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.

中文翻译：

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与运算符和应用程序相关的加权 Besov 和 TRIEBEL-LIZORKIN 空间

让 $X$ 是一个齐次类型的空间和 $L$ 是一个非负自伴算子 $L^{2}(X)$ 满足其热核的高斯上界。在本文中，我们发展了加权 Besov 空间理论 ${\dot{B}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ 和加权 Triebel-Lizorkin 空间 ${\dot{F}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ 与运营商相关联 $L$ 全系列 $0<p,q\leqslant \infty$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ 和 $w$ 在 Muckenhoupt 重量级 $A_{\infty }$ . 在相当弱的假设下 $L$ 如上所述，我们证明了我们的新空间满足重要特征，例如平方函数、原子分解的连续表征以及与一些著名的函数空间（如 Hardy 型空间和 Sobolev 型空间）的识别。我们的结果的亮点之一是通过非紧凑支持的函数演算来表征这些空间。这种表征的一个重要副产品是通过热核表征所有指数。此外，对操作员有额外的假设 $L$ ，我们证明了与 $L$ 与经典函数空间相吻合。最后，我们应用我们的结果来证明分数幂的有界性 $L$ , 的频谱乘数 $L$ 在我们的新函数空间和波动方程的色散估计中。