Sharp \(L^p\) estimates on BMO (Q2841848)

For a cube \(Q\) and a locally integrable real valued function \(\varphi\), let NEWLINE\[NEWLINE \langle \varphi \rangle_{Q}=\frac{1}{|Q|}\int_{Q}\varphi. NEWLINE\]NEWLINE Let \(\text{BMO}^p(Q)\) be the space NEWLINE\[NEWLINE \text{BMO}^p(Q)=\{\;\varphi \in L^1(Q):\;\;\langle |\varphi - \langle \varphi\rangle_{J}|^p\rangle_{J}^{1/p} \leq C < \infty, \;\;\text{for all cubes}\;J \subset Q\;\} NEWLINE\]NEWLINE with the (quasi-)norm NEWLINE\[NEWLINE ||\varphi||_{\text{BMO}^p(Q)}=\sup_{\text{cube}\;J \subset Q} \langle |\varphi - \langle \varphi\rangle_{J}|^p\rangle_{J}^{1/p}. NEWLINE\]NEWLINE It is known that all \(p\)-based norms \(||\varphi||_{\text{BMO}^p(Q)}\) are equivalent for all \(p>0\). The main goal of this paper is to quantify this equivalence precisely, in dimension 1. To this end, the authors try to find the best constants \(c_{p}\) and \(C_{p}\) in the double inequalities NEWLINE\[NEWLINE c_{p}||\varphi||_{\text{BMO}(Q)} \leq ||\varphi||_{\text{BMO}^p(Q)} \leq C_{p}||\varphi||_{\text{BMO}(Q)},\;\;p>0, NEWLINE\]NEWLINE where \(\text{BMO}(Q)=\text{BMO}^2(Q)\).NEWLINENEWLINEThe knowledge of the Bellman functions leads to find the best constants \(c_{p}\) and \(C_{p}\) in dimension 1.NEWLINENEWLINEHence they investigate the upper and lower Bellman functions : for \(\;p>0\), \(\;\varepsilon >0\) and \(x \in \Omega_{\varepsilon}=\{\;x=(x_{1},\;x_{2}):\;\;x_{1}^2 \leq x_{2} \leq x_{1}^2 + \varepsilon^2\;\}\), let NEWLINE\[NEWLINE B_{\varepsilon, p}(x)=\sup_{\;\varphi \in \text{BMO}_{\varepsilon}(Q)}\{\;\langle |\varphi|^p\rangle_{Q}:\;\;\langle \varphi \rangle_{Q}=x_{1},\;\langle \varphi^2 \rangle_{Q}=x_{2}\;\} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE b_{\varepsilon, p}(x)=\inf_{\;\varphi \in \text{BMO}_{\varepsilon}(Q)}\{\;\langle |\varphi|^p\rangle_{Q}:\;\;\langle \varphi \rangle_{Q}=x_{1},\;\langle \varphi^2 \rangle_{Q}=x_{2}\;\} NEWLINE\]NEWLINE whre \(\text{BMO}_{\varepsilon}(Q)=\{\;\varphi \in \text{BMO}(Q): \;\;||\varphi||_{\text{BMO}(Q)} \leq \varepsilon\;\}\).NEWLINENEWLINEThese appear as solutions to a series of Monge-Ampère boundary value problem on a non-convex plane domain.NEWLINENEWLINEThey give the explicit expressions for the Bellman functions \(B_{\varepsilon, p}(x)\) and \(b_{\varepsilon, p}(x)\) for all \(p>0\) and obtain the sharp inequalities that are the main purpose of the paper.