Quasi-extremals for convolution with surface measure on the sphere (Q2267708)

Let \(T\) be the operator given by the convolution with surface measure on the sphere, that is for a continuous function \(f\) on \(\mathbb{R}^d\), \(Tf\) is defined by \[ Tf(x)=\int_{S^{d-1}}f(x-\omega)d\sigma(\omega). \] If \(E\) and \(F\) are Borel sets having positive Lebesgue measure, then \((E,F)\) is an \(\varepsilon\)-quasi-extremal or a quasi-extremal pair if \[ |\langle T\chi_E, \chi_F \rangle|\geq\varepsilon ||\chi_E||_{L^{(d+1)/d}}||\chi_F||_{L^{(d+1)/d}}, \] where \(\langle \cdot, \cdot\rangle\) is the \(L^2\) inner product. In the article, the authors explicitly define a family of quasi-extremal pairs of sets for \(T\). This extends work carried out by \textit{M. Christ} [Lebesgue space bounds for one-dimensional generalized Radon transforms, preprint; Quasi-extremals for a Radon-like transform, preprint] for convolution with surface measure on the paraboloid.