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 - <p style="text-align: justify;">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&gt;1,$ $n&gt;1,$ $\alpha&gt;0$ and $\gamma$ is convex on $(0,\infty)$. <br/>We prove that there exists $A(p,n)&gt;0$ such that if $\beta&gt;A(p,n)(1+\alpha)$, 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 $\beta&gt;2\alpha$ and the constant is independent of $\gamma$.</p>