𝐻𝑝 spaces for generalized Schrödinger operators and applications

Abstract

AbstractIn this paper, we study the Hardy type space {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup {e^{-t\mathcal{L}}} generated by {-\mathcal{L}}, where {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in {{\mathbb{R}^{n}}} ({n\geq 3}) and {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for {\frac{n}{n+\delta^{\prime}}<p\leq 1\leq q\leq\infty} ({p\neq q}) with some {\delta^{\prime}>0}. As applications, we prove that Calderón–Zygmund operators related to the auxiliary function {m(x,\mu)} are bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {L^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {\gamma_{1}>0}. In particular, we show that the Riesz transform {\nabla(-\Delta+\mu)^{-\frac{1}{2}}}, which is a special example of the above Calderón–Zygmund operators, is bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {{H}^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {0<\gamma_{1}<1}.