An $A_2$ Theorem for One-Sided Calderón-Zygmund Operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908680173125632 |
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| author | Green, A. Walton Petrov, Ljupcho Wick, Brett D. |
| author_facet | Green, A. Walton Petrov, Ljupcho Wick, Brett D. |
| contents | We present a proof of the one-sided $A_2$ theorem in dimension one, with a logarithmic loss. This theorem concerns one-sided Calderón-Zygmund operators (CZOs) whose kernels $K(x,y)$ vanish whenever $x < y$. These operators are bounded on $L^2(w)$ provided that the weight $w$ belongs to the one-sided class $A_2^{\uparrow}$. The argument reduces the norm estimate to testing on indicator functions via a two-weight testing theorem. By combining this with the weak-type $(1,1)$ estimate in the one-sided setting and an extrapolation theorem, we obtain the one-sided $A_2^{\uparrow}$ theorem with a logarithmic loss. We develop a localized theory on fixed intervals by introducing adapted weight classes and showing that the same quantitative bound holds locally for one-sided operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_23313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An $A_2$ Theorem for One-Sided Calderón-Zygmund Operators Green, A. Walton Petrov, Ljupcho Wick, Brett D. Classical Analysis and ODEs We present a proof of the one-sided $A_2$ theorem in dimension one, with a logarithmic loss. This theorem concerns one-sided Calderón-Zygmund operators (CZOs) whose kernels $K(x,y)$ vanish whenever $x < y$. These operators are bounded on $L^2(w)$ provided that the weight $w$ belongs to the one-sided class $A_2^{\uparrow}$. The argument reduces the norm estimate to testing on indicator functions via a two-weight testing theorem. By combining this with the weak-type $(1,1)$ estimate in the one-sided setting and an extrapolation theorem, we obtain the one-sided $A_2^{\uparrow}$ theorem with a logarithmic loss. We develop a localized theory on fixed intervals by introducing adapted weight classes and showing that the same quantitative bound holds locally for one-sided operators. |
| title | An $A_2$ Theorem for One-Sided Calderón-Zygmund Operators |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.23313 |