Weighted BMO spaces associated to operators

Details Weighted BMO spaces associated to operators Weighted BMO spaces associated to operators

2012-01-27

arxiv.org/abs/1201.5828v1

Mathematics

Functional Analysis

29 pages

Scientific paper

Let $(X, d, \mu)$ be a space of homogeneous type and $\{{\cal A}_t\}_{t>0}$, be a generalized approximations to the identity, for example $\{{\cal A}_t\}$ is a holomorphic semigroup $e^{-tL}$ with Gaussian upper bounds generated by an operators $L$ on $L^2(X)$. In this paper, we introduce and study the weighted BMO space BMO$_{\cal A}(X,w)$ associated to the the family $\{{\cal A}_t\}$. We show that for these spaces, the weighted John-Nirenberg inequality holds and we establish an interpolation theorem in scale of weighted $L^p$ spaces. Then, we show that the dual space of the weighted Hardy space $H_L(X,w)$ associated to $L$ as in \cite{SY} is certain weighted BMO space BMO$_{L^*}(X,w)$ associated to the adjoint operator $L^*$. As applications, we prove the boundedness of two singular integrals with non-smooth kernels on the weighted BMO space BMO$_L(X,w)$.

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