$L^\infty$ to $L^p$ constants for Riesz projections

TL;DR

This paper investigates the norms of Riesz projections from $L^( ^n)$ to $L^p( ^n)$, revealing exact thresholds for $n=1$ and bounds for higher dimensions, with asymptotic behavior as $p$ grows large.

Contribution

It establishes the precise norm behavior for $n=1$ and bounds for the critical exponent $p_n$ in higher dimensions, advancing understanding of Riesz projection norms.

Findings

For $n=1$, the norm equals 1 if and only if $p\, extless= 4$.

As $p o \infty$, the norm behaves asymptotically as $p/(\pi e)$.

For $n>1$, the critical exponent $p_n$ satisfies $2+2/(2^n-1)\, extless p_n\, extless 4$.

Abstract

The norm of the Riesz projection from $L^\infty(\T^n)$ to $L^p(\T^n)$ is considered. It is shown that for $n=1$, the norm equals $1$ if and only if $p\le 4$ and that the norm behaves asymptotically as $p/(\pi e)$ when $p\to \infty$. The critical exponent $p_n$ is the supremum of those $p$ for which the norm equals $1$. It is proved that $2+2/(2^n-1)\le p_n <4$ for $n>1$; it is unknown whether the critical exponent for $n=\infty$ exceeds $2$.