Weighted Hardy spaces associated with elliptic operators. Part III: Characterizations of $H_L^p(w)$ and the weighted Hardy space associated with the Riesz transform

Abstract

We consider Muckenhoupt weights $w$, and define weighted Hardy spaces $H_{\mathcal{T}}^p(w)$, where $\mathcal{T}$ denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second order divergence form elliptic operator $L$. In the range $0<p<1$, we give a molecular characterization of these spaces. Additionally, in the range $p\in {\mathcal{W}}_w(p_{-}(L),p_{+}(L))$ we see that these spaces are isomorphic to the $L^p(w)$ spaces. We also consider the Riesz transform $\nabla L^{-\frac{1}{2}}$, associated with $L$, and show that the Hardy spaces $H_{\nabla L^{-1/2},q}^p(w)$ and $H_{{\mathcal{S}}_{\mathrm{H}},q}^p(w)$ are isomorphic, in some range of $p^{\prime }$s, and $q\in {\mathcal{W}}_w(q_{-}(L),q_{+}(L))$.

References in corpus (2)

• Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions
• Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H_L^1(w)$

Cited by in corpus (4)

• Weighted Hardy spaces associated with elliptic operators. Part I: Weighted norm inequalities for conical square functions
• Weighted Hardy spaces associated with elliptic operators. Part II: Characterizations of $H_L^1(w)$
• The regularity problem for uniformly elliptic operators in weighted spaces
• Vertical square functions and other operators associated with an elliptic operator