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Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L2(X) satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces Ḃα,Lp,q,w and weighted Triebel–Lizorkin spaces Ḟα,Lp,q,w (X) associated with the operator L for the full range 0 < p,q ≤ ∞, α ∈ R and w being in the Muckenhoupt weight class A∞. Under rather weak assumptions on L as stated above, we prove that our new spaces satisfy important features such as continuous characterizations in terms of square functions, atomic decompositions and the identifications with some well-known function spaces such as Hardytype spaces and Sobolev-type spaces. One of the highlights of our result is the characterization of these spaces via noncompactly supported functional calculus. An important by-product of this characterization is the characterization via the heat kernel for the full range of indices. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated with L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L, the spectral multiplier of L in our new function spaces and the dispersive estimates of wave equations.
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