Hardy Spaces $H_L^p({\mathbb{R}}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates

Abstract

Let $L$ be a one to one operator of type $\omega$ having a bounded $H_{\infty }$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in \mathbb{N}$. In this paper, the authors introduce the Hardy space $H_L^p({\mathbb{R}}^n)$ with $p\in (0,1]$ associated to $L$ in terms of square functions defined via $\{e^{-t^{2k}L}{\}}_{t>0}$ and establish their molecular and generalized square function characterizations. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1$ with complex bounded measurable coefficients and the $2k$-order Schrödinger type operator $L_2\equiv (-\Delta {)}^k+V^k$, where $\Delta$ is the Laplacian and $0\le V\in L_{\mathrm{loc}}^k({\mathbb{R}}^n)$. Moreover, as applications, for $i\in \{1,2\}$, the authors prove that the associated Riesz transform ${\nabla }^k(L_i^{-1/2})$ is bounded from $H_{L_i}^p({\mathbb{R}}^n)$ to $H^p({\mathbb{R}}^n)$ for $p\in (n/(n+k),1]$ and establish the Riesz transform characterizations of $H_{L_1}^p({\mathbb{R}}^n)$ for $p\in (rn/(n+kr),1]$ if $\{e^{-t{L}_1}{\}}_{t>0}$ satisfies the $L^r-L^2$ $k$-off-diagonal estimates with $r\in (1,2]$. These results when $k\equiv 1$ and $L\equiv L_1$ are known.

arXiv commentSci. China Math. (to appear)

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Cited by in corpus (3)

• Musielak-Orlicz-Hardy Spaces Associated with Operators Satisfying Reinforced Off-Diagonal Estimates
• Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications
• Endpoint Boundedness of Riesz Transforms on Hardy Spaces Associated with Operators