Obstacle problems generated by the estimates of square function

TL;DR

This paper derives the explicit Bellman function for the dyadic square function's $L^p$ estimates, connecting it to obstacle problems and confirming the hypergeometric function as its core component.

Contribution

It provides the exact Bellman function for the unweighted $L^p$ estimate of the dyadic square function, filling a gap in the literature and linking it to obstacle problem techniques.

Findings

Explicit formula for the Bellman function of the dyadic square function's $L^p$ estimate.

Connection of the Bellman function to confluent hypergeometric functions.

Comparison of Bellman functions for strong $L^p$ and weak $(1,1)$ estimates.

Abstract

In this note we give the formula for the Bellman function associated with the problem considered by B. Davis in \cite{Davis} in 1976. In this article the estimates of the type $\|Sf\|_p \le C_p \|f\|_p$, $p\ge 2$, were considered for the dyadic square function operator $S$, and Davis found the sharp values of constants $C_p$. However, along with the sharp constants one can consider a more subtle characteristic of the above estimate. This quantity is called the Bellman function of the problem, and it seems to us that it was never proved that the confluent hypergeometric function from Davis' paper (second page) basically gives this Bellman function. Here we fill out this gap by finding the exact Bellman function of the unweighted $L^p$ estimate for operator $S$. We cast the proofs in the language of obstacle problems. For the sake of comparison, we also find the Bellman function of weak $(1,1)$ estimate of $S$. This formula was suggested by Bollobas \cite{Bollobas} and proved by Osekowski \cite{Os2009}, so it is not new, but we like to emphasize the common approach to those two Bellman functions descriptions.