Old and New Morrey Spaces with Heat Kernel Bounds

Abstract

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space $L^{p,\lambda}({\Bbb R}^n)$ of all locally integrable complex-valued functions f on ${\Bbb R}^n$ such that for every open Euclidean ball B ⊂ ${\Bbb R}^n$ with radius r_B there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying

$r^{-\lambda}_B\sum_B \vert f(x) -c\vert^p dx\leq C$

and derive old and new, two essentially different cases arising from either choosing $c = f_B = \vert B\vert^{−1} \sum_B f (y)dy$ or replacing c by $P_{t_B} (x) = \sum_{t_B} p_{t_B} (x, y)f (y) dy$—where t_B is scaled to r_B and p_t(·, ·) is the kernel of the infinitesimal generator L of an analytic semigroup $\{e^{−tL}\}_{t\geq 0}$ on $L^2({\Bbb R}^n).$ 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.