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This paper concerns the universality of the two-layer neural network with the k-rectified linear unit activation function with k=1,2,… with a suitable norm without any restriction on the shape of the domain in the real line. This type of result is called global universality, which extends the previous result for k=1 by the present authors. This paper covers k-sigmoidal functions as an application of the fundamental result on k-rectified linear unit functions.

Abstract 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 this 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.

Abstract This work extends the theory of Rychkov, who developed the theory of A p loc $A_{p}^{\mathrm{loc}}$ weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class A p ( ⋅ ) loc $A_{p(\cdot )}^{\mathrm{loc}}$ is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

Abstract Our goal is to obtain the John–Nirenberg inequality for ball Banach function spaces X, provided that the Hardy–Littlewood maximal operator M is bounded on the associate space X′ $X'$ by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.

In the present study, we investigate a universality of neural networks, which concerns a density of the set of two-layer neural networks in 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.

The spaces of pointwise multipliers on Morrey spaces are described in terms of Morrey spaces, their preduals, and vector-valued Morrey spaces introduced by Ho. This paper covers weak Morrey spaces as well. The result in the present paper completes the characterization of the earlier works of the first author’s papers written in 1997 and 2000, as well as Lemarié-Rieusset’s 2013 paper. As a corollary, the main result in the present paper shows that different quasi-Banach lattices can create the same vector-valued Morrey spaces. The goal of the present paper is to provide a complete picture of the pointwise multiplier spaces.

In this paper we describe the function space M s,w p,q with w ∈ A loc∞ together with some related results of weighted modulation spaces.En este artículo describimos el espacio de la funciones M s,w p,q con w ∈ A loc∞ junto con algunos resultados relacionados a espacios de modulación con peso.

We introduce generalized Morrey-Campanato spaces of martingales, which generalize both martingale Lipschitz spaces introduced by Weisz (1990) and martingale Morrey-Campanato spaces introduced in 2012. We also introduce generalized Morrey-Hardy and Campanato-Hardy spaces of martingales and study Burkholder-type equivalence. We give some results on the boundedness of fractional integrals of martingales on these spaces.

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces over nondoubling measure spaces.

The boundedness of the various operators on B˙σ-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of B˙σ-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of B˙σ-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on B˙σ-Lipschitz spaces is discussed.

We show some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces.

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in n-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi [3]. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper. [ABSTRACT FROM AUTHOR]

This paper considers the atomic decomposition in martingale Hardy spaces generated by quasi-Banach function lattices. The results in this paper unify and extend the existing ones. Based on the Doob inequality, we make the paper self-contained except for weighted inequalities of martingale transforms. We obtain the vector-valued Doob maximal inequality for a large class of Banach function lattices over probability spaces as a by-product. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the maximal regularity of the heat equation in Morrey spaces based on the 2010 paper by Ogawa and Shimizu. What differs from their work is that the Fourier multipliers are used instead of interpolation. Some recent studies on the real interpolation show that Morrey spaces do not interpolate well. The estimate needed in this paper depends on the local means. The norm estimate of local means is transformed into the form which is suitable in the context of the maximal regularity. As an application, the Cauchy problems for the Keller-Segel system are studied. The function spaces used in this paper correspond to the scaling critical case for the Keller-Segel system. An observation shows why Besov-Morrey spaces are suitable for the study of maximal regularity and that some other related spaces are not. [ABSTRACT FROM AUTHOR]