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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2502.06322 |
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| _version_ | 1866929706859757568 |
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| author | Wu, Huoxiong Wu, Lin |
| author_facet | Wu, Huoxiong Wu, Lin |
| contents | Suppose that $Ω\in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$μ_{Ω,β}f(x) = \left ( \int_{0}^{\infty } \left | \int_{\left | x-y \right |\le t }^{} \frac{Ω(x-y)}{\left | x-y \right |^{n-1-β} } f(y)dy \right | ^{2}\frac{dt}{t^3} \right )^{\frac{1}{2} },\quad 0<β<n.$$ Our main contribution is the quantitive weighted result of the classical Marcinkiewicz integral $μ_Ω$ proved by Hu and Qu [Math. Ineq. appl., 22(2019), 885-899] can be recovered from the quantitative weighted estimates of $μ_{Ω,β}$ in this paper when $β\to 0^+$. As inference, we also gives the uniform quantitive weighted bounds for the corresponding fractional commutators of $μ_{Ω,β}$ when $β\rightarrow 0^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The uniform quantitive weighted boundedness of fractional Marcinkiewicz integral and its commutator Wu, Huoxiong Wu, Lin Classical Analysis and ODEs 42B20, 42B25 Suppose that $Ω\in L^{\infty}(\mathbb{S} ^{n-1})$ is homogeneous of degree zero with mean value zero. Then we consider a fractional type Marcinkiewicz integral operator $$μ_{Ω,β}f(x) = \left ( \int_{0}^{\infty } \left | \int_{\left | x-y \right |\le t }^{} \frac{Ω(x-y)}{\left | x-y \right |^{n-1-β} } f(y)dy \right | ^{2}\frac{dt}{t^3} \right )^{\frac{1}{2} },\quad 0<β<n.$$ Our main contribution is the quantitive weighted result of the classical Marcinkiewicz integral $μ_Ω$ proved by Hu and Qu [Math. Ineq. appl., 22(2019), 885-899] can be recovered from the quantitative weighted estimates of $μ_{Ω,β}$ in this paper when $β\to 0^+$. As inference, we also gives the uniform quantitive weighted bounds for the corresponding fractional commutators of $μ_{Ω,β}$ when $β\rightarrow 0^+$. |
| title | The uniform quantitive weighted boundedness of fractional Marcinkiewicz integral and its commutator |
| topic | Classical Analysis and ODEs 42B20, 42B25 |
| url | https://arxiv.org/abs/2502.06322 |