2. The basic abstract result

Throughout the paper, we consider complex linear spaces. Let $H$ be an infinite-dimensional (not necessarily separable) Hilbert space, and let $N$ be a normed space. Suppose that $H$ and $N$ are algebraically embedded in a certain linear space. As usual, $\|\cdot \|_{H}$ and $(\cdot,\,\cdot )_{H}$ respectively denote the norm and inner product in $H$, and $\|\cdot \|_{N}$ stands for the norm in $N$. Thus, the linear space $H\cap N$ is endowed with two norms.

Let $L$ be a normal (specifically, self-adjoint) linear operator in $H$. We allow the case where $L$ is unbounded in $H$ as well as the case where $L$ is bounded on $H$. We suppose that $L$ has pure point spectrum, i.e. the Hilbert space $H$ has an orthonormal basis $\{e_{j}:j\in \Theta \}$ formed by certain eigenvectors $e_{j}$ of $L$. Here, $\Theta$ is an index set whose cardinality coincides with the Hilbert dimension of $H$. Thus, every vector $f\in H$ expands in the series

(2.1)\begin{equation} f=\sum_{j\in\Theta}(f,e_j)_{H}\,e_j \end{equation}

in the topology of $H$. (Of course, this series contains only a countable number of non-zero terms, and its sum does not depend on their order.) Let $\lambda _{j}$ denote the eigenvalue of $L$ such that $Le_j=\lambda _{j}e_j$. As usual, $\sigma (L)$ stands for the spectrum of $L$, and $\sigma _{p}(L)$ denotes the point spectrum of $L$, i.e. $\sigma _{p}(L)$ is the set of all eigenvalues of $L$. Note that eigenvalues of $L$ may be of infinite geometric multiplicity and that the set of all eigenvalues of $L$ may have accumulation points.

Let $\mathcal {Q}$ denote the system of all finite subsets of $\Theta$. We interpret $\mathcal {Q}$ as an upper directed set with respect to the relation $\subseteq$. Equality (2.1) means for $f\in H$ that the net

(2.2)\begin{equation} \biggl\{\sum_{j\in\Upsilon}(f,e_j)_{H}\,e_j:\Upsilon\in\mathcal{Q}\biggr\} \end{equation}

converges to $f$ in $H$.

Remark 2.2 Providing $H$ is separable, eigenvector expansion (2.1) becomes

(2.3)\begin{equation} f=\sum_{j=1}^{\infty}(f,e_j)_{H}\,e_j. \end{equation}

In this case, the expansions of $f$ in eigenvectors of $L$ converge unconditionally in $N$ if and only if series (2.3) converges to $f$ in $N$ for every orthonormal basis $\{e_{j}:j\in \mathbb {N}\}$ formed by eigenvectors of $L$ (i.e. for any choices of orthonormal bases of eigenspaces of $L$ associated with non-degenerate eigenvalues). It follows from this condition that series (2.3) converges under an arbitrary permutation of its terms.

Theorem 2.3 Let $\omega,\,\eta :\sigma (L)\to \mathbb {C}\setminus \{0\}$ be Borel measurable bounded functions, and define the bounded operators $\omega (L)$ and $\eta (L)$ on $H$ as functions of the normal operator $L$. Assume that the mapping $u\mapsto \omega (L)u$ sets a bounded operator from $H$ to $N$, and denote this operator by $R$. Let $f$ be an arbitrary vector from the image of the operator $(\omega \eta )(L)$. Then the expansions of $f$ in eigenvectors of $L$ converge unconditionally in the space $N$. Moreover, the degree of this convergence admits the estimate

(2.4)\begin{equation} \biggl\|f-\sum_{j\in\Upsilon}(f,e_j)_{H}\,e_j\biggr\|_{N}\leq \|R\|_{H\to N}\cdot\|g\|_{H}\cdot \sup_{j\in\Theta\setminus\Upsilon}|\eta(\lambda_j)| \cdot r_{g,e}(\Upsilon) \end{equation}

for each set $\Upsilon \in \mathcal {Q}$ and every orthonormal basis $e:=\{e_{j}:j\in \Theta \}$ indicated in definition 2.1 and with some net $\{r_{g,e}(\Upsilon ):\Upsilon \in \mathcal {Q}\}\subseteq [0,\,1]$ that tends monotonically to zero and depends neither on $\omega$ nor on $\eta$. Here, $g\in H$ is the unique vector such that $(\omega \eta )(L)g=f$.

Proof. Since $\omega (L)\eta (L)e_j=(\omega \eta )(\lambda _j)e_j$ whenever $j\in \Theta$ and since $(\omega \eta )(t)\neq 0$ whenever $t\in \sigma (L)$, we conclude that each $e_j\in N$ because the image of $\omega (L)$ lies in $N$. Thus, the left-hand side of (2.4) makes sense. Since the operator $(\omega \eta )(L)$ is algebraically reversible, the vector $g$ is well defined for every above-mentioned $f$. Suppose that $f\neq 0$, otherwise the conclusion of this theorem is trivial. Given $\Upsilon \in \mathcal {Q}$, we let $P_{\Upsilon }$ denote the orthoprojector of $H$ on the span of $\{e_j:j\in \Upsilon \}$ and obtain the following estimate:

(2.5)\begin{equation} \begin{aligned} \|f-P_{\Upsilon}f\|_{N} & = \|(\omega\eta)(L)g-P_{\Upsilon}(\omega\eta)(L)g\|_{N}\\ & =\|(\omega\eta)(L)(I-P_{\Upsilon})g\|_{N}= \|\omega(L)\eta(L)(I-P_{\Upsilon})^{2}g\|_{N}\\ & \leq\|R\|_{H\to N}\cdot\|\eta(L)(I-P_{\Upsilon})\|_{H\to H}\cdot \|(I-P_{\Upsilon})g\|_{H}, \end{aligned} \end{equation}

with $I$ being the identity operator on $H$. We put

(2.6)\begin{equation} r_{g,e}(\Upsilon):=\|(I-P_{\Upsilon})g\|_{H}\cdot\|g\|_{H}^{{-}1} \end{equation}

and see that the net $\{r_{g,e}(\Upsilon ):\Upsilon \in \mathcal {Q}\}$ is required. Note also that

(2.7)\begin{equation} \|\eta(L)(I-P_{\Upsilon})\|_{H\to H}\leq \sup_{j\in\Theta\setminus\Upsilon}|\eta(\lambda_j)|. \end{equation}

Indeed, since

\[ \eta(L)(I-P_{\Upsilon})h= \eta(L)\sum_{j\in\Theta\setminus\Upsilon}(h,e_{j})_{H}\,e_{j}= \sum_{j\in\Theta\setminus\Upsilon}(h,e_{j})_{H}\,\eta(\lambda_j)e_{j} \]

for every $h\in H$ (the convergence holds in $H$), we have the inequality

\[ \|\eta(L)(I-P_{\Upsilon})h\|_{H}^{2}= \sum_{j\in\Theta\setminus\Upsilon}|(h,e_{j})_{H}\,\eta(\lambda_j)|^{2}\leq \|h\|_{H}^{2}\cdot\sup_{j\in\Theta\setminus\Upsilon}|\eta(\lambda_j)|^{2}, \]

which gives (2.7). Now (2.5)–(2.7) yield estimate (2.4), which implies the required unconditional convergence.

Remark 2.4 Assume that the norms in $H$ and $N$ are compatible on the linear space $H\cap N$. (By definition [Reference Gel'fand and Shilov15, Chapter I, Section 2.2], this compatibility means the following: if a sequence lies in $H\cap N$ and converges to zero in one of these norms and is a Cauchy sequence with respect to the second norm, then it converges to zero in the second norm.) Let $R$ be a bounded linear operator on $H$ (specifically, $R:u\mapsto \omega (L)u$ where $u\in H$, as in theorem 2.3). Then the inclusion $R(H)\subseteq N$ implies that $R$ acts continuously from $H$ to $N$. Indeed, suppose that $R(H)\subseteq N$. Then, owing to the closed graph theorem, the operator $R:H\to \widetilde {N}$ is bounded if and only if it is closed; here, $\widetilde {N}$ is the completion of the normed space $N$. We therefore have to prove that this operator is closable. Assume that a sequence $(u_{k})_{k=1}^{\infty }\subset H$ satisfies the following two conditions: $u_{k}\to 0$ in $H$ and

(2.8)\begin{equation} Ru_{k}\to v\text{ in }\widetilde{N}\text{ for certain } v\in\widetilde{N}, \end{equation}

as $k\to \infty$. Then $Ru_{k}\to 0$ in $H$ because $R$ is bounded on $H$. Besides, $(Ru_{k})_{k=1}^{\infty }$ is a Cauchy sequence in $N$. Hence, $Ru_{k}\to 0$ in $N$ because the norms in $N$ and $H$ are compatible on $H\cap N$. Thus, $v=0$ in view of (2.8), which means that the operator $R:H\to \widetilde {N}$ is closable.

Theorem 2.3 contains Krein's result [Reference Krein23, Theorem 4] according to which series (2.3) converges in $N$ for every $f\in L(H)$ if $L$ is a compact self-adjoint operator in $H$ and if $L$ acts continuously from $H$ to a Banach space $N$. The latter theorem generalizes (to abstract operators) the Hilbert–Schmidt theorem about the uniform decomposability of sourcewise representable functions in eigenfunctions of a symmetric integral operator.

If $L$ is a positive definite self-adjoint operator with discrete spectrum and if $\omega (t)\equiv t^{-\delta }$ and $\eta (t)\equiv t^{-\tau }$ for certain numbers $\delta,\,\tau \geq 0$ and if $R:=L^{-\delta }$ acts continuously from $H$ to a Banach space $N$, Krasnosel'skiĭ and Pustyl'nik [Reference Krasnosel'skiĭ, Zabreĭko, Pustyl'nik and Sobolevskiĭ22, Theorem 22.1] proved that

\[ \biggl\|f-\sum_{j=1}^{k}(f,e_j)_{H}\,e_j\biggr\|_{N}= o(\lambda_{k}^{-\tau})\text{ as }k\to\infty \]

for any $f$ from the domain of $L^{\delta +\tau }$ (the Hilbert space $H$ is assumed to be separable). Here, the eigenvalues $\lambda _{k}>0$ of $L$ are numbered so that $\lambda _{k}\leq \lambda _{k+1}$. Formula (2.4) implies a more general result for normal operators with discrete spectrum.

3. Hörmander spaces and elliptic operators

We will use them to estimate the degree of the convergence of expansions in eigenfunctions of some elliptic operators given on a closed manifold. Such spaces were introduced over $\mathbb {R}^{n}$ and investigated by Hörmander [Reference Hörmander17, Section 2.2]. Their smoothness index (or differentiation order) is a general enough function of the frequency variables, which allows us to obtain finer estimates than those [Reference Agranovich1, Subsection 6.1 a] received with the help of classical function spaces such as Sobolev spaces, whose smoothness index is a number. We use a broad class of isotropic inner product Hörmander spaces that admit a reasonable definition on smooth closed manifolds. The class was introduced and investigated in [Reference Mikhailets and Murach27], [Reference Mikhailets and Murach28, Section 2.4.2] and [Reference Mikhailets and Murach29] (note that these works develop an approach proposed in [Reference Mikhailets and Murach25, Reference Mikhailets and Murach26]). It has various applications to elliptic operators [Reference Mikhailets and Murach28, Section 2.4.3] and elliptic boundary-value problems [Reference Anop, Chepurukhina and Murach4, Reference Anop, Denk and Murach5].

This class is formed by certain function spaces $H^{\alpha }$ whose smoothness index $\alpha$ ranges over a set denoted by $\mathrm {OR}$. By definition, $\mathrm {OR}$ consists of all Borel functions $\alpha : [1,\,\infty )\rightarrow (0,\,\infty )$ such that

\[ c^{{-}1}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq c \text{ whenever } t\geq1\text{ and }1\leq\lambda\leq b, \]

with the numbers $b>1$ and $c\geq 1$ being independent of $t$ and $\lambda$ (but may depend on $\alpha$). Such functions were introduced by Avakumović [Reference Avakumović7], are called OR-varying (or O-regularly varying) at infinity, and are well investigated [Reference Bingham, Goldie and Teugels9, Reference Buldygin, Indlekofer, Klesov and Steinebach10].

The class $\mathrm {OR}$ admits the following description [Reference Bingham, Goldie and Teugels9, Theorem 2.2.7]: $\alpha \in \mathrm {OR}$ if and only if

(3.1)\begin{equation} \alpha(t)= \exp\Biggl(\beta(t)+ \int\limits_{1}^{t}\frac{\gamma(\tau)}{\tau}\;{\rm d}\tau\Biggr) \text{ whenever }t\geq1 \end{equation}

for some bounded Borel functions $\beta,\,\gamma :[1,\,\infty )\to \mathbb {R}$.

For every function $\alpha \in \mathrm {OR}$ there exist numbers $s_{0},\,s_{1}\in \mathbb {R}$, with $s_{0}\leq s_{1}$, and numbers $c_{0},\,c_{1}>0$ such that

(3.2)\begin{equation} c_{0}\lambda^{s_{0}}\leq\frac{\alpha(\lambda t)}{\alpha(t)}\leq c_{1}\lambda^{s_{1}}\quad\mbox{whenever}\quad t\geq1\text{ and }\lambda\geq1. \end{equation}

Given $\alpha \in \mathrm {OR}$, we let $s_{\ast }(\alpha )$ denote the supremum of the set of all numbers $s_{0}$ that the left-hand inequality in (3.2) holds true (for certain $c_{0}$ depending on $s_{0}$), and we let $s^{\ast }(\alpha )$ denote the infimum of the set of all numbers $s_{1}$ that the right-hand inequality in (3.2) holds true (for certain $c_{1}$ depending on $s_{1}$). Of course, $-\infty < s_{\ast }(\alpha )\leq s^{\ast }(\alpha )<\infty$. The numbers $s_{\ast }(\alpha )$ and $s^{\ast }(\alpha )$ are equal to the lower and upper Orlicz–Matuszewska indices of $\alpha$, resp. These facts follow from [Reference Bingham, Goldie and Teugels9, Theorem 2.1.7 and Proposition 2.2.1].

For instance, every continuous function $\alpha :[1,\,\infty )\rightarrow (0,\,\infty )$ such that

\[ \alpha(t)=t^{s}(\log t)^{s_{1}}(\log\log t)^{s_{2}}\ldots (\underbrace{\log\ldots\log}_{k\;\mbox{{times}}} t)^{s_{k}}\quad\mbox{whenever}\quad t\gg1 \]

belongs to $\mathrm {OR}$; here, $1\leq k\in \mathbb {Z}$ and $s,\,s_{1},\,\ldots,\,s_{k}\in \mathbb {R}$. It is well known that $s_{\ast }(\alpha )=s^{\ast }(\alpha )=s$ for this function.

An example of a function $\alpha \in \mathrm {OR}$ with the different Orlicz–Matuszewska indices is given by formula (3.1) in which

(3.3)\begin{equation} \gamma(\tau):=\left\{ \begin{array}{ll} r & \hbox{if}\;\;\tau\in[\theta_{1}\cdots\theta_{2j-1}, \theta_{1}\cdots\theta_{2j-1}\theta_{2j}] \text{ for certain integer }j\geq1,\\ s & \hbox{otherwise} \end{array}\right. \end{equation}

provided that $r< s$, $1=\theta _{1}<\theta _{2}<\theta _{3}<\dots$, and $\theta _{k}\to \infty$ as $k\to \infty$. Then ${s_{\ast }(\alpha )=r}$ and $s^{\ast }(\alpha )=s$, as is noticed, e.g., in [Reference Anop, Chepurukhina and Murach4, Section 3].

Let $\alpha \in \mathrm {OR}$ and $1\leq n\in \mathbb {Z}$. By definition, the linear space $H^{\alpha }(\mathbb {R}^{n})$ consists of all tempered distributions $w$ on $\mathbb {R}^{n}$ that their Fourier transform $\widehat {w}:=\mathcal {F}w$ is locally Lebesgue integrable over $\mathbb {R}^{n}$ and satisfies the condition

\[ \int\limits_{\mathbb{R}^{n}}\alpha^{2}(\langle\xi\rangle)\, |\widehat{w}(\xi)|^{2}\,{\rm d}\xi<\infty. \]

As usual, $\langle \xi \rangle :=(1+|\xi |^{2})^{1/2}$ is the smoothed absolute value of $\xi \in \mathbb {R}^{n}$. The space $H^{\alpha }(\mathbb {R}^{n})$ is endowed with the inner product

\[ (w_{1},w_{2})_{H^{\alpha}(\mathbb{R}^{n})}:= \int\limits_{\mathbb{R}^{n}}\alpha^{2}(\langle\xi\rangle)\, \widehat{w_{1}}(\xi)\,\overline{\widehat{w_{2}}(\xi)}\,d\xi \]

and the corresponding norm. This space is an isotropic Hilbert case of the Hörmander spaces $\mathcal {B}_{p,k}$ considered in [Reference Hörmander17, Section 2.2] and [Reference Hörmander18, Section 10.1]. Namely, $H^{\alpha }(\mathbb {R}^{n})=\mathcal {B}_{2,k}$ provided that $k(\xi )=\alpha (\langle \xi \rangle )$ whenever $\xi \in \mathbb {R}^{n}$.

Consider a version of the space $H^{\alpha }(\mathbb {R}^{n})$ for smooth closed manifolds. In the sequel, $\Gamma$ is a closed (i.e. compact and boundaryless) manifold of dimension $n\geq 1$ and class $C^{\infty }$. Suppose that a positive $C^{\infty }$-density ${\rm d}x$ is given on $\Gamma$. We arbitrarily choose a finite atlas from the $C^{\infty }$-structure on $\Gamma$; let this atlas be formed by $\varkappa$ local charts $\pi _{j}: \mathbb {R}^{n}\leftrightarrow \Gamma _{j}$, with $j=1,\,\ldots,\,\varkappa$. Here, the open sets $\Gamma _{1},\,\ldots,\,\Gamma _{\varkappa }$ form a covering of $\Gamma$. We also arbitrarily choose functions $\chi _j\in C^{\infty }(\Gamma )$, with $j=1,\,\ldots,\,\varkappa$, such that $\chi _{1}(x)+\cdots +\chi _{\varkappa }(x)=1$ whenever $x\in \Gamma$ and that $\mathrm {supp}\,\chi _j\subset \Gamma _j$ whenever $1\leq j\leq \varkappa$. Thus, these functions form a partition of unity on $\Gamma$ subordinate to the above covering.

By definition, the linear space $H^{\alpha }(\Gamma )$ consists of all distributions $f$ on $\Gamma$ such that $(\chi _{j}f)\circ \pi _{j}\in H^{\alpha }(\mathbb {R}^{n})$ for each $j\in \{1,\,\ldots,\,\varkappa \}$. Here, $(\chi _{j}f)\circ \pi _{j}$ is the representation of the distribution $\chi _{j}f$ in the local chart $\pi _{j}$. The space $H^{\alpha }(\Gamma )$ is endowed with the inner product

\[ (f_{1},f_{2})_{H,\alpha}:= \sum_{j=1}^{\varkappa}\,((\chi_{j}f_{1})\circ\pi_{j}, (\chi_{j}f_{2})\circ\pi_{j})_{H^{\alpha}(\mathbb{R}^{n})} \]

and the corresponding norm $\|f\|_{H,\alpha }:=(f,\,f)_{H,\alpha }^{1/2}$. This space is complete (i.e. Hilbert) and separable and does not depend up to equivalence of norms on the choice of the atlas and partition of unity on $\Gamma$; the set $C^{\infty }(\Gamma )$ is dense in $H^{\alpha }(\Gamma )$. These properties are due to [Reference Mikhailets and Murach28, Theorem 2.21].

If $\alpha (t)\equiv t^{s}$ for some $s\in \mathbb {R}$, then $H^{\alpha }(G)$, where $G=\mathbb {R}^{n}$ or $G=\Gamma$, becomes the inner product Sobolev space $W^{s}_{2}(G)$ of order $s$.

We also need an equivalent definition of the space $H^{\alpha }(\Gamma )$, with $\alpha \in \mathrm {OR}$, in terms of a function of some elliptic pseudodifferential operators (PsDOs). Following [Reference Agranovich1, Section 2.1], we let $\Psi _{\mathrm {ph}}^{m}(\Gamma )$ denote the class of all classical (i.e. polyhomogeneous) PsDOs on $\Gamma$ of order $m\in \mathbb {R}$. Suppose that $m>0$ and that a PsDO $A\in \Psi _{\mathrm {ph}}^{m}(\Gamma )$ is elliptic on $\Gamma$, i.e. $a_{0}(x,\,\xi )\neq 0$ whenever $x\in \Gamma$ and $0\neq \xi \in T_{x}^{\ast }\Gamma$, with $a_{0}(x,\,\xi )$ denoting the principal symbol of $A$ and with $T_{x}^{\ast }\Gamma$ standing (as usual) for the cotangent space to $\Gamma$ at $x$. We may and will consider $A$ as a closed unbounded operator in the Hilbert Lebesgue space $L_{2}(\Gamma ):=L_{2}(\Gamma,\,{\rm d}x)$ and with the domain $\mathrm {Dom}\,A=W^{m}_{2}(\Gamma )$ (see [Reference Agranovich1, Theorem 2.3.5]). Suppose that $A$ is a self-adjoint operator in $L_{2}(\Gamma )$ and that $\sigma (A)\subseteq [1,\,\infty )$. Then the operator $\alpha (A^{1/m})$ is well defined in the Hilbert space $L_{2}(\Gamma,\,{\rm d}x)$ via Spectral Theorem as the Borel function $\alpha (t^{1/m})$ of $A$.

Proposition 3.1 Let $\alpha \in \mathrm {OR}$. Then the norm in the space $H^{\alpha }(\Gamma )$ is equivalent to the norm

(3.4)\begin{equation} f\mapsto\|\alpha(A^{1/m})f\|_{L_{2}(\Gamma)} \end{equation}

on the dense set $C^{\infty }(\Gamma )$. Thus, $H^{\alpha }(\Gamma )$ coincides with the completion of $C^{\infty }(\Gamma )$ with respect to norm (3.4). Hence, if the function $1/\alpha$ is bounded on $[1,\,\infty )$, then $H^{\alpha }(\Gamma )$ is the domain of the operator $\alpha (A^{1/m})$ and this operator sets an isomorphism between $H^{\alpha }(\Gamma )$ and $L_{2}(\Gamma )$.

This proposition is due to [Reference Mikhailets and Murach28, Theorem 2.23]. We may put $A:=1-\Delta _{\Gamma }$ and $m=2$, where $\Delta _{\Gamma }$ is the Laplace–Beltrami operator on $\Gamma$, with $\Gamma$ being endowed with the Riemannian metric inducing ${\rm d}x$.

4. Mean convergence

Given $s\in \mathbb {R}$ and $p\in (1,\,\infty )$, we let $W^{s}_{p}(\Gamma )$ denote the Sobolev space over $\Gamma$ with the smoothness index $s$ and the integral-exponent $p$ and let $\|\cdot \|_{W,s,p}$ stand for the norm in this space. The space $W^{s}_{p}(\Gamma )$ is defined on the base of $W^{s}_{p}(\mathbb {R}^{n})$ with the help of the above atlas and partition of unity like the definition of $H^{\alpha }(\Gamma )$ on the base of $H^{\alpha }(\mathbb {R}^{n})$. Recall that the Sobolev space $W^{s}_{p}(\mathbb {R}^{n})$ consists of all tempered distributions $w$ on $\mathbb {R}^{n}$ such that the distribution $v:=\mathcal {F}^{-1}[\langle \xi \rangle ^{s}\,\widehat {w}(\xi )]$ belongs to the Lebesgue space $L_{p}(\mathbb {R}^{n})$, with the norm of $w$ in $W^{s}_{p}(\mathbb {R}^{n})$ being equal to the norm of $v$ in $L_{p}(\mathbb {R}^{n})$. Of course, $\mathcal {F}^{-1}$ stands for the inverse Fourier transform. Note that $W^{0}_{p}(\Gamma )$ coincides with the Lebesgue space $L_{p}(\Gamma ):=L_{p}(\Gamma,\,{\rm d}x)$, and assume that this holds with equality of norms.

We use the term ‘mean convergence’ to refer to the convergence in the normed Sobolev space $W^{\ell }_{p}(\Gamma )$ subject to $0\leq \ell \in \mathbb {Z}$ and $1< p<\infty$, specifically in the Lebesgue space $L_{p}(\Gamma )$.

Let real $m>0$, and let $A\in \Psi _{\mathrm {ph}}^{m}(\Gamma )$. Suppose that the PsDO $A$ is elliptic on $\Gamma$. If $A$ is a normal operator in $L_{2}(\Gamma )$, the separable Hilbert space $L_{2}(\Gamma )$ has an orthonormal basis formed by some eigenfunctions of $A$ (see e.g. [Reference Shubin32, Section 15.3]), every eigenfunction of $A$ pertaining to $C^{\infty }(\Gamma )$ and each eigenvalue of $A$ being of finite multiplicity. Then every vector $f\in L_{2}(\Gamma )$ admits the spectral decomposition

(4.1)\begin{equation} f=\lim_{r\to\infty} \sum_{\substack{\lambda\in\sigma_{p}(A)\\|\lambda|\leq r}}P(\lambda)f, \end{equation}

the series converging in $L_{2}(\Gamma )$. As usual, $P(\lambda )$ denotes the orthoprojector on the eigenspace of $A$ associated with the eigenvalue $\lambda$. Note that the sum in (4.1) contains only a finite number of terms for every $r>0$. The convergence in $L_{2}(\Gamma )$ implies one in each normed space $L_{p}(\Gamma )$ with $1\leq p<2$; hence, we are interested in conditions for the convergence of (4.1) in the mean with index $p>2$.

Given real $m>0$, we let $\mathrm {NE}\Psi _{\mathrm {ph}}^{m}(\Gamma )$ denote the set of all PsDOs of class $\Psi _{\mathrm {ph}}^{m}(\Gamma )$ that are elliptic on $\Gamma$ and normal in $L_{2}(\Gamma )$. Specifically, every elliptic constant-coefficient partial differential operator on $\Gamma$ of order $m$ belongs to $\mathrm {NE}\Psi _{\mathrm {ph}}^{m}(\Gamma )$. Recall that $n=\dim \Gamma \geq 1$.

Let $m>0$, and let $A\in \mathrm {NE}\Psi _{\mathrm {ph}}^{m}(\Gamma )$. Consider an orthonormal basis $e:=(e_{j})_{j=1}^{\infty }$ in $L_{2}(\Gamma )$ formed by eigenfunctions $e_{j}\in C^{\infty }(\Gamma )$ of $A$. Let $\lambda _{j}$ denote the eigenvalue of $A$ such that $Ae_j=\lambda _{j}e_j$. We enumerate the eigenfunctions so that $|\lambda _{j}|\leq |\lambda _{j+1}|$ whenever $j\geq 1$. Then

(4.2)\begin{equation} |\lambda_{j}|\sim\widetilde{c}\,j^{m/n}\quad\mbox{as}\quad j\to\infty, \end{equation}

where $\widetilde {c}$ is a certain positive number that does not depend on $j$ (see e.g. [Reference Shubin32, Section 15.3]). According to theorem 4.1, the series (2.3), where $H=L_{2}(\Gamma )$, converges unconditionally in $W^{\ell }_{p}(\Gamma )$ on the class $W^{\ell +n/2-n/p}_{2}(\Gamma )$. This class is the broadest one among the spaces $W^{s}_{q}(\Gamma )$ indicated by theorem 4.1 as classes of convergence of the above series. Using the Hörmander spaces $H^{\alpha }(\Gamma )\subset W^{\ell +n/2-n/p}_{2}(\Gamma )$ as classes of the convergence, we can estimate its degree.

Theorem 4.2 Let $0\leq \ell \in \mathbb {Z}$, $2< p\in \mathbb {R}$ and $\alpha \in \mathrm {OR}$. Suppose that

\[ h(t):=t^{\ell+n/2-n/p}\,(\alpha(t))^{{-}1}\to0 \quad\mbox{as}\quad t\to\infty \]

and that the function $h$ decreases on $[1,\,\infty )$. Then

(4.3)\begin{equation} \biggl\|f-\sum_{j:|\lambda_{j}|\leq\lambda} (f,e_j)_{H}\,e_j\biggr\|_{W,\ell,p} \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda^{1/m}) \end{equation}

and

(4.4)\begin{equation} \biggl\|f-\sum_{j=1}^{k}(f,e_j)_{H}\,e_j\biggr\|_{W,\ell,p} \leq c\cdot\|f\|_{H,\alpha}\cdot h(k^{1/n}) \end{equation}

for all $f\in H^{\alpha }(\Gamma )$, $\lambda \geq 1$ and integer-valued $k\geq 1$. Here, $c$ is a certain positive number that does not depend on $f$, $\lambda$ and $k$.

Proof. Put $\sigma :=l+n/2-n/p$, $L:=(I+A^{\ast }A)$ and $\omega (t):=t^{-\sigma /(2m)}$ and $\eta (t):=h(t^{1/(2m)})$ whenever $t\geq 1$. Since $(\omega \eta )(t)=(\alpha (t^{1/(2m)}))^{-1}$ whenever $t\geq 1$, the operator $(\omega \eta )(L)$ sets an isomorphism between $L_{2}(\Gamma )$ and $H^{\alpha }(\Gamma )$ due to proposition 3.1. Hence, according to theorem 2.3 and since $h$ decreases, we have the estimates

(4.5)\begin{equation} \biggl\|f-\sum_{j:\lambda_{j}'\leq1+\lambda^{2}} (f,e_j)_{H}\,e_j\biggr\|_{W,\ell,p}\leq c_{0}\cdot\|g\|_{H}\cdot \eta(1+\lambda^{2}) \end{equation}

and

(4.6)\begin{equation} \biggl\|f-\sum_{j=1}^{k}(f,e_j)_{H}\,e_j\biggr\|_{W,\ell,p} \leq c_{0}\cdot\|g\|_{H}\cdot\eta(\lambda_{k}'), \end{equation}

where $c_{0}$ is the norm of $\omega (L)$ considered as a bounded operator from $H=L_{2}(\Gamma )$ to $N=W^{\ell }_{p}(\Gamma )$ (see the proof of theorem 4.1), $g:=((\omega \eta )(L))^{-1}f$ and the number $\lambda _{j}'$ satisfies $Le_{j}=\lambda _{j}'e_{j}$, i.e. $\lambda _{j}'=1+|\lambda _{j}|^{2}$. Hence, $\|g\|_{H}\asymp \|f\|_{H,\alpha }$, $\eta (1+\lambda ^{2})\asymp h(\lambda ^{1/m})$ as $\lambda \geq 1$, and

\[ \eta(\lambda_{k}')=h((1+|\lambda_{k}|^{2})^{1/(2m)})\asymp h(k^{1/n}) \quad\mbox{as}\quad k\geq1, \]

due to $h\in \mathrm {OR}$ and (4.2). (As usual, the symbol $\asymp$ means the weak equivalence of positive values.) Thus, (4.5) and (4.6) imply the required estimates (4.3) and (4.4), resp.

Remark 4.3 If $A$ is a positive definite operator in $L_{2}(\Gamma )$, we have the equivalence of norms

\[ \|f\|_{H,\alpha}\asymp\biggl(\,\sum_{j=1}^{\infty} \alpha^{2}(j^{1/n})\,|(f,e_j)_{H}|^{2}\biggr)^{1/2}\asymp \biggl(\,\sum_{j=1}^{\infty} \alpha^{2}((1+\lambda_{j})^{1/m})\,|(f,e_j)_{H}|^{2}\biggr)^{1/2}, \]

which follows from proposition 3.1 (cf. [Reference Mikhailets and Murach28, Theorem 2.7]).

Let us consider three examples of a function $\alpha$ that satisfy hypotheses of theorem 4.2. As above, $0\leq \ell \in \mathbb {Z}$ and $2< p\in \mathbb {R}$. We arbitrarily choose a number $\varepsilon >0$.

Putting $\alpha (t):=t^{l+\varepsilon +n/2-n/p}$ whenever $t\geq 1$, we obtain the power estimates (4.3) and (4.4) with $h(\lambda ^{1/m})=\lambda ^{-\varepsilon /m}$ and $h(k^{1/n})=k^{-\varepsilon /n}$, resp. In this case, $H^{\alpha }(\Gamma )$ is a Sobolev space.

Taking $s:=l+n/2-n/p$ and $r:=s+\varepsilon$ in (3.3) and defining $\alpha$ by formula (3.1) with $\beta (t)\equiv 1$, we receive the Hörmander space $H^{\alpha }(\Gamma )$ that is not a part of the union of all Sobolev spaces $W^{s}_{2}(\Gamma )$ such that $s>\ell +n/2-n/p$. In this case, $h(t)\searrow 0$ as $t\to \infty$.

The third example is given by any function $\alpha \in \mathrm {OR}$ such that

\[ \alpha(t):=t^{l+n/2-n/p}\, (\underbrace{\log\ldots\log}_{k\;\mbox{{times}}} t)^{\varepsilon} \quad\text{ whenever }t\gg1. \]

In this case, estimate (4.4) is of a logarithmic kind, and the space $H^{\alpha }(\Gamma )$ is broader than the above union of Sobolev spaces.

Remark 4.5 Let $0\leq \ell \in \mathbb {R}$, $1\leq q<2< p$ and $s=\ell +n/q-n/p$. Assertions (ii) and (iii) of theorem 4.1 hold true if we replace the normed space $W^{\ell }_{p}(\Gamma )$ with the Triebel–Lizorkin space $F^{\ell }_{p,\theta }(\Gamma )$, where $1\leq \theta \leq \infty$, or with the Besov space $B^{\ell }_{p,\theta }(\Gamma )$, where $2\leq \theta \leq \infty$, and replace the convergence class $W^{s}_{q}(\Gamma )$ with $F^{s}_{q,\eta }(\Gamma )$, where $1\leq \eta \leq \infty$, or with $B^{s}_{q,\eta }(\Gamma )$, where $1\leq \eta \leq 2$. This follows from the continuous embeddings

\[ F^{s}_{q,\eta}(\Gamma)\hookrightarrow W^{\ell+n/2-n/p}_{2}(\Gamma)\hookrightarrow F^{\ell}_{p,\theta}(\Gamma) \]

and

\[ B^{s}_{q,\eta}(\Gamma)\hookrightarrow W^{\ell+n/2-n/p}_{2}(\Gamma)\hookrightarrow B^{\ell}_{p,\theta}(\Gamma) \]

[Reference Triebel34, p. 60, properties (ii) and (iii)], as we can conclude analysing the proof of this theorem. Of course, it is natural to restrict ourselves to the marginal cases where $\eta =\infty$ and $\theta =1$ for $F$-spaces and where $\eta =\theta =2$ for $B$-spaces. In these cases we obtain narrower normed spaces than $W^{\ell }_{p}(\Gamma )$ and get broader convergence classes than $W^{s}_{q}(\Gamma )$. Of course, theorem 4.2 is also valid under this replacement of $W^{\ell }_{p}(\Gamma )$.

5. Uniform convergence

We use this term to refer to the convergence in the normed space $C^{\ell }(\Gamma )$ of $\ell$ times continuously differentiable functions on the manifold $\Gamma$, with $0\leq \ell \in \mathbb {Z}$. Specifically, if $\ell =0$, we get the convergence in the space $C(\Gamma )$ of continuous functions on $\Gamma$, i.e. the uniform convergence on $\Gamma$. Let $\|\cdot \|_{C,\ell }$ denote the norm in $C^{\ell }(\Gamma )$. If $\ell \geq 1$, we use the norm

\[ \|f\|_{C,\ell}:=\sum_{j=1}^{\varkappa}\,\max_{|\varrho|\leq\ell}\, \sup_{y\in\mathbb{R}^{n}}|\partial^{\varrho}((\chi_{j}f)\circ\pi_{j})(y)|, \]

with $\pi _{j}$ and $\chi _{j}$ being taking from the definition of Hörmander spaces over $\Gamma$. Here, of course, $\varrho =(\varrho _1,\,\ldots,\,\varrho _n)$ is a multi-index, $|\varrho |=\varrho _1+\cdots +\varrho _n$, and $\partial ^{\varrho }$ is the partial derivative corresponding to $\varrho$.

Proof. It follows from Hörmander's embedding theorem [Reference Hörmander17, Theorem 2.2.7] that

(5.1)\begin{equation} \mbox{condition (i)}\;\Longleftrightarrow\; H^{\alpha}(\Gamma)\subset C^{\ell}(\Gamma), \end{equation}

as is seen from [Reference Mikhailets and Murach28, Proposition 2.6 (vi)].

Let us now prove that $\mathrm {(i)}\Rightarrow \mathrm {(iii)}$. We assume (i) to hold true and arbitrarily choose a PsDO $A\in \mathrm {NE}\Psi _{\mathrm {ph}}^{m}(\Gamma )$. Owing to (5.1), the space $H^{\alpha }(\Gamma )$ lies in $L_{2}(\Gamma )$. Hence, the function $1/\alpha$ is bounded on $[1,\,\infty )$ in view of [Reference Hörmander17, Theorem 2.2.2]. We put $\omega (t):=1/\alpha (t^{1/(2m)})$ whenever $t\geq 1$ and consider the function $\omega (L)$ of the operator $L=(I+A^{\ast }A)\in \mathrm {NE}\Psi _{\mathrm {ph}}^{2m}(\Gamma )$ acting in $L_{2}(\Gamma )$. According to proposition 3.1, the operator $\omega (L)$ sets an isomorphism between $L_{2}(\Gamma )$ and $H^{\alpha }(\Gamma )$. Hence, owing to (5.1), this operator acts continuously from $H=L_{2}(\Gamma )$ to $N:=C^{\ell }(\Gamma )$, as noticed in remark 2.4. We therefore conclude by theorem 2.3, where $\eta (t)\equiv 1$, that the expansions in eigenfunctions of $L$ converge unconditionally in $C^{\ell }(\Gamma )$ on the class $H^{\alpha }(\Gamma )$. We have proved that $\mathrm {(i)}\Rightarrow \mathrm {(iii)}$.

The implication $\mathrm {(iii)}\Rightarrow \mathrm {(ii)}$ is obvious.

The implication $\mathrm {(ii)}\Rightarrow \mathrm {(i)}$ is true because assertion (ii) entails the inclusion $H^{\alpha }(\Gamma )\subset C^{\ell }(\Gamma )$ and hence implies (i) by (5.1).

Let $m>0$ and $A\in \mathrm {NE}\Psi _{\mathrm {ph}}^{m}(\Gamma )$. As in the previous section, $e:=(e_{j})_{j=1}^{\infty }$ is an orthonormal basis in $H=L_{2}(\Gamma )$ formed by eigenfunctions $e_{j}$ of $A$, with $Ae_j=\lambda _{j}e_j$ and $|\lambda _{j}|\leq |\lambda _{j+1}|$ whenever $j\geq 1$. We complement theorem 5.1 by estimating the degree of the uniform convergence of series (2.3) on Hörmander classes.

Theorem 5.2 Let $0\leq \ell \in \mathbb {Z}$, and suppose that certain functions $h,\,\beta \in \mathrm {OR}$ satisfy the following conditions: $h(t)\to 0$ as $t\to \infty$, $h$ decreases on $[1,\,\infty )$, and

\[ \int\limits_{1}^{\infty}\frac{t^{2\ell+n-1}}{\beta^{2}(t)}\,{\rm d}t<\infty. \]

Put $\alpha :=\beta /h$, and note that the function $\alpha$ belongs to $\mathrm {OR}$ and satisfies hypothesis (i) of theorem 5.1. Then

(5.2)\begin{equation} \biggl\|f-\sum_{j:|\lambda_{j}|\leq\lambda} (f,e_j)_{H}\,e_j\biggr\|_{C,\ell} \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda^{1/m}) \end{equation}

and

(5.3)\begin{equation} \biggl\|f-\sum_{j=1}^{k}(f,e_j)_{H}\,e_j\biggr\|_{C,\ell} \leq c\cdot\|f\|_{H,\alpha}\cdot h(k^{1/n}) \end{equation}

for all $f\in H^{\alpha }(\Gamma )$, $\lambda \geq 1$, and integer-valued $k\geq 1$. Here, $c$ is a certain positive number that does not depend on $f$, $\lambda$ and $k$.

Let us indicate important examples of the function $\alpha$ satisfying hypotheses of theorem 5.2. We let $0\leq \ell \in \mathbb {Z}$ and arbitrarily choose numbers $\varepsilon$ and $\delta$ such that $0<\delta <\varepsilon$.

Putting $\alpha (t):=t^{l+n/2+\varepsilon +\delta }$ whenever $t\geq 1$, we obtain power estimates (5.2) and (5.3) with $h(t)\equiv t^{-\delta }$. Such power estimates hold true for functions

\[ f\in\bigcup_{s>l+n/2}W^{s}_{2}(\Gamma)\subsetneqq W^{l+n/2}_{2}(\Gamma). \]

To achieve the limiting value $s=l+n/2$, we put $\alpha (t):=t^{l+n/2}\log ^{1/2+\varepsilon +\delta }(t+1)$ whenever $t\geq 1$ and receive the estimates of a logarithmic kind with ${h(t)\equiv \log ^{-\delta }(t+1)}$.

6. Applications to ordinary differential operators

Let us discuss results of § 4 and 5 in the case where $\Gamma$ is a circle $\mathbb {T}$ of the unit radius and when $A$ is an (ordinary) differential operator on $\mathbb {T}$ of order $m$ with infinitely smooth complex-valued coefficients. Let $\tau$, with $0\leq \tau \leq 2\pi$, set a natural parametrization of $\mathbb {T}$. We suppose that the leading coefficient of $A$ does not equal zero for any $\tau$, which is equivalent to the ellipticity of $A$ on $\mathbb {T}$. We also assume that $A$ is a normal operator in the Hilbert space $H:=L_{2}(\mathbb {T})$. If all coefficients of $A$ are constant, this assumption holds true. In the case of variables coefficients, we recall the following necessary and sufficient condition for $A$ to be self-adjoint in $L_{2}(\mathbb {T})$ [Reference Naimark31, Chapter I, Section 1, Subsection 5]: $A$ is a sum of differential operators of the form

\[ A_{2k}u:=(\mu_{k}u^{(k)})^{(k)}\quad\mbox{and}\quad A_{2k-1}u:=\frac{i}{2}\bigl((\nu_{k}u^{(k)})^{(k-1)}+ (\nu_{k}u^{(k-1)})^{(k)}\bigr) \]

provided that $\mu _{k}$ and $\nu _{k}$ are real-valued functions of class $C^{\infty }(\mathbb {T})$. Specifically, if all coefficients of $A$ are real-valued, the above condition means that $m$ is even and that $A$ is the sum of $A_{2k}$ with $k=0,\,1,\,\ldots,\,m/2$.

Let $e:=(e_{j})_{j=1}^{\infty }$ be an orthonormal basis in $H=L_{2}(\mathbb {T})$ formed by eigenfunctions $e_{j}$ of $A$, with $Ae_j=\lambda _{j}e_j$ and $|\lambda _{j}|\leq |\lambda _{j+1}|$ whenever $j\geq 1$. Given a function $f\in L_{2}(\mathbb {T})$, we consider its expansion

(6.1)\begin{equation} f(\tau)=\sum_{j=1}^{\infty}(f,e_j)_{H}\,e_j(\tau), \quad\mbox{with}\quad 0\leq\tau\leq2\pi, \end{equation}

in these eigenfunctions, the expansion converging in $L_{2}(\mathbb {T})$. Let $0\leq \ell \in \mathbb {Z}$ and $2< p\in \mathbb {R}$. If $1< q\leq 2$ and $s\geq \ell +1/q-1/p$, then theorem 4.1 implies that, for every $f\in W^{s}_{q}(\mathbb {T})$, this expansion converges in the $p$-th mean unconditionally and preserves this convergence after the termwise differentiation up to $\ell$ times. If a function parameter $\alpha \in \mathrm {OR}$ satisfies

\[ \int\limits_{1}^{\infty}\frac{t^{2\ell}}{\alpha^{2}(t)}\,{\rm d}t<\infty, \]

then theorem 5.1 implies that, for every $f\in H^{\alpha }(\mathbb {T})$, the expansion converges uniformly and unconditionally and preserve this convergence after the same differentiation. Note that the derivatives of $e_j(\tau )$ may not form an orthogonal basis in $L_{2}(\mathbb {T})$. Assuming $\alpha$ to satisfy the hypotheses of theorem 4.2 or 5.2, we see that estimates (4.3) and (5.2) become

\[ \sum_{k=0}^{\ell}\biggl(\,\int\limits_{0}^{2\pi}\, \biggl|f^{(k)}(\tau)-\sum_{j:|\lambda_{j}|\leq\lambda} (f,e_j)_{H}\,e_j^{(k)}(\tau)\biggr|^{p}{\rm d}\tau\biggr)^{1/p} \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda^{1/m}) \]

and

\[ \sum_{k=0}^{\ell}\,\sup_{0\leq\tau\leq2\pi}\, \biggl|f^{(k)}(\tau)-\sum_{j:|\lambda_{j}|\leq\lambda} (f,e_j)_{H}\,e_j^{(k)}(\tau)\biggr| \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda^{1/m}) \]

whenever $f\in H^{\alpha }(\mathbb {T})$ and $\lambda \geq 1$; here, $h(\lambda ^{1/m})\to 0$ as $\lambda \to \infty$. Formulas (4.4) and (5.3) are rewritten quite similarly.

Considering remark 4.3 in the $A:=1-d^{2}/d^{2}\tau$ case, we conclude that the norm $\|f\|_{H,\alpha }$ in the space $H^{\alpha }(\mathbb {T})$ is equivalent to the norm

\[ \biggl(|c_{0}(f)|^{2}+\sum_{j=1}^{\infty} \alpha^{2}(|j|)\,\bigl(|c_{j}(f)|^{2}+|c_{{-}j}(f)|^{2}\bigr)\biggr)^{1/2}, \]

with $c_{j}(f)$ being the Fourier coefficient of $f$ with respect to the eigenfunction $e_{j}(\tau ):=(2\pi )^{-1}e^{ij\tau }$ of the differential operator ${1-d^{2}/d^{2}\tau }$.

In this case or if $A=d/d\tau$, expansion (6.1) becomes the Fourier series with respect to the complex trigonometric system on $[0,\,2\pi ]$. The above results on the mean convergence are new even for this series.

Ending this section, we remark that theorems 4.1, 4.2, 5.1 and 5.2 remain valid in the case where the coefficients of the ordinary differential operator $A$ have finite smoothness depending, of course, on the parameters involved in these theorems. We will not specify the smoothness.

7. Applications to multiple Fourier series

Let $2\leq n\in \mathbb {Z}$, and consider the $n$-dimensional torus $\mathbb {T}^{n}$; as above, $\mathbb {T}$ is a circle of the unit radius. Let $A$ denote the Laplace–Beltrami operator on $\mathbb {T}^{n}$. It belongs to $\mathrm {NE}\Psi _{\mathrm {ph}}^{2}(\Gamma )$, is a self-adjoint operator in the Hilbert space $H:=L_{2}(\mathbb {T}^{n})$ and takes the form $A=\partial ^{2}/\partial ^{2}\tau _{1}+\cdots + \partial ^{2}/\partial ^{2}\tau _{n}$, where $\tau _{k}$, with $0\leq \tau _{k}\leq 2\pi$, sets a natural parametrization of the $k$-th specimen of $\mathbb {T}$. The eigenfunctions $e_{j}(\tau ):=(2\pi )^{-n}e^{i(j_1\tau _1+\cdots +j_n\tau _n)}$ of $A$, where $\tau =(\tau _1,\,\ldots,\,\tau _n)\in [0,\,2\pi ]^{n}$ and $j=(j_1,\,\ldots,\,j_n)\in \mathbb {Z}^{n}$, form an orthonormal basis in the Hilbert space $L_{2}(\mathbb {T}^{n})$. Given a function $f\in L_{2}(\mathbb {T}^{n})$, we consider its expansion in the multiple Fourier series

(7.1)\begin{equation} f(\tau)=\sum_{j\in\mathbb{Z}^{n}}c_j(f)\,e_j(\tau), \quad\mbox{with}\quad\tau\in[0,2\pi]^{n}, \end{equation}

the expansion converging in $L_{2}(\mathbb {T}^{n})$. Here, $c_j(f)$ denotes the Fourier coefficient of $f$ with respect to $e_{j}$,

Since the termwise differentiation of series (7.1) gives the multiple Fourier series again, we restrict ourselves to the $\ell =0$ case applying results of § 4 and 5 to expansion (7.1). Let $2< p<\infty$. If $1< q\leq 2$ and $s\geq n/q-n/p$, it follows from theorem 4.1 that, whatever $f\in W^{s}_{q}(\mathbb {T}^{n})$, expansion (7.1) is unconditionally convergent in the $p$-th mean. If a function parameter $\alpha \in \mathrm {OR}$ satisfies

(7.2)\begin{equation} \int\limits_{1}^{\infty}\frac{t^{n-1}}{\alpha^{2}(t)}\,{\rm d}t<\infty, \end{equation}

it follows from theorem 5.1 that, for every $f\in H^{\alpha }(\mathbb {T}^{n})$, this expansion converges unconditionally uniformly (on $\mathbb {T}^{n}$). Considering the last result in the case of power functions $\alpha (t)\equiv t^{s}$, we conclude that expansion (7.1) converges unconditionally uniformly on each class $W^{s}_{2}(\mathbb {T}^{n})$ where $s>n/2$. If $n$ is odd, such a conclusion is substantiated in [Reference Ilin20, Theorem A.1] concerning the uniform convergence over the torus.

Assuming $\alpha$ to satisfy hypotheses of theorem 4.2 or 5.2, we rewrite estimates (4.3) and (5.2) as follows:

\[ \biggl(\int\limits_{[0,2\pi]^{n}}\, \biggl|f(\tau)-\sum_{j\in\mathbb{Z}^{n}:\|j\|\leq\lambda} c_j(f)\,e_{j}(\tau)\biggr|^{p}{\rm d}\tau\biggr)^{1/p} \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda) \]

and

\[ \sup_{\tau\in[0,2\pi]^{n}}\, \biggl|f(\tau)-\sum_{j\in\mathbb{Z}^{n}:\|j\|\leq\lambda} c_j(f)\,e_{j}(\tau)\biggr| \leq c\cdot\|f\|_{H,\alpha}\cdot h(\lambda), \]

whenever $f\in H^{\alpha }(\mathbb {T})$ and $\lambda \geq 1$. Here, $h(\lambda )\to 0$ as $\lambda \to \infty$, and we take into account that $Ae_{j}=-\|j\|^{2}e_{j}$, with $\|j\|:=(j_{1}^{2}+\cdots +j_{n}^{2})^{1/2}$. According to remark 4.3, the norm $\|f\|_{H,\alpha }$ in $H^{\alpha }(\mathbb {T}^{n})$ is equivalent to the norm

\[ \biggl(|c_{0}(f)|^{2}+\sum_{j\in\mathbb{Z}^{n},j\neq0} \alpha^{2}(\|j\|)\,|c_{j}(f)|^{2}\biggr)^{1/2}. \]

If $\alpha$ belongs to $\mathrm {OR}$ and satisfies (7.2), then

(7.3)\begin{equation} \sum_{j\in\mathbb{Z}^{n}}|c_{j}(f)|<\infty \end{equation}

for every $f\in H^{\alpha }(\mathbb {T}^{n})$, which means the absolute convergence of the Fourier series (7.1) of $f$ and follows from the unconditional uniform convergence of this series. As is known [Reference Alimov, Il'in and Nikishin2, Section 6, Subsection 3], property (7.3) is fulfilled provided that $f$ belongs to the Hölder space $\mathcal {C}^{\ell }(\mathbb {T}^{n})$ for some $\ell >n/2$ and is not fulfilled for certain functions $f\in \mathcal {C}^{n/2}(\mathbb {T}^{n})$. Our sufficient condition for (7.3) to hold true is weaker than that indicated in terms of Hölder spaces. This is demonstrated by the example when $\alpha (t)\equiv t^{n/2}\log ^{1/2+\varepsilon }(t+1)$ for some $\varepsilon >0$, which gives the broader space $H^{\alpha }(\mathbb {T}^{n})$ (satisfying (7.2)) than the union of all $\mathcal {C}^{\ell }(\mathbb {T}^{n})$ with $\ell >n/2$.

Note that of considerable interest is also the study of conditions for multiple Fourier series to converge almost everywhere. Relevant results are expounded in [Reference Alimov, Il'in and Nikishin2, Reference Alimov, Il'in and Nikishin3, Reference Ashurov6, Reference Carbery and Soria11, Reference Golubov16] and references therein.

8. Concluding remarks

Remark 8.2 According to theorem 2.3, the left-hand side of (2.4) (i.e. the norm of a remainder of series (2.1)) depends monotonically on the corresponding set of eigenvectors of $L$. Hence, the estimates in theorems 4.2 and 5.2 remain true if the written sums additionally contain some other terms of the eigenfunction expansion. Specifically, estimates (4.3) and (5.2) are valid if we replace the sum

\[ \sum_{j:|\lambda_{j}|\leq\lambda} (f,e_j)_{H}\,e_j \]

with any partial sum of the series in the arbitrarily numbered eigenfunctions $e_{j}$ provided that the later sum contains all $e_{j}$ subject to $|\lambda _{j}|\leq \lambda$.