Smoothness of Extremizers of a Convolution Inequality

Let $d\ge 2$ and $T$ be the convolution operator $Tf(x)=\int_{\reals^{d-1}} f(x'-t,x_d-|t|^2)\,dt$, which is is bounded from $L^{(d+1)/d}(\reals^d)$ to $L^{d+1}(\reals^d)$. 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 $T$.