Let (X,μ) be a space of homogeneous type satisfying μ(X)=∞, the doubling property and the reverse doubling condition. Let L be a nonnegative self-adjoint operator on L2(X)
Nearby in the stack
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.