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

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