一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~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.

Boundedness Properties of Certain Oscillatory Integrals on Wiener Amalgam Space

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.