Old and New Morry Spaces via Heat Kernel Bounds

Abstract

Given $p\in [1,\infty )$ and $\lambda \in (0,n)$, we study Morrey space ${\mathrm{L}}^{p,\lambda }({\mathbb{R}}^n)$ of all locally integrable complex-valued functions $f$ on ${\mathbb{R}}^n$ such that for every open Euclidean ball $B\subset {\mathbb{R}}^n$ with radius $r_B$ there are numbers $C=C(f)$ (depending on $f$) and $c=c(f,B)$ (relying upon $f$ and $B$) satisfying $$r_B^{-\lambda }{\int }_B|f(x)-c{|}^{p}dx\le C$$ and derive old and new, two essentially different cases arising from either choosing $c=f_B=|B{|}^{-1}{\int }_{B}f(y)dy$ or replacing $c$ by $P_{t_B}(x)={\int }_{t_B}{p}_{t_B}(x,y)f(y)dy$ -- where $t_B$ is scaled to $r_B$ and $p_t(\cdot ,\cdot )$ is the kernel of the infinitesimal generator $L$ of an analytic semigroup $\{e^{-tL}{\}}_{t\ge 0}$ on ${\mathrm{L}}^2({\mathbb{R}}^n)$. Consequently, we are led to simultaneously characterize the old and new Morrey spaces, but also to show that for a suitable operator $L$, the new Morrey space is equivalent to the old one.