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 $A_{p}^{ρ,θ,\mathcal{R}}$, which is larger than the classical $A_{p}^{\mathcal{R}}$ weights. Meanwhile, several characterizations of $A_{p,q,α}^{ρ,θ,\mathcal{R}}$ in terms of restricted weak type estimates of maximal operators are established.

Theorems & Definitions (56)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10