Smoothness of Extremizers of a Convolution Inequality

Abstract

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 that any critical point $f\in L^{(d+1)/d}$ of the functional $\norm{Tf}_{d+1}/\norm{f}_{(d+1)/d}$ is infinitely differentiable, and that $|x{|}^{\delta }f\in L^{(d+1)/d}$ for some $\delta >0$. In particular, this holds for all extremizers of the associated inequality. This is done by exploiting a generalized Euler-Lagrange equation, and certain weighted norm inequalities for $T$.

25 pages