Old and new Morrey spaces with heat Kernel bounds

Given p ∈ [1,∞) and λ (0, n), we study Morrey space L^p,λ(ℝⁿ) of all locally integrable complex-valued functions f on ℝⁿ such that for every open Euclidean ball B ⊂ ℝⁿ 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^-λ_B ∫_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 ∫_B f(y)dy or replacing c by P_tB (x) = ∫_tB p_tB (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 L²(ℝⁿ). 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.