Weighted Bourgain–Morrey‐Besov–Triebel–Lizorkin spaces associated with operators

Let X$X$ be a space of homogeneous type and L$L$ be a nonnegative self‐adjoint operator on L2(X)$L^2(X)$ satisfying a Gaussian upper bound on its heat kernel. First, we obtain the boundedness of the Hardy–Littlewood maximal function and its variant on weighted Bourgain–Morrey spaces. The Hardy‐type inequality on sequence Bourgain–Morrey spaces are also given. Then, we introduce the weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces associated with the operator L$L$. We obtain characterizations of these spaces in terms of Peetre maximal functions, noncompactly supported functional calculus, and heat kernel. Atomic decompositions and molecular decompositions of weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces are also proved. Finally, we apply our results to prove the boundedness of the fractional power of L$L$ and the spectral multiplier of L$L$ on Bourgain–Morrey Besov and Triebel–Lizorkin spaces.