Weighted Bourgain-Morrey-Besov type and Triebel-Lizorkin type spaces associated with operators

Let $(X,μ)$ be a space of homogeneous type satisfying $μ(X) =\infty$, the doubling property and the reverse doubling condition. Let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel enjoys a Gaussian upper bound. We introduce the weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces associated with the operator $L$. We obtain their continuous characterizations in terms of Peetre maximal functions, noncompactly supported functional calculus, heat kernel. Atomic and molecular decompositions of weighted homogeneous Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on Bourgain-Morrey-Besov type spaces and Triebel-Lizorkin type spaces.