Syarat Cukup Ketaksamaan Hӧlder dan Ketaksamaan Minkowski di Perumuman Ruang Morrey

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.

In this paper, we prove that the inclusions between Morrey spaces, between weak Morrey spaces, and between a Morrey space and a weak Morrey space are all proper. The proper inclusion between a Morrey space and a weak Morrey space is established via the unboundedness of the Hardy-Littlewood maximal operator on Morrey spaces of exponent 1. In addition, we also give a necessary condition for each inclusion. Our results refine previous inclusion properties studied in [Gunawan et al, \emph{Math. Nachr.} {\bf 290} (2017), 332--340].

This paper aims to explore the embedding relation between Bourgain-Morrey spaces and weak Bourgain-Morrey spaces. Bourgain-Morrey Spaces are important function spaces in harmonic analysis. In this article, we introduce weak Bourgain-Morrey spaces as certain generalization of weak Morrey spaces. We investigate the necessary and sufficient conditions for functions in weak Bourgain-Morrey spaces to also be elements in Bourgain-Morrey spaces, and vice versa. Our result are inclusion between weak Bourgain-Morrey spaces and inclusion of weak Bourgain- Morrey spaces into a certain Bourgain-Morrey spaces. These result generelize the inclusion result in Morrey spaces, weak Morrey spaces, and Bourgain- Morrey spaces.

Orlicz–Morrey spaces are generalizations of Orlicz spaces and Morrey spaces which were first introduced by Nakai. There are three versions of Orlicz–Morrey spaces. In this article, we discussed the third version of weak Orlicz–Morrey space, which is an enlargement of third version of (strong) Orlicz– Morrey space. Similar to its first version and second version, the third version of weak Orlicz-Morrey space is considered as a generalization of weak Orlicz spaces, weak Morrey spaces, and generalized weak Morrey spaces. This study investigated some properties of the third version of weak Orlicz–Morrey spaces, especially the sufficient and necessary conditions for inclusion relations between two these spaces. One of the keys to get our result is to estimate the quasi- norm of characteristics function of open balls in ℝ.

On inclusion relation between weak Morrey spaces and Morrey spaces

In this paper, we shall establish a theory of interpolation of generalized Morrey spaces. We use the complex interpolation methods. Our results extend the interpolation results for Morrey spaces which are discussed by Lu et al. (Can Math Bull 57:598–608, 2014), and also Lemarie-Rieusset (2014). We establish the interpolation of generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also consider the closure of the functions which are essentially bounded and have compact support. The second interpolation of such spaces will yield a class of closed spaces; we describe the second interpolation of the closure of the functions which are essentially bounded and have compact support. This result will carry over to generalized Morrey spaces, generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also give several examples that explain the subtlety of proving the interpolation of Morrey spaces.

This paper discusses the structure of Morrey spaces, weak Morrey spaces, generalized Morrey spaces, and generalized weak Morrey spaces. Some necessary and sufficient conditions for the inclusion property of these spaces are obtained through a norm estimate for the characteristic functions of balls.

11 - ALGEBRAIC INEQUALITIES

Morrey spaces are the powerful tool for the study of partial differential equations. Recently, weak Morrey spaces and weak Herz spaces are known to be useful for the study of Navier–Stokes equations. In this paper we introduce weak central Herz–Morrey spaces W\mathcal H^{p(\cdot),q,\omega}(\mathbf B) and establish the weak estimate for the maximal and Riesz potential operators in the central Herz–Morrey space \mathcal H^{p(\cdot),q,\omega}(\mathbf B) . We also treat generalized Riesz potential operators. Further we obtain the strong estimate for Sobolev functions.

Compact embeddings of radial and subradial subspaces of some Besov-type spaces related to Morrey spaces

We study embeddings of weighted local and consequently global generalized Morrey spaces defined on a quasi-metric measure set (X, d, μ) of general nature which may be unbounded, into Lebesgue spaces Ls (X), 1 ≤ s ≤ p < ∞. The main motivation for obtaining such an embedding is to have an embedding of non-separable Morrey space into a separable space. In the general setting of quasi-metric measure spaces and arbitrary weights we give a sufficient condition for such an embedding. In the case of radial weights related to the center of local Morrey space, we obtain an effective sufficient condition in terms of (fractional in general) upper Ahlfors dimensions of the set X. In the case of radial weights we also obtain necessary conditions for such embeddings of local and global Morrey spaces, with the use of (fractional in general) lower and upper Ahlfors dimensions. In the case of power-logarithmic-type weights we obtain a criterion for such embeddings when these dimensions coincide.

We present some characterizations for the boundedness of the generalized fractional integral operators on Morrey spaces. The characterizations follow from two key estimates, one for the norm of some functions in Morrey spaces, and another for the values of the corresponding fractional integrals. We prove three theorems about necessary and sufficient conditions. We show that these theorems are independent by giving some examples. We also obtain counterparts for the weak generalized Morrey spaces.

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.

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝn)) on Morrey spaces.

We study the pointwise multipliers from one Morrey space to another Morrey space. We give a necessary and sufficient condition to grant that the space of those multipliers is a Morrey space as well.