A note on fractional type integrals associated to operators

期刊论文
发表日期：

2024年11月12日

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Yongming Wen,
Xianming Hou,
Jing Zhang

摘要

Let $\mathcal {L}$ $L$ be a non-negative self-adjoint operator on $L^2(\mathbb {R}^d)$ $L^{2} (R^{d})$ and let $e^{-t\mathcal {L}}$ $e^{- t L}$ be a semigroup generated by $-\mathcal {L}$ $- L$ . Assume that the kernels of $e^{-t\mathcal {L}}$ $e^{- t L}$ satisfy the upper bound related to a critical radius function but do not possess any regularity conditions on spacial variables. We consider the class of $A_{p,q}$ $A_{p, q}$ weights associated to critical radius function, denoted by $A_{p,q}^{\rho }(\mathbb {R}^d)$ $A_{p, q}^{ρ} (R^{d})$ , which include the classical Muckenhoupt $A_{p,q}(\mathbb {R}^d)$ $A_{p, q} (R^{d})$ weights. We obtain the quantitative $A_{p,q}^{\rho }(\mathbb {R}^d)$ $A_{p, q}^{ρ} (R^{d})$ estimates for fractional integrals associated to $\mathcal {L}$ $L$ . Particularly, the quantitative weighted endpoint bound for fractional integrals associated to $\mathcal {L}$ $L$ is first established, which was missing in the literature of Li, Rahm and Wick (Math Z 293(1-2): 259-283, 2019) . Moreover, we generalize weighted endpoint inequalities to weighted mixed weak type inequalities for fractional type integrals associated to $\mathcal {L}$ $L$ . As applications, our results can be applied to settings of magnetic Schrödinger operator, Laguerre operators, etc.