Old and new Morrey spaees with heat kernel bounds

Given p is an element of [1, infinity) and lambda is an element of (0, n), we study Morrey space L-p,L-lambda(R-n) of all locally integrable complex-valued functions f on R-n such that for every open Euclidean ball B subset of 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) integral(B)\f(x) - c\(p) dx <= C and derive old and new, two essentially different cases arising from either choosing c = f(B) = \B\(-1) integral(B) f(y) dy or replacing c by Pt-B (x) = integral(tB) pt(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 >= 0) on L-2 (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.