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

Let $(X,\mu)$ be a space of homogeneous type satisfying $\mu(X) =\infty$, 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, and heat kernel. Atomic and molecular decompositions of these spaces are also given. As an application, we obtain the boundedness of the fractional power of $L$, the spectral multiplier of $L$ on these spaces.