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

TL;DR

This paper proves new $L^p$-improving bounds for $k$-simplex averaging operators in lower dimensions, extending previous results and establishing bounds that include nontrivial mappings into quasi-Banach spaces.

Contribution

It introduces novel $L^p$-improving bounds for simplex averaging operators in dimensions $d \,\geq\, k$, extending prior work and including nontrivial quasi-Banach bounds.

Findings

Established new $L^p$-improving bounds for $S^k$ in dimensions $d \,\geq\, k$

Derived nontrivial bounds for $S^k$ mapping into $L^r$ with $r<1$

Improved bounds for the triangle averaging operator $S^2$ in dimensions $d \geq 2$

Abstract

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 $k = d = 2$.