Some interpolation inequalities in Lorentz, Morrey and BMO spaces

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 $LM^{p,r;\kappa}(\mathbb R^n)$ with indices $p$, $r$ and $\kappa$, and then shows that the space $LM^{p,r;\kappa}(\mathbb R^n)\cap \mathrm{BMO}(\mathbb R^n)$ is continuously embedded into $L^{q;\kappa}(\mathbb R^n)$ for all $q$ with $p<q<\infty$, where $1<p<\infty$, $p\leq r\leq\infty$ and $0<\kappa<1$. 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.