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Naoya Hatano

Publications and source records attributed to Naoya Hatano.

17 recordsLinked to original sources
arXiv2026

Generalization for Poincaré--Sobolev inequalities with local weights

It is well known the local integral inequality which is called Poincaré--Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.

math.CA↗
arXiv2026

Characterization for Campanato norm via quasi-Banach function spaces not assuming the Fatou property

It is well known that the BMO and Campanato norms can be characterized using the $L^p$-average. These characterizations were later generalized to averages taken over various types of function spaces. In particular, generalizations using Banach function spaces were provided by Ho, Izuki, Noi, and Sawano. In this paper, as a further generalization, we provide similar characterizations using quasi-Banach function spaces that do not assume the Fatou property. Note that the duality argument is not available in this setting.

math.FA↗
arXiv2026

The Kerman-Sawyer trace theorem for product Morrey spaces

By using parallel corona decomposition, the Kerman-Sawyer trace theorem is extended from Lebesgue spaces to \textit{product Morrey spaces}. By discretizing the multilinear fractional integral operator based on dyadic analysis, the framework of \textit{product Morrey spaces} naturally arises in the course of estimating the operator. Within this natural setting, by establishing Sawyer-type testing estimates (to the setting of measures), we obtain an extension of the Kerman-Sawyer trace theorem. The classical approach to the Kerman-Sawyer trace theorem typically relies on a reduction to Carleson's embedding theorem. In contrast, in this paper we employ a parallel corona decomposition, which allows us to overcome the difficulties inherent in the multilinear setting and to provide a transparent and streamlined proof. By incorporating recent developments in the theory of weights, this work clarifies the relationship between trace inequalities and Morrey spaces and contributes to a deeper understanding of these topics.

math.FA↗
arXiv2026

Unconditional uniqueness of Hardy--Hénon parabolic equations on Herz spaces

In this paper, we introduce the unconditional uniqueness of solutions in Herz spaces for the Hardy--Hénon parabolic equation, which is a semilinear heat equation with a power-type weight in the nonlinear term $|x|^γ|u|^{α-1}u$. It is expected that the power-type weight in the nonlinear term can be effectively handled within Herz spaces. In fact, our result in Herz spaces $\dot{K}^s_{q,r}({\mathbb R}^n)$ relaxes the endpoint case $q=α$ and the large interpolation exponent case $r\ge q$ compared to previous results.

math.AP↗
arXiv2026

Multilinear embedding theorem for fractional sparse operators

We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient conditions for which the $L^p \to L^q$, $1<p,q<\infty$, infinitesimal relative bounds hold for Schrödinger operators of the form $(-Δ)^{α/2}+v$.

math.FA↗
arXiv2024

Boundedness of composition operators from Lorentz spaces to Orlicz spaces

The boundedness (continuity) of composition operators from some function space to another one is significant, though there are few results about this problem. Thus, in this study, we provide necessary and sufficient conditions on the boundedness of composition operators from Lorentz spaces to Orlicz spaces. We also give a counter example of a mapping which implies unboundedness of the composition operators from a Lebesgue space $L^p$ to another Lebesgue space $L^q$ with $p>q$. We emphasize that the measure spaces associated with the Lorentz space may be different from those associated with the Orlicz spaces. We give more examples and counterexamples of the composed mappings in the conditions satisfying our main results.

math.FA↗
arXiv2024

Endpoint estimates for commutators with respect to the fractional integral operators on Orlicz-Morrey spaces

It is known that the necessary and sufficient conditions of the boundedness of commutators on Morrey spaces are given by Di Fazio, Ragusa and Shirai. Moreover, according to the result of Cruz-Uribe and Fiorenza in 2003, it is given that the weak-type boundedness of the commutators of the fractional integral operators on the Orlicz spaces as the endpoint estimates. In this paper, we gave the extention to the weak-type boundedness on the Orlicz-Morrey spaces.

math.FA↗
arXiv2023

Choquet integrals, Hausdorff content and sparse operators

Let $H^d$, $0 0$. In this paper we verify that the sparse operator ${\mathcal A}_{\mathcal S}$ maps ${\mathcal L}^p(H^d)$, $1\le p<\infty$, into an associate space of Orlicz-Morrey space ${{\mathcal M}^{p'}_{Φ_0}(H^d)}'$, $Φ_0(t)=t\log(e+t)$. We also give another characterizations of those associate spaces using the tiling ${\mathcal T}$ of ${\mathbb R}^n$.

math.FA↗
arXiv2023

Choquet integrals, Hausdorff content and fractional operators

It is shown that the fractional integral operator $I_α$, $0<α<n$, and the fractional maximal operator $M_α$, $0\leα<n$, are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet-Morrey spaces. These results are extensions of the previous works due to Adams, Orobitg and Verdera, and Tang. The results for the fractional integral operator $I_α$ are essentially new.

math.FA↗
arXiv2021

Variants of $q$-hypergeometric equation

We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell equation of two variables.

math.CA↗
arXiv2021

Predual of weak Orlicz spaces

In this paper, we consider the predual spaces of weak Orlicz spaces. As an application, we provide the Fefferman-Stein vector-valued maximal inequality for the weak Orlicz spaces. In order to prove this statement, we introduced the Orlicz-Lorentz spaces, and showed the boundedness of the Hardy-Littlewood maximal operator on these spaces.

math.FA↗
arXiv2020

A global universality of two-layer neural networks with ReLU activations

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.

cs.LG↗
arXiv2020

Boundedness of composition operators on Morrey spaces and weak Morrey spaces

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.

math.FA↗
arXiv2019

Bilinear estimates on Morrey spaces by using average

This paper is a follow up of [6]. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3]. When the local integrability index $s$ falls 1 with weights and $t$ exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7]. Later in the paper [6], we considered the case $t=1$. This paper handles the remaining case $0<t<1$.

math.FA↗
arXiv2019

A note on the bilinear fractional integral operator acting on Morrey spaces

The boundedness of the bilinear fractional integral operator is investigated. This bilinear fractional integral operator goes back to Kenig and Stein. This paper is oriented to the boundedness of this operator on products of Morrey spaces. Compared to the earlier work by He and Yan, the local integrability condition of the domain is expanded. The local integrability condition can be relaxed with the help of the averaging technique.

math.FA↗