Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions

Abstract

Let $\mathcal{L}$ be a Schrödinger operator and ${\mathcal{V}}_{\varrho }(e^{-t\mathcal{L}})$ be the variation operator of heat semigroup associated to $\mathcal{L}$ with $\varrho >2$. In this paper, we first obtain the quantitative weighted $L^p$ bounds for ${\mathcal{V}}_{\varrho }(e^{-t\mathcal{L}})$ with a class of weights related to critical radius functions, which contains the classical Muckenhoupt weights as a proper subset. Next, a new bump condition, which is weaker than the classical bump condition, is given for two-weight inequality of ${\mathcal{V}}_{\varrho }(e^{-t\mathcal{L}})$, and the weighted mixed weak type inequality corresponding to Sawyer's conjecture for ${\mathcal{V}}_{\varrho }(e^{-t\mathcal{L}})$ are obtained. Furthermore, the quantitative restricted weak type $(p,p)$ bounds for ${\mathcal{V}}_{\varrho }(e^{-t\mathcal{L}})$ are also given with a new class of weights $A_p^{\ddot{I},\hat{I}¸,\mathcal{R}}$, which is larger than the classical $A_p^{\mathcal{R}}$ weights. Meanwhile, several characterizations of $A_{p,q,\hat{I}\pm }^{\ddot{I},\hat{I}¸,\mathcal{R}}$ in terms of restricted weak type estimates of maximal operators are established.

35pages