$L^p$ boundness of Oscillatory singular integral with Calderón Type Commutators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918410046144512 |
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| author | Shen, Jiawei Jie, Yang |
| author_facet | Shen, Jiawei Jie, Yang |
| contents | In the paper, we study a kind of Oscillatory singular integral operator with Calderón Type Commutators $T_{P,K,A} $ defined by
\[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y) e^{i P(x-y)} d y, \]
where $P(t)$ is a real polynomial on $\mathbb{R},$ and $K$ is a function on $\mathbb{R}^{n},$ satisfies the vanishing moment and $CZ(δ)$ conditions. Under these conditions, we show that $T_{P,K,A}$ is bounded on $L^p(\mathbb{R}^{n})$ with uniform boundedness, which improve and extend the previous result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L^p$ boundness of Oscillatory singular integral with Calderón Type Commutators Shen, Jiawei Jie, Yang Classical Analysis and ODEs 42B20, 42B25 In the paper, we study a kind of Oscillatory singular integral operator with Calderón Type Commutators $T_{P,K,A} $ defined by \[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y) e^{i P(x-y)} d y, \] where $P(t)$ is a real polynomial on $\mathbb{R},$ and $K$ is a function on $\mathbb{R}^{n},$ satisfies the vanishing moment and $CZ(δ)$ conditions. Under these conditions, we show that $T_{P,K,A}$ is bounded on $L^p(\mathbb{R}^{n})$ with uniform boundedness, which improve and extend the previous result. |
| title | $L^p$ boundness of Oscillatory singular integral with Calderón Type Commutators |
| topic | Classical Analysis and ODEs 42B20, 42B25 |
| url | https://arxiv.org/abs/2506.22879 |