Smoothness of Extremizers of a Convolution Inequality

Michael Christ, Qingying Xue

Let $d\ge 2$ and $T$ be the convolution operator $Tf(x)={\int }_{{\mathbb{R}}^{d-1}}f(x^{\prime }-t,x_d-|t{|}^2)dt$, which is is bounded from $L^{(d+1)/d}({\mathbb{R}}^d)$ to $L^{d+1}({\mathbb{R}}^d)$. We show…