Some interpolation inequalities in Lorentz, Morrey and BMO spaces
Hua Wang
TL;DR
The paper addresses interpolation inequalities between Lorentz, Morrey, and BMO spaces by introducing Lorentz–Morrey spaces $LM^{p,r;\kappa}$ and proving sharp embeddings into $L^q$ and $M^{q;\kappa}$ for $p<q<\infty$, including asymptotically optimal growth rates as $q\to\infty$. The main approach leverages rearrangement techniques, distribution function estimates, and threshold-based decompositions to derive explicit constants, with extensions to the Morrey setting and a parallel John–Nirenberg-type framework in these spaces. Key contributions include precise Lorentz and Lorentz–Morrey interpolation inequalities, bilinear estimates in Lorentz/Lorentz–Morrey spaces, and new Morrey-space John–Nirenberg-type inequalities, all with PDE-analytic implications for elliptic and parabolic problems. The results provide tools for establishing global existence and regularity results for weak solutions, by connecting Sobolev-type embeddings with the nuanced scale of Lorentz and Morrey spaces, and they flesh out asymptotic sharpness and embedding constants that are relevant for applications.
Abstract
In this paper, the author establishes some interpolation results between Lorentz, Morrey and BMO spaces. Let $1<p<\infty$ and $p\leq r\leq\infty$. It is proved that the space $L^{p,r}(\mathbb R^n)\cap\mathrm{BMO}(\mathbb R^n)$ is continuously embedded into $L^q(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $L^{p,r}(\mathbb R^n)$ denotes the classical Lorentz space with indices $p$ and $r$. Moreover, the author establishes the optimal growth rate of this embedding constant as $q\to\infty$. Based on Morrey spaces, the author introduces a new family of function spaces called Lorentz--Morrey spaces $LM^{p,r;κ}(\mathbb R^n)$ with indices $p$, $r$ and $κ$, and then shows that the space $LM^{p,r;κ}(\mathbb R^n)\cap \mathrm{BMO}(\mathbb R^n)$ is continuously embedded into $L^{q;κ}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1<p<\infty$, $p\leq r\leq\infty$ and $0<κ<1$. Furthermore, the asymptotically optimal growth order of this embedding constant is also established. As an application of the above interpolation results, some new bilinear estimates in the setting of Lorentz and Lorentz--Morrey spaces are also obtained, which can be used in the study of the global existence and regularity of weak solutions to elliptic and parabolic partial differential equations of the second order.