On inequalities between norms of partial derivatives on convex domains

TL;DR

This paper investigates bounds on the ratios of $L_p$-norms of partial derivatives for concave functions on convex domains, revealing boundedness for $p=1$ and domain-dependent bounds for $p>1$.

Contribution

It provides sharp estimates for the ratio of norms when $p=1$ and characterizes the dependence on domain geometry for $p>1$.

Findings

The ratio of $L_1$-norms is always bounded with sharp estimates.

For $p>1$, the ratio bounds depend on the convex domain's geometry.

The results clarify inequalities between derivatives' norms on convex domains.

Abstract

We consider inequalities between $L_p$-norms of partial derivatives, $p\in [1,+\infty]$, for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? If not, what is the upper bound? We show that for $p=1$, the ratio is always bounded and find sharp estimates, while for $p>1$, the answer depends on the geometry of the domain.