Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions
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| Format: | Preprint |
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2025
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| _version_ | 1866915144431304704 |
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| author | Wen, Yongming Wu, Huoxiong |
| author_facet | Wen, Yongming Wu, Huoxiong |
| contents | Let $\mathcal{L}$ be a Schrödinger operator and $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ be the variation operator of heat semigroup associated to $\mathcal{L}$ with $\varrho>2$. In this paper, we first obtain the quantitative weighted $L^p$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ 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 $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$, and the weighted mixed weak type inequality corresponding to Sawyer's conjecture for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are obtained. Furthermore, the quantitative restricted weak type $(p,p)$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are also given with a new class of weights $A_{p}^{ρ,θ,\mathcal{R}}$, which is larger than the classical $A_{p}^{\mathcal{R}}$ weights. Meanwhile, several characterizations of $A_{p,q,α}^{ρ,θ,\mathcal{R}}$ in terms of restricted weak type estimates of maximal operators are established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_05862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions Wen, Yongming Wu, Huoxiong Classical Analysis and ODEs 42B20, 42B25, 35J10 Let $\mathcal{L}$ be a Schrödinger operator and $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ be the variation operator of heat semigroup associated to $\mathcal{L}$ with $\varrho>2$. In this paper, we first obtain the quantitative weighted $L^p$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ 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 $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$, and the weighted mixed weak type inequality corresponding to Sawyer's conjecture for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are obtained. Furthermore, the quantitative restricted weak type $(p,p)$ bounds for $\mathcal{V}_\varrho(e^{-t\mathcal{L}})$ are also given with a new class of weights $A_{p}^{ρ,θ,\mathcal{R}}$, which is larger than the classical $A_{p}^{\mathcal{R}}$ weights. Meanwhile, several characterizations of $A_{p,q,α}^{ρ,θ,\mathcal{R}}$ in terms of restricted weak type estimates of maximal operators are established. |
| title | Weighted variational inequalities for heat semigroups associated with Schrödinger operators related to critical radius functions |
| topic | Classical Analysis and ODEs 42B20, 42B25, 35J10 |
| url | https://arxiv.org/abs/2502.05862 |