Simplex Averaging Operators: Quasi-Banach and $L^p$-Improving Bounds in Lower Dimensions

Abstract

We establish some new $L^p$-improving bounds for the $k$-simplex averaging operators $S^k$ that hold in dimensions $d\ge k$. As a consequence of these $L^p$-improving bounds we obtain nontrivial bounds $S^k:L^{p_1}\times \cdots \times L^{p_k}\to L^r$ with $r<1$. In particular we show that the triangle averaging operator $S^2$ maps $L^{\frac{d+1}{d}}\times L^{\frac{d+1}{d}}\to L^{\frac{d+1}{2d}}$ in dimensions $d\ge 2$. This improves quasi-Banach bounds obtained by Palsson and Sovine and extends bounds obtained by Greenleaf, Iosevich, Krauss, and Liu for the case of $k=d=2$.

11 pages