Embeddings of local generalized Morrey spaces between weighted Lebesgue spaces
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 decreasing on \((r_0,\infty)\), and such that \(\int^\infty_0 w(t)/t \;dt <\infty\). Then \[ L^p\Big(\mathbb R^n, \frac{1}{\varphi_{x_0}}\Big)\hookrightarrow L^{p,\varphi}_{x_0}(\mathbb R^n) \hookrightarrow L^p\Big(\mathbb R^n, \frac{w_{x_0}}{\varphi_{x_0}}\Big). \] Moreover, the norms of these embeddings are independent of \(x_0\) provided that the function \(w/\varphi\) is decreasing and the function \(\varphi\) satisfies the doubling condition. If \(\varphi\) is a positive measurable function on the interval \((0,\infty)\), the authors also introduce the generalized Stummel space \(\mathfrak S^{p,\varphi}(\mathbb R^n)\) defined as the collection of those functions \(f\) on \(\mathbb R^n\) that satisfy \[ \|f\|_{\mathfrak S^{p,\varphi}(\mathbb R^n)}:=\sup_{x\in \mathbb R^n} \Big( \int_{\mathbb R^n} \frac{|f(y)|^p}{\varphi(|x-y|)} \,dy\Big)^{1/p}<\infty. \] The vanishing generalized Stummel space \(V_0\mathfrak S^{p,\varphi}(\mathbb R^n)\) is given as the subset of all \(f \in \mathfrak S^{p,\varphi}(\mathbb R^n)\) such that \[ \lim_{r\rightarrow \,0}\,\sup_{x\in \mathbb R^n}\int_{B(x,r)} \frac{|f(y)|^p}{\varphi(|x-y|)} \,dy=0, \] and the vanishing Stummel class \(V_\infty\mathfrak S^{p,\varphi}(\mathbb R^n)\) consists of all \(f\in \mathfrak S^{p,\varphi}(\mathbb R^n)\) for which \[ \lim_{r\rightarrow \infty}\,\sup_{x\in \mathbb R^n}\int_{\mathbb R^n\setminus B(x,r)} \frac{|f(y)|^p}{\varphi(|x-y|)} \,dy=0. \] Further embeddings investigated in the paper are embeddings between generalized Morrey spaces and Stummel spaces, as well as embeddings among generalized Morrey spaces, vanishing Stummel spaces and Stummel classes.
scientific article; zbMATH DE number 6798947
scientific article