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• W2143451625 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 $kin{mathbb N}$. In this paper, the authors introduce the weak Hardy space $WH_L^p(mathbb{R}^n)$ associated to $L$ for $pin (0,,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $WH_L^p(mathbb{R}^n)$. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1:=(-1)^ksum_{|alpha|=k=|beta|}partial^beta(a_{alpha,beta}partial^alpha)$, where ${a_{alpha,beta}}_{|alpha|=k=|beta|}$ are complex bounded measurable functions, and the $2k$-order Schrodinger type operator $L_2:= (-Delta)^k+V^k$, where $Delta$ is the Laplacian operator and $0le Vin L^k_{mathopmathrm{loc}}(mathbb{R}^n)$. As applications, for $iin{1,,2}$ and $pin(frac{n}{n+k},,1]$, the authors prove that the associated Riesz transform $nabla^k (L_i^{-1/2})$ is bounded from $WH^p_{L_i}(mathbb{R}^n)$ to the classical weak Hardy space $WH^p(mathbb{R}^n)$ and, for all $0<p<rle1$ and $alpha=n(frac{1}{p}-frac{1}{r})$, the fractional power $L_i^{-frac{alpha}{2k}}$ is bounded from $WH_{L_i}^p(mathbb{R}^n)$ to $WH_{L_i}^r(mathbb{R}^n)$. Furthermore, the authors find the dual space of $WH_L^p(mathbb{R}^n)$ for $pin(0,,1]$, which can be defined via mean oscillations based on some subtle coverings of bounded open sets and, even when $L:=-Delta$, are also previously unknown. In particular, if $L$ is a nonnegative self-adjoint operator in $L^2({mathbb R}^n)$ satisfying the Davies-Gaffney estimates, the authors further establish the weak atomic characterization of $WH_L^p(mathbb{R}^n)$." @default.
• W2143451625 created "2016-06-24" @default.
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• W2143451625 date "2013-08-25" @default.
• W2143451625 modified "2023-10-17" @default.
• W2143451625 title "Weak Hardy Spaces $WH_L^p({mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates" @default.
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• W2143451625 doi "https://doi.org/10.48550/arxiv.1308.5385" @default.
• W2143451625 hasPublicationYear "2013" @default.
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