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 rB 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 \(c = f_B = \vert B\vert^{−1} \sum_B f (y)dy\) or replacing c by \(P_{t_B} (x) = \sum_{t_B} p_{t_B} (x, y)f (y) dy\)—where tB is scaled to rB and pt(·, ·) 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Duong, X., Xiao, J. & Yan, L. Old and New Morrey Spaces with Heat Kernel Bounds. J Fourier Anal Appl 13, 87–111 (2007). https://doi.org/10.1007/s00041-006-6057-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-006-6057-2