Weighted Hardy Spaces Associated with Elliptic Operators. Part III: Characterisations of $$H_{L}^{p}({w})$$ 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 \(H^p_{\mathcal {T}}(w)\), 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 characterisation 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 \(H^p_{\nabla L^{-1/2},q}(w)\) and \(H^p_{\mathcal {S}_{ \mathrm {H} },q}(w)\) are isomorphic, in some range of \(p'\)s, and \(q\in \mathcal {W}_w(q_-(L),q_+(L))\).