Old and New Morrey Spaces with Heat Kernel Bounds

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space of all locally integrable complex-valued functions f on such that for every open Euclidean ball B ⊂ with radius r_B there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying and derive old and new, two essentially different cases arising from either choosing or replacing c by —where t_B is scaled to r_B and p_t(·, ·) is the kernel of the infinitesimal generator L of an analytic semigroup on 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.