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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.00261 |
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| _version_ | 1866910851788701696 |
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| author | Wen, Yongming Wu, Huoxiong |
| author_facet | Wen, Yongming Wu, Huoxiong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00261 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matrix weighted inequalities for fractional type integrals associated to operators with new classes of weights Wen, Yongming Wu, Huoxiong Classical Analysis and ODEs 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. |
| title | Matrix weighted inequalities for fractional type integrals associated to operators with new classes of weights |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.00261 |