Volume 32, Issue 4
Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution

Anal. Theory Appl., 32 (2016), pp. 396-404.

Published online: 2016-10

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• Abstract

Here we consider the following strongly singular integral $$T_{\Omega,\gamma,\alpha,\beta}f(x,t)=\int_{R^n} e^{i|y|^{-\beta}}\frac {\Omega(\frac{y}{|y|})}{|y|^{n+\alpha}}f(x-y,t-\gamma(|y|))dy,$$ where $\Omega\in L^p(S^{n-1}),$ $p>1,$ $n>1,$ $\alpha>0$ and $\gamma$ is convex on $(0,\infty)$.
We prove that there exists $A(p,n)>0$ such that if $\beta>A(p,n)(1+\alpha)$, then $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself and the constant is independent of $\gamma$. Furthermore, when $\Omega\in C^\infty(S^{n-1})$, we will show that $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself only if $\beta>2\alpha$ and the constant is independent of $\gamma$.

• Keywords

Oscillatory strongly rough singular integral, rough kernel, surfaces of revolution.

42B20, 42B35

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@Article{ATA-32-396, author = {}, title = {Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution}, journal = {Analysis in Theory and Applications}, year = {2016}, volume = {32}, number = {4}, pages = {396--404}, abstract = {

Here we consider the following strongly singular integral $$T_{\Omega,\gamma,\alpha,\beta}f(x,t)=\int_{R^n} e^{i|y|^{-\beta}}\frac {\Omega(\frac{y}{|y|})}{|y|^{n+\alpha}}f(x-y,t-\gamma(|y|))dy,$$ where $\Omega\in L^p(S^{n-1}),$ $p>1,$ $n>1,$ $\alpha>0$ and $\gamma$ is convex on $(0,\infty)$.
We prove that there exists $A(p,n)>0$ such that if $\beta>A(p,n)(1+\alpha)$, then $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself and the constant is independent of $\gamma$. Furthermore, when $\Omega\in C^\infty(S^{n-1})$, we will show that $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself only if $\beta>2\alpha$ and the constant is independent of $\gamma$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2016.v32.n4.7}, url = {http://global-sci.org/intro/article_detail/ata/4679.html} }
TY - JOUR T1 - Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution JO - Analysis in Theory and Applications VL - 4 SP - 396 EP - 404 PY - 2016 DA - 2016/10 SN - 32 DO - http://doi.org/10.4208/ata.2016.v32.n4.7 UR - https://global-sci.org/intro/article_detail/ata/4679.html KW - Oscillatory strongly rough singular integral, rough kernel, surfaces of revolution. AB -

Here we consider the following strongly singular integral $$T_{\Omega,\gamma,\alpha,\beta}f(x,t)=\int_{R^n} e^{i|y|^{-\beta}}\frac {\Omega(\frac{y}{|y|})}{|y|^{n+\alpha}}f(x-y,t-\gamma(|y|))dy,$$ where $\Omega\in L^p(S^{n-1}),$ $p>1,$ $n>1,$ $\alpha>0$ and $\gamma$ is convex on $(0,\infty)$.
We prove that there exists $A(p,n)>0$ such that if $\beta>A(p,n)(1+\alpha)$, then $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself and the constant is independent of $\gamma$. Furthermore, when $\Omega\in C^\infty(S^{n-1})$, we will show that $T_{\Omega,\gamma,\alpha,\beta}$ is bounded from $L^2(R^{n+1})$ to itself only if $\beta>2\alpha$ and the constant is independent of $\gamma$.

J. C. Chen, S. Y. He & X. R. Zhu. (1970). Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution. Analysis in Theory and Applications. 32 (4). 396-404. doi:10.4208/ata.2016.v32.n4.7
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