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

2025/02/09 by Yongming Wen, Wen, Yongming, Huoxiong Wu +1 · 2 citations
Mathematics · #35J10 #42B20 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2502.05862

openalex publication_date 2025/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let L be a Schrödinger operator and V_\varrho(e^-tL) be the variation operator of heat semigroup associated to L with \varrho>2. In this paper, we first obtain the quantitative weighted L^p bounds for V_\varrho(e^-tL) 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 V_\varrho(e^-tL), and the weighted mixed weak type inequality corresponding to Sawyer's conjecture for V_\varrho(e^-tL) are obtained. Furthermore, the quantitative restricted weak type (p,p) bounds for V_\varrho(e^-tL) are also given with a new class of weights A_p^ρ,θ,R, which is larger than the classical A_p^R weights. Meanwhile, several characterizations of A_p,q,α^ρ,θ,R in terms of restricted weak type estimates of maximal operators are established.

Cited by

• Weighted inequalities for variation operators associated with fractional Schrödinger semigroups
• Perturbed Dyadic Cubes and Quantitative Estimates for Schrödinger Operators with Potentials in RH^n/2

Related

• Weighted inequalities for variation operators associated with fractional Schrödinger semigroups
• Variation operators associated with the semigroups generated by Schrödinger operators with inverse square potentials
• A_p weights and Quantitative Estimates in the Schrödinger Setting
• Boundedness of variation operators associated with the heat semigroup generated by high order Schrödinger type operators
• Mixed inequalities for operators associated to critical radius functions with applications to Schrödinger type operators