x)-\mathcal{R}f\big) (K_{n,xy}^\gamma)\phi_x^\lambda(y) dy\, \end{equation*}

and thus one obtains the identity $\mathcal{R}\mathcal{K}_\gamma f = K(\mathcal{R} f)$.

The proof of the difference bound follows by similar steps as to those presented here and applied in the proof of [Reference Hairer41].

3.4.1. Schauder estimates for singular modelled distributions

Since our main application will be to semi-linear evolution equations we will often require a Schauder estimate for modelled distributions with permissible singularities near a given subgroup $P\subset \mathbb{G}$, as in $\S$ 3.3 on singular modelled distributions.

PROPOSITION 3.29.

In the setting of $\S$ 3.3 and Theorem 3.25, given a sector V of regularity $\alpha \in A$, the operator $\mathcal{K}_\gamma$ is well defined on $\mathscr{D}^{\gamma,\eta}_P(V)$ for $\eta \lt \gamma$ provided that γ > 0 and $\eta\wedge \alpha \gt -|\mathfrak{m}|$. Furthermore, if $\gamma+\beta\notin \triangle$ and $\eta+\beta\notin \triangle$, one has $\mathcal{K}_\gamma f\in \mathscr{D}^{\gamma+\beta,(\eta\wedge \alpha)+\beta}_P$ continuity bound

(3.38)\begin{equation} {\left\vert\left\vert\left\vert{\mathcal{K}_\gamma f; \bar{\mathcal{K}}_\gamma\bar{f}}\right\vert\right\vert\right\vert}_{\gamma+\beta,(\eta\wedge \alpha)+\beta;\mathfrak{K}} \lesssim {\left\vert\left\vert\left\vert{f; \bar{f}}\right\vert\right\vert\right\vert}_{\gamma,\eta;\bar{\mathfrak{K}}} +\|\Pi-\bar{\Pi}\|_{\gamma;\bar{\mathfrak{K}}} +\|\Gamma-\bar{\Gamma}\|_{\gamma;\bar{\mathfrak{K}}} \ \end{equation}

over any compact set $\mathfrak{K} \subset \mathbb{G}$ and admissible models, where the implicit constant is uniform in the semi-norms of $M, \bar M$ and $f\in \mathscr{D}_{P,M}^{\gamma,\eta}(V),\,\bar{f}\in \mathscr{D}_{P,\bar{M}}^{\gamma,\eta}(V)$ on $\bar{\mathfrak{K}}$.

Proof. The proof follows exactly along the same lines as the one of [Reference Hairer41, Prop. 6.16], where as in the proof of Theorem 3.25 the only modification needed is due to the fact that our Taylor expansions are not explicit. Using the same notation as in the proof of Theorem 3.25, we recall (3.35)

(3.39)\begin{equation} P:=\underbrace{\Gamma_{x,y}\mathcal{N}_\gamma f(y)}_{=:P_1}- \underbrace{\mathcal{N}_\gamma f(x)}_{=:P_2} + \underbrace{\mathcal{J}(x)(\Gamma_{x,y}f(y)-f(x) )}_{=:P_3}. \end{equation}

This time, in order to show that $\mathcal{K}_\gamma f \in \mathscr{D}^{\gamma+\beta,(\eta\wedge \alpha)+\beta}_{P}$, by Lemma 3.6, we are required to show that for $p(e):=\Pi_e P = \sum_{i=1}^3 \Pi_e P_i$,

(3.40)\begin{equation} |X^I p(e)| \lesssim {|y^{-1}x|}^{\gamma+\beta-d(I)} |x,y|^{(\eta\wedge \alpha)+\beta-\gamma}_P\,, \end{equation}

There are three scales, which need separate arguments:

• • In the case $2^{-n}\leqslant |y^{-1}x|$ one proceeds exactly as in the proof of Theorem 3.25 using the decomposition $p^n = \sum_{i=1}^3 p^n_i$.

• • In the cases $2^{-n}\in \left(|y^{-1}x|, \frac{1}{2}|x,y|_P\right)$ and $2^{-n}\geqslant \frac{1}{2}|x,y|_P$ one uses Eq. (3.37) and proceeds exactly as in [Reference Hairer41] from there onwards.

Again, the proof of the difference bound follows analogously.

3.5. Local operations

To handle SPDEs using regularity structures we require suitable extensions to modelled distributions of standard local operations. Since these extensions adapt mutatis mutandis from [Reference Hairer41], we only provide pointers to the relevant literature.

1. (i) Differential operators can be lifted to the regularity structure and the model exactly as in [Reference Friz and Hairer33, Def. 14.1] and one retains [Reference Hairer41, Prop. 5.28 & Prop. 6.15]. We note that any left invariant differential operator $\mathfrak{L}$ of homogeneous degree β on $\mathbb{G}$ is of the form

   \begin{equation*}\mathfrak{L}=\sum_{d(I)= \beta} a_I X^I,\end{equation*}

   for some $I\in \mathbb{N}^d$. Thus, in order to lift such an $\mathfrak{L}$, it suffices to lift each of the differential operators $\{X_i\}_{i=1}^d$ to abstract differential operators. In $\S$ 3.1.1 we lifted the differential operators X_i to the polynomial regularity structure.

2. (ii) Next, recall the notion of a product on a regularity structure, cf. [Reference Friz and Hairer33, Def. 14.3]. In particular [Reference Hairer41, Thm. 4.7 & Prop. 6.12] still holds.

3. (iii) Lastly, we mention that composition of modelled distributions with smooth functions, cf. [Reference Hairer41, Section 4.1], as well as the coresponding results adapt ad verbatim to the setting of homogeneous Lie groups.

3.6. Symmetries

As in [Reference Hairer41, $\S$ 3.6], we shall consider modelled distributions which respect certain symmetries of $\mathbb{G}$. In this work we will only consider the symmetries of $\mathbb{G}$ under the canonical left action of a discrete subgroup $\mathfrak{G}\subset \mathbb{G}$ as in $\S$ 2.3 acting by $\mathfrak{G}\times\mathbb{G}\ni (n, x)\mapsto (n x)\in \mathbb{G}$. We extend this to an action on function $\psi:\mathbb{G}\rightarrow \mathbb{R}$ by pull-back, i.e.

\begin{equation*} (n^* \psi)(x):= \psi(n^{-1}x) = \psi_n(x). \end{equation*}

For a regularity structure $\mathcal{T}=(\operatorname{T},\operatorname{G})$ we give the following definition, by analogy with [Reference Hairer41, Def. 3.33] but restricted to this more straightforward setting.

Definition 13. Given a discrete sub-group $\mathfrak{G}\subset \mathbb{G}$ as above, we say that a model $M=(\Pi,\Gamma)$ is adapted to the action of $\mathfrak{G}$ if, for every test function, $\phi \in C^\infty_c(\mathbb{G})$, $x \in \mathbb{G}$, $\tau \in \operatorname{T}$ and $n \in \mathfrak{G}$ one has

\begin{equation*}(\Pi_{nx}\tau)(n^* \psi) = (\Pi_x \tau)(\psi)\quad \text{and}\quad \Gamma_{nx,ny} = \Gamma_{x,y}.\end{equation*}

A modelled distribution $f :\mathbb{G}\rightarrow \operatorname{T}$ is said to be symmetric if $ f( nx )= f(x)$ for every $x \in \mathbb{G}$ and $n \in \mathfrak{G}$.

The following proposition is an amalgam of [Reference Hairer41, Prop.  3.38 & Prop.  5.23].