Title: Matrix weighted inequalities for fractional type integrals associated to operators with new classes of weights Show Chinese title

Title: 关于与算子相关的分数阶积分的矩阵加权不等式及其新类权重

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}}{\rho(x)}+ \frac{\sqrt{t}}{\rho(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 Schr\"{o}dinger operator, Laguerre operators, etc. Show Chinese abstract

Abstract: 设 $e^{-tL}$ 是由 $-L$ 生成的解析半群，其中 $L$ 是定义在 $L^2(\mathbb{R}^d)$ 上的非负自伴算子。 假设$e^{-tL}$的核，记作$p_t(x,y)$，仅满足上界：对于所有的$N>0$，存在常数$c,C>0$使得\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}}{\rho(x)}+ \frac{\sqrt{t}}{\rho(y)}\Big)^{-N} \end{align}对所有$x,y\in\mathbb{R}^d$和$t>0$成立。 我们首先针对与$L$相关的分数阶积分建立了定量矩阵权不等式，这些结果利用了新的矩阵权类，这些新类是 Li、Rahm 和 Wick [23] 所得结果的非平凡推广。 其次，我们给出了新的双权 bump 条件，其中满足更宽松条件的 Young 函数适用于与$L$相关的分数阶积分，这涵盖了由 Cruz-Uribe、Isralowitz 和 Moen [6] 所得的结果。 我们指出，新的矩阵权类和 bump 条件比 [17] 和 [6] 中的经典权类更大且更弱。 作为应用，我们的结果可以应用于磁性薛定谔算子、Laguerre 算子等情形。