arXiv 1809.027952018-09-08math.FA

与算子相关的加权Besov和Triebel-Lizorkin空间

Weighted Besov and Triebel--Lizorkin spaces associated to operators

Huy-Qui Bui, The Anh Bui, Xuan Thinh Duong

备注：: 57 pages

摘要：

设$X$是齐型空间，$L$是$L^2(X)$上的非负自伴算子，其热核满足高斯上界.本文对满值域$0<p,q\le \infty$，$α\in \mathbb R$和$w$在Muckenhoupt权类$A_\infty$中，发展了与算子$L$相联系的加权Besov空间$\dot{B}^{α,L}_{p,q,w}(X)$和加权Triebel-Lizorkin空间$\dot{F}^{α,L}_{p,q,w}(X)$的理论.类似于经典的情况下，在欧几里德设置，我们证明了我们的新空间满足重要的功能，如连续charaterizations平方函数，原子分解和一些著名的功能空间，如Hardy型空间和Sobolev型空间的标识。此外，通过对算子$L$的额外假设，我们证明了与$L$相关的新函数空间与经典函数空间是一致的.最后，我们应用这些结果证明了$L$的分数幂和$L$的谱乘子在我们的新函数空间中的有界性.

英文摘要：

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 $\dot{B}^{α,L}_{p,q,w}(X)$ and weighted Triebel--Lizorkin spaces $\dot{F}^{α,L}_{p,q,w}(X)$ associated to the operator $L$ for the full range $0<p,q\le \infty$, $α\in \mathbb R$ and $w$ being in the Muckenhoupt weight class $A_\infty$. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations 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. Moreover, with extra assumptions on the operator $L$, we prove that the new function spaces associated to $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ and the spectral multiplier of $L$ in our new function spaces.

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