Matrix weighted inequalities for fractional type integrals associated to operators with new classes of weights

TL;DR

The paper develops a quantitative framework for matrix weighted inequalities of fractional-type integrals tied to a general operator $L$ whose heat kernel satisfies a precise upper bound. It introduces and analyzes new matrix weight classes adapted to a critical radius function, and proves sharp-type bounds for $L^{-{rac{eta}{2}}}$ and related fractional maximal operators in these weighted settings. It also establishes two-weight bump conditions with Young functions for matrix weights, providing sufficient and (in many cases) necessary criteria for boundedness, along with a characterization of these weights via an averaging operator. The results extend and unify prior scalar and matrix-weight theories, with applications to operators such as magnetic Schrödinger and Laguerre operators, and offer a pathway toward resolving conjectures about fractional integrals in this broader operator framework.

Abstract

Let $e^{-tL}$ be a analytic semigroup generated by $-L$, where $L$ is a non-negative self-adjoint operator on $L^2(\mathbb{R}^d)$. Assume that the kernels of $e^{-tL}$, denoted by $p_t(x,y)$, only satisfy the upper bound: for all $N>0$, there are constants $c,C>0$ such that \begin{align}\label{upper bound} |p_t(x,y)|\leq\frac{C}{t^{d/2}}e^{-\frac{|x-y|^2}{ct}}\Big(1+\frac{\sqrt{t}}{ρ(x)}+ \frac{\sqrt{t}}{ρ(y)}\Big)^{-N} \end{align} holds for all $x,y\in\mathbb{R}^d$ and $t>0$. We first establish the quantitative matrix weighted inequalities for fractional type integrals associated to $L$ with new classes of matrix weights, which are nontrivial extension of the results established by Li, Rahm and Wick [23]. Next, we give new two-weight bump conditions with Young functions satisfying wider conditions for fractional type integrals associated to $L$, which cover the result obtained by Cruz-Uribe, Isralowitz and Moen [6]. We point out that the new classes of matrix weights and bump conditions are larger and weaker than the classical ones given in [17] and [6], respectively. As applications, our results can be applied to settings of magnetic Schrödinger operator, Laguerre operators, etc.

Theorems & Definitions (45)

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