Weak Hardy Spaces $W{H}_L^p({\mathbb{R}}^n)$ 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 weak Hardy space $W{H}_L^p({\mathbb{R}}^n)$ associated to $L$ for $p\in (0,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $W{H}_L^p({\mathbb{R}}^n)$. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1:=(-1{)}^k{\sum }_{|\alpha |=k=|\beta |}{\partial }^{\beta }(a_{\alpha ,\beta }{\partial }^{\alpha })$, where $\{a_{\alpha ,\beta }{\}}_{|\alpha |=k=|\beta |}$ are complex bounded measurable functions, and the $2k$-order Schrödinger type operator $L_2:=(-\Delta {)}^k+V^k$, where $\Delta$ is the Laplacian operator and $0\le V\in L_{\mathrm{loc}}^k({\mathbb{R}}^n)$. As applications, for $i\in \{1,2\}$ and $p\in (\frac{n}{n+k},1]$, the authors prove that the associated Riesz transform ${\nabla }^k(L_i^{-1/2})$ is bounded from $W{H}_{L_i}^p({\mathbb{R}}^n)$ to the classical weak Hardy space $W{H}^p({\mathbb{R}}^n)$ and, for all $0<p<r\le 1$ and $\alpha =n(\frac{1}{p}-\frac{1}{r})$, the fractional power $L_i^{-\frac{\alpha }{2k}}$ is bounded from $W{H}_{L_i}^p({\mathbb{R}}^n)$ to $W{H}_{L_i}^r({\mathbb{R}}^n)$. Furthermore, the authors find the dual space of $W{H}_L^p({\mathbb{R}}^n)$ for $p\in (0,1]$, 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 $W{H}_L^p({\mathbb{R}}^n)$.

46 pages; J. Nonlinear Convex Anal. (to appear)

References in corpus (1)

• Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators"