Old and new Morrey spaces without heat kernel bounds on RD-spaces

An RD-space \(\mathcal{X}\) is a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse (volume) doubling condition. Let \(L\) be a non-negative self-adjoint operator acting on \(L^2(\mathcal{X})\). Assume that \(L\) generates an analytic semigroup \(\{\mathrm{e}^{-tL}\}_{t>0}\) whose kernels \(\{h_t(x,y)\}_{t>0}\) satisfy a generalized Gaussian heat kernel upper estimate. Roughly speaking, the heat kernel behavior is a mixture of locally Gaussian and sub-Gaussian at infinity. With the help of this Gaussian heat kernel, we first introduce a novel Morrey space and then prove that it coincides with the classical Morrey space. As applications, some new characterizations of square Morrey space are established via a Carleson measure condition.