Embeddings of local generalized Morrey spaces between weighted Lebesgue spaces
type: Item
instance of: scholarly article
title: Embeddings of local generalized Morrey spaces between weighted Lebesgue spaces
zbMATH Open document ID: 1388.46023
author: Alexandre Almeida
author: S. G. Samko
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2017-10-27T00:00:00Z
full work available at URL: http://hdl.handle.net/10773/18638
review text: Let \(p\in (0,\infty)\). Given a weight \(w\) (i.e., a positive measurable function) on \(\mathbb R^n\), denote by \(L^p(\mathbb R^n, w)\) the weighted Lebesgue space and equip it with the (quasi)norm \[ \|f\|_{L^p(\mathbb R^n, w)}:=\Big(\int_{\mathbb R^n}|f(x)|^p w(x)\,dx\Big)^{1/p}. \] If \(\varphi\) is a positive measurable function on \((0,\infty)\), the generalized global Morrey space \(L^{p,\varphi}(\mathbb R^n)\) is the family of all functions \(f\) on \(\mathbb R^n\) which satisfy \[ \|f\|_{L^{p,\varphi}(\mathbb R^n)}:=\sup_{x\in \mathbb R^n, r>0} \Big( \frac{1}{\varphi(r)} \int_{B(x,r)}|f(y)|^p \,dy\Big)^{1/p}<\infty, \] where the symbol \( B(x,r)\) denotes the open ball in \(\mathbb R^n\) centered at \(x\in \mathbb R^n\) and with radius \(r>0.\) The localized version of this space is defined as follows: Given a~fixed point \(x_0 \in \mathbb R^n\), the local generalized Morrey space \(L^{p,\varphi}_{x_0}(\mathbb R^n)\) consists of all functions \(f\) on \(\mathbb R^n\) satisfying \[ \|f\|_{L^{p,\varphi}_{x_0}(\mathbb R^n)}:=\sup_{r>0} \Big( \frac{1}{\varphi(r)} \int_{B(x_0,r)}|f(y)|^p \,dy\Big)^{1/p}<\infty. \] First the authors prove that, if the function \(\varphi\) is increasing, the function \(\varphi(t)/t^\varepsilon\) is almost increasing for some \(\varepsilon>0\) and the function \(\varphi(t)/t^n\) is almost decreasing, then there is a function \(f \in L^{p,\varphi}(\mathbb R^n)\) which does not belong to the space \(L^p(\mathbb R^n)\). In the sequel, for a fixed \(x_0 \in \mathbb R^n\) and positive measurable functions \(w, \varphi\), we put \[ w_{x_0}(y):=w(|y-x_0|), \quad \varphi(y):=\varphi(|y-x_0|), \quad y\in \mathbb R^n. \] The main result of the paper concerns embeddings between weighted Lebesgue spaces and local generalized Morrey spaces, it reads as follows: Theorem. Let \(0<p<\infty\) and \(x_0 \in \mathbb R^n\). Assume that the function \(\varphi\) is increasing and the weight \(w\) is almost increasing on \((0,r_0)\) and almost decreasi…
reviewed by: Bohumír Opic
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 6798947
zbMATH Keywords: local Morrey space
zbMATH Keywords: global Morrey space
zbMATH Keywords: generalized Morrey space
zbMATH Keywords: weighted Lebesgue space
zbMATH Keywords: embeddings
zbMATH Keywords: Stummel space
zbMATH Keywords: Stummel classes
MaRDI profile type: Publication
OpenAlex ID: W2756473009
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DOI: 10.1016/J.NA.2017.08.006
description: scientific article; zbMATH DE number 6798947