Abstract

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces ${\dot{B}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ and weighted Triebel–Lizorkin spaces ${\dot{F}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ associated with the operator $L$ for the full range $0<p,q\leqslant \infty$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ and $w$ being in the Muckenhoupt weight class $A_{\infty }$ . 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 Hardy-type 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.

Highlights

  • 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

  • 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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Summary

Introduction

Let X be a space of homogeneous type, with quasidistance d and μ being a nonnegative Borel measure on X , which satisfies the doubling property below. (iv) Recently, under the assumption that L is a nonnegative self-adjoint operator satisfying Gaussian upper bounds, Holder continuity and Markov semigroup properties, the frame decompositions of Besov and Triebel–Lizorkin spaces associated with L with full range of indices were studied in [26, 36, 50]. This theory has a wide range of applications from the setting of Lie groups to Riemannian manifolds. Besov and Triebel–Lizorkin spaces associated with L: properties and characterizations

Definitions of Besov and Triebel–Lizorkin spaces associated with L
Atomic decompositions
Comparison with classical Besov and Triebel–Lizorkin spaces
Applications
Fractional powers
Dispersive estimates and Strichartz estimates
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