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 $WH_L^p(\mathbb{ R}^n)$ associated to $L$ for $p\in (0,\,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $WH_L^p(\mathbb{ R}^n)$. 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 Schr\"odinger type operator $L_2:= (-\Delta)^k+V^k$, where $\Delta$ is the Laplacian operator and $0\le V\in L^k_{ \mathop\mathrm{ loc}}(\mathbb{ R}^n)$. 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 $WH^p_{ L_i}(\mathbb{ R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{ R}^n)$ and, for all $0<p<r\le1$ 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 $p\in(0,\,1]$, 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 $WH_L^p(\mathbb{ R}^n)$.
Categories: math.CA math.FA