A class of subspaces of Morrey spaces and norm inequalities on Riesz potential operators
The Riesz potential \(I_{\gamma}\) defined by \[ I_{\gamma}f(x) = \int \limits _{\mathbb{R}^{d}} |x-y|^{d(\gamma-1)}f(y)dy, \] for \(f \in C_{c}(\mathbb{R}^{d})\), \(x \in \mathbb{R}^{d}\), and \(\gamma \in (0,1)\), is known to map \(L^{p}\) to \(L^{p^{*}}\) when \(p \in [1,\frac{1}{\gamma})\) and \(\frac{1}{p^{*}}=\frac{1}{p}-\gamma\). The paper under review deals with refinements of this result, aimed at describing the space \(B=\{f \in L^{0}(\mathbb{R}^{d}))\;;\; I_{\gamma}(|f|) \in L^{p^{*}}\}\). This space contains \(L^{p}\), but only as a proper subspace (a counterexample is constructed in Proposition 1.14). To describe \(B\), the authors introduce a space \(F\) of measurable functions \(f\) such that \[ \sum \limits _{Q \in S}(|Q|^{-\frac{1}{p'}}\|f.1_{Q}\|_{1})^{p^{*}} \] is finite for all countable partitions \(S\) of \(\mathbb{R}^{d}\) by disjoint cubes. Their main result gives a description of \(B\) that is close to optimal: \(B\) is contained in the closure of \(L^{p^{*}}\) in \(F\), and, for \(f\) in that closure, \(I_{\gamma}(f) \in L^{p^{*},\infty}\). To shed some light on the nature of \(F\), the authors also compare their space to more classical function spaces used in harmonic analysis. They show, for instance, that \(F\) is a subspace of the Herz space of measurable functions such that \[ \sum \limits _{n \in \mathbb{Z}} 2^{-\frac{dnp^{*}}{p'}}\|f.1_{C_{n}}\|_{1}^{p^{*}} \] is finite, where \(C_{n} = \{x \in \mathbb{R}^{d} \;;\; 2^{n}<|x|\leq 2^{n+1}\}\). They also compare \(F\) to certain amalgam spaces introduced by the first author in an earlier work. In the second half of the paper, they generalise their results to analogous spaces of measures \(\mu\), and to the corresponding extension of the Riesz potential defined by \( I_{\gamma}\mu(x) = \int \limits _{\mathbb{R}^{d}} |x-y|^{d(\gamma-1)}d\mu(y) \).
scientific article; zbMATH DE number 6484524
scientific article