一类振荡积分算子在Wiener 共合空间上的有界性
Boundedness Properties of Certain Oscillatory Integrals on Wiener Amalgam Space

假设 $a, b>0$ 并且 $$ K_{a,b}(x)=\dfrac{\rme^{\rmi |x|^{-b}}}{|x|^{n+a}}. $$ 定义强奇异卷积算子$T$如下: $$ Tf(x)=(K_{a,b}\ast f)(x), $$ 本文主要考虑了如上定义的算子$T$在Wiener共合空间$W(\mathcal{F}L^{p},L^{q})({\mathbb{R}}^{n})$上的有界性. 另一方面, 设$\alpha,\beta>0$ 并且 $\gamma(t)=|t|^{k}$ 或 $\gamma(t)={\rm sgn}(t)|t|^{k}$. 利用振荡积分估计, 本文还研究了算子 $$ T_{\alpha,\beta}f(x,y)=\text{p.v.}\int_{-1}^{1}f(x-t,y-\gamma(t))\frac{\rme^{-2\pi \rmi|t|^{-\beta}}}{t|t|^{\alpha}}\rmd t $$ 及其推广形式 $$ \Lambda_{\alpha,\beta}f(x,y,z)=\int_{Q^{2}}f(x-t,y-s,z-t^{k}s^{j})\rme^{-2\pi\rmi t^{-\beta_1}s^{-\beta_2}}t^{-\alpha_1-1}s^{-\alpha_2-1}\rmd t\rmd s $$ 在Wiener共合空间$W(\mathcal{F}L^{p},L^{q})$上的映射性质. 本文的结论足以表明, Wiener共合空间是Lebesgue空间的一个很好的替代.
Suppose $a, b>0$ and $$ K_{a,b}(x)=\dfrac{\rme^{\rmi |x|^{-b}}}{|x|^{n+a}}. $$ The first task in this paper is to study the boundedness properties of the strongly singular convolution operator $Tf(x)=(K_{a,b}\ast f)(x)$ on Wiener amalgam spaces $W(\mathcal{F}L^{p},L^{q})({\mathbb{R}}^{n})$. If $\alpha,\beta>0$ and $\gamma(t)=|t|^{k}$ or $\gamma(t)={\rm sgn}(t)|t|^{k}$, the second task of this paper is to investigate the mapping properties of the operator defined by $$ T_{\alpha,\beta}f(x,y)=\text{p.v.}\int_{-1}^{1}f(x-t,y-\gamma(t))\frac{\rme^{-2\pi\rmi |t|^{-\beta}}}{t|t|^{\alpha}}\rmd t $$ and its general form given by $$ \Lambda_{\alpha,\beta}f(x,y,z)=\int_{Q^{2}}f(x-t,y-s,z-t^{k}s^{j})\rme^{-2\pi\rmi t^{-\beta_1}s^{-\beta_2}}t^{-\alpha_1-1}s^{-\alpha_2-1}\rmd t\rmd s $$ on Wiener amalgam spaces $W(\mathcal{F}L^{p},L^{q})$. The essential tool of this paper is the oscillatory integral estimation. The results of this paper show that Wiener amalgam spaces are good substitutions for Lebesgue spaces.