arXiv subjects
Publications and source records attributed to Yoshihiro Sawano.
The goal of this paper is to characterize the boundedness property of the uncentered Hardy--Littlewood maximal operator on ball Banach function spaces and their Köthe associate spaces in terms of the vector-valued maximal inequality. The main result of this paper supplements the recent results by Nieraeth.
We prove a Kolmogorov--Riesz compactness theorem for Banach lattices over spaces of homogeneous type. Especially, we consider the one for $L^1$. Our characterization is formulated in terms of boundedness, tightness, and approximation by averaging operators, which replace translations in the absence of a group structure. The result requires neither the Vitali covering property nor continuity assumptions on the measure of balls, lower volume bounds, or finiteness of the underlying measure space, thereby extending several recent compactness criteria in the literature. Our approach begins with a new proof of the Kolmogorov--Riesz theorem in $L^1$, which avoids reflexivity and yields a unified treatment of $L^p$-spaces for $1\le p<\infty$. We then establish an approximation theorem based on Christ's dyadic cubes and use it to characterize the closure of compactly supported continuous functions in general ball Banach function spaces. Applications include compactness criteria for rearrangement-invariant Banach function spaces and Morrey spaces. In the Euclidean setting, we also identify the resulting approximation space with the classical heat-semigroup (equivalently, mollifier) closure.
In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.
In this article, we study the weighted homogeneous Sobolev--Morrey spaces on domains in $\mathbb{R}^n$ with higher co-dimensional boundaries. Precisely, we systematically establish a real-variable theory of these spaces, including completeness, embedding theorems, Riesz potential characterizations, continuity, trace and extension theorems, and complex interpolation. Applying the boundedness of the trace and the extension operators, we obtain sharp weighted a priori estimates for solutions to the Dirichlet problem of divergence-form degenerate second-order elliptic equations on such domains in weighted Lebesgue spaces. The absence of a boundary manifold structure of these domains poses some essential difficulties, which are overcome by using some tools, such as the intrinsic properties of distance weights and the geometric structure of domains, different from those available in Lipschitz domains.
We establish a reproducing formula for the ridgelet transform on $\mathbb{R}^n$ in the framework of Banach lattices introduced in a recent paper by Nieraeth. Our approach is based on the $k$-plane Radon transform and a wavelet-type reconstruction operator acting on functions defined on the Grassmannian of $k$-dimensional affine planes. Under mild structural assumptions on the underlying Banach lattice, we prove that the ridgelet reconstruction converges both in the lattice norm and almost everywhere. The admissibility conditions on the wavelet function are formulated in terms of the Riemann--Liouville fractional integral. As a consequence, we obtain explicit inversion formulas for functions in a Banach lattice $X$ which is contained in $L^1({\mathbb R}^n)+L^p(\mathbb{R}^n)$ with some constant $1 \le p < \frac{n}{k}$, together with precise expressions for the reconstruction constant. These results provide a unified framework for ridgelet-type reproducing formulas in a broad class of function spaces beyond the classical $L^p$ setting.
We study smooth function spaces of Gelfand-Shilov type, with global behavior governed through a translation-invariant Banach function space and localized via a weight function system. We clarify the roles of the translation-invariant Banach function space, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces-as well as the corresponding convolutor and multiplier spaces-through mollification. For this purpose, we introduce the moment-wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.
In this article, the authors provide some new characterizations of several vanishing Campanato spaces using a type of oscillation defined within the general framework of ball Banach function spaces. This approach yields fresh insights even in the special case of the vanishing BMO space. The characterization reveals a self-improvement phenomenon inherent in vanishing Campanato spaces. A key innovation of this approach lies in using higher-order differences to dominate oscillations. Instead of directly estimating these differences, the authors achieve the domination by smoothing the function via convolution. As additional outcomes, the authors also obtain new characterizations of vanishing Campanato spaces in terms of higher-order differences. Finally, the authors present several examples to show that these vanishing Campanato spaces naturally arise in the study on the compactness of fractional integral commutators in Morrey spaces.
The aim of this paper is to apply an extrapolation result without relying on convexification. We characterize ball Banach function spaces in terms of wavelets, formulated in a way that takes into account the smoothness properties of the spaces under consideration. The same technique can also be applied to prove vector-valued inequalities, for example. Furthermore, the result presented here refines a recent extension operator result by Zhu, Yang, and Yuan.
In this paper, we extend several approximation theorems, originally formulated in the context of the standard $L^p$ norm, to the more general framework of variable exponent spaces. Our study is motivated by applications in neural networks, where function approximation plays a crucial role. In addition to these generalizations, we provide alternative proofs for certain well-known results concerning the universal approximation property. In particular, we highlight spaces with variable exponents as illustrative examples, demonstrating the broader applicability of our approach.
In this note, notwithstanding the generalization, we simplify and shorten the proofs of the main results of the third author's paper \cite{SXY} significantly. In particular, the new proof for \cite[Theorem 1.1]{SXY} is quite short and, unlike the original proof, does not rely on the properties of the "Marcinkiewicz function". This allows us to get a precise linear dependence on Dini constants with a subsequent application to Littlewood--Paley operators by well-known techniques. In other words, we relax the log-Dini condition in the pointwise bound to the classical Dini condition. %$\int\limits_{0}^{1} \frac{φ(t)}{t}dt<\infty$. This solves an open problem (see e.g. \cite[pp. 37--38]{CY}). Our method can be applied to the multilinear case.
This paper defines local weighted Hardy spaces with variable exponent. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As an application, we obtain the boundedness of singular integral operators, the Littlewood--Paley characterization and wavelet decomposition.
In this paper, we show that under the condition $ 1<p_-, q_-, p_+, q_+<\infty$, the space $\ell^{q(\cdot)} (L^{p(\cdot)})$ is reflexive. In this way we give an answer to open problem posed by Hästö in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$. What is important here is that the dual space of $\ell^{q(\cdot)} (L^{p(\cdot)})$ is specified. As its direct corollary, we show that the corresponding Besov space $B^{s(\cdot)}_{p(\cdot)q(\cdot)}$ is reflexive.
In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on $\mathbf{R}^d$. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the Paley-Wiener space on the real line, studied in previous works, but also much more general RKHSs corresponding to analytic positive definite functions where existing methods do not work. Our method only relies on an intrinsic properties of the RKHSs, and we establish a connection between the behavior of composition operators and the asymptotic properties of the greatest zeros of orthogonal polynomials on a weighted $L^2$-spaces on the real line. We also investigate the compactness of the composition operators and show that any bounded composition operators cannot be compact in our situation.
In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in a function spaces. There are many works that handle the convergence over compact sets. In the present paper, we consider a global convergence by introducing a norm suitably, so that our results will be uniform over any compact set.
In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As the specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.
We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.
Our aim in this paper is to characterize local Muckenhoupt weighted Lebesgue spaces with variable exponent by compactly supported smooth wavelets. We also investigate necessary and sufficient conditions for the corresponding modular inequalities to hold. One big achievement is that the weights with exponetial growth can be handled in the framework of variable exponents.
We define $A_{p(\cdot)}^{\rm loc}$ and show that the weighted inequality for local Hardy--Littlewood maximal operator on the Lebesgue spaces with variable exponent. This work will extend the theory of Rychkov, who developed the theory of $A_p^{\rm loc}$ weights. It will also extend the work by Cruz-Uribe. SFO, Fiorenza and Neugebaucer, who considered the Muckenhoupt class for Lebesgue spaces with variable exponents. Due to the setting of variable exponents, a new method of extension of weights will be needed; the extension method is different from the one by Rychkov. A passage to the vector-valued inequality is also done by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. We develop the theory of extrapolation adapted to our class of weights.