Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space E. The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space E is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

We consider the nonlocal differential equation

\begin{equation*}-A\left(\int_0^1b(1-s)\big(u(s)\big)^{p(s)}\ ds\right)u''(t)=\lambda f\big(t,u(t)\big)\text{, }t\in(0,1),\end{equation*}

which is a one-dimensional Kirchhoff-like equation with a nonlocal convolution coefficient. The novelty of our work involves allowing a variable growth term in the nonlocal coefficient. By relating the variable growth problem to a constant growth problem, we are able to deduce the existence of at least one positive solution to the differential equation when equipped with boundary data. Our methodology relies on topological fixed point theory. Because our results treat both the convex and concave regimes, together with both the variable growth and constant growth regimes, our results provide a unified framework for one-dimensional Kirchhoff-type problems.

Motivated by the study of multiplicative linear functionals in reproducing kernel Hilbert space (RKHS) with normalized complete Pick kernel, we define and study the multiplicative linear map between two RKHS. We identify the conditions under which such maps are continuous. Additionally, we prove that any unital cyclicity-preserving linear map is multiplicative. Conversely, we also characterize when a multiplicative linear map is unital cyclicity preserving. These results serve as a generalization of the Gleason–Kahane–Żelazko theorem to the setting of multiplicative maps between two RKHS. We present the composition operator as a natural class of examples of multiplicative linear maps on an RKHS. We also prove that every continuous multiplicative linear operator can be realized as a composition operator on various analytic Hilbert spaces over the unit disc $\mathbb {D}.$

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\mathrm{Gr}({\kern2pt}f))$ of the graph of a real-valued Hölder or Sobolev function f defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-Hölder continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

In this article, the authors establish an extension-restriction theorem between homogeneous weighted Besov spaces and weighted mixed-Riesz potential spaces. This general frame covers both the classical Besov spaces and their logarithmic analogs.

In this article, we introduce a new logarithmic Q-type space $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to study the well-posedness of the classical/fractional Naiver–Stokes equations. We show that $\nabla \cdot (Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n}))^{n}$ covers the well-known critical spaces $BMO^{-1}(\mathbb R^{n}), Q_{\alpha }^{-1}(\mathbb R^{n})$ and $\mathcal {Q}_{0}^{-1}(\mathbb R^{n})$ for the classical Naiver–Stokes equations. Moreover, it covers the fractional counterparts $BMO^{-(2\beta -1)}(\mathbb R^{n}), Q_{\alpha }^{\beta ,-1}(\mathbb R^{n})$ and even the largest critical space $\dot {B}^{-(2\beta -1)}_{\infty ,\infty }(\mathbb R^{n}).$ In doing so, we first establish some basic properties of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb {R}^{n}).$ Then, via the fractional heat semigroups, we characterize the extension of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to $\mathscr H_{K_{\ln }^{(l,k)}}^{p,\lambda }(\mathbb R_+^{n+1})$ which is a function space related to the weight function $K_{\ln }^{(l,k)}(\cdot )$. This extension provides a semigroup characterization of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$. With this in hand, we establish the well-posedness of mild solutions to fractional Naiver–Stokes equations and fractional magneto-hydrodynamic equations, respectively, with small data in $\nabla \cdot \left (Q_{\ln ,\frac {4(1-\beta )}{n}}^{2,k,l+2(1-\beta )}(\mathbb {R}^{n})\right )^{n}$ for $k\in \mathbb {N}$ and $l>n+2\beta -4.$

We define the Schur–Agler class in infinite variables to consist of functions whose restrictions to finite-dimensional polydisks belong to the Schur–Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell ^\infty $. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur–Agler class in infinite variables via the Bohr correspondence.

We study p-Wasserstein spaces over the branching spaces $\mathbb {R}^2$ and $[-1,1]^2$ equipped with the maximum norm metric. We show that these spaces are isometrically rigid for all $p\geq 1,$ meaning that all isometries of these spaces are induced by isometries of the underlying space via the push-forward operation. This is in contrast to the case of the Euclidean metric since with that distance the $2$-Wasserstein space over $\mathbb {R}^2$ is not rigid. Also, we highlight that the $1$-Wasserstein space is not rigid over the closed interval $[-1,1]$, while according to our result, its two-dimensional analog, the closed unit ball $[-1,1]^2$ with the more complicated geodesic structure is rigid.

In this paper, we prove a new uncertainty principle for functions with radial symmetry by differentiating a radial version of the Stein–Weiss inequality. The difficulty is to prove the differentiability in the limit of the best constant that unlike the general case it is not known. We provide also an integral alternative formula for the logarithmic weight $(\log|\xi|)$ in Fourier domain.

We establish Trudinger-type inequalities for variable Riesz potentials $J_{\alpha (\cdot ), \tau }f$ of functions in Musielak–Orlicz spaces $L^{\Phi }(X)$ over bounded metric measure spaces X equipped with lower Ahlfors $Q(x)$-regular measures under conditions on $\Phi $ which are weaker than conditions in the previous paper (Houston J. Math. 48 (2022), no. 3, 479–497). We also deal with the case $\Phi $ is the double phase functional with variable exponents. As an application, Trudinger-type inequalities are discussed for Sobolev functions.

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

Let $2\leq p<\infty $ and X be a complex infinite-dimensional Banach space. It is proved that if X is p-uniformly PL-convex, then there is no nontrivial bounded Volterra operator from the weak Hardy space $\mathscr {H}^{\text {weak}}_p(X)$ to the Hardy space $\mathscr {H}^+_p(X)$ of vector-valued Dirichlet series. To obtain this, a Littlewood–Paley inequality for Dirichlet series is established.

We characterize the functions with ‘small’ bounded mean oscillation (BMO) norm by establishing the precise connection between the space BMO and class $A_\infty$ of Muckenhoupt weights. We prove that there exists a universal constant $c^*_2$ such that $\Vert f \Vert_{BMO} \lt c^*_2$ if and only if $\exp f \in A_2$, where $c^*_2$ is the sharp constant in the John and Nirenberg inequality. Similarly, in dimension one, we prove that $\Vert f \Vert_{BLO} \lt 1$ if and only if $\exp f \in A_1$. As application we introduce a structure of metric space in $A_\infty$ and prove that the closed unit ball of $A_\infty$ is a Banach space.

McCullough and Trent generalize Beurling–Lax–Halmos invariant subspace theorem for the shift on Hardy space of the unit disk to the multi-shift on Drury–Arveson space of the unit ball by representing an invariant subspace of the multi-shift as the range of a multiplication operator that is a partial isometry. By using their method, we obtain similar representations for a class of invariant subspaces of the multi-shifts on Hardy and Bergman spaces of the unit ball or polydisk. Our results are surprisingly general and include several important classes of invariant subspaces on the unit ball or polydisk.

We introduce a relaxed version of the metric definition of quasiconformality that is natural also for mappings of low regularity, including $W_{\mathrm{loc}}^{1,1}({\mathbb R}^n;{\mathbb R}^n)$-mappings. Then we show on the plane that this relaxed definition can be used to prove Sobolev regularity, and that these ‘finely quasiconformal’ mappings are in fact quasiconformal.

In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I, we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$, and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$. For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$, we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$, where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$. Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$. We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$, where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$, and the coefficients γm and αj can be solved by a system of linear equations.

We use a special tiling for the hyperbolic d-space $\mathbb {H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(P)\oplus \mathcal {F}(\mathcal {N})$, where P is a polytope in $\mathbb {R}^d$ and $\mathcal {N}$ a net in $\mathbb {H}^d$ coming from the tiling. This implies that the spaces $\mathcal {F}(\mathbb {H}^d)$ and $\mathcal {F}(\mathbb {R}^d)\oplus \mathcal {F}(\mathcal {M})$ are isomorphic for every net $\mathcal {M}$ in $\mathbb {H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal {F}(\mathbb {H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm {Lip}(\mathbb {H}^d)$ and $\mathrm {Lip}(\mathbb {R}^d)$.

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited—namely the absolute continuity of the target quasinorm.

We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis–Wainger type are so-called maximally noncompact, i.e. their ball measure of noncompactness is the worst possible.

We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.