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 $k = d = 2$.
Categories: math.CA