Bellman function for extremal problems in BMO (Q447876)

One method for finding inequalities for functions in \(\mathrm{BMO}\) or sharp constants for those inequalities has been to find a Bellman function for the inequality. The technique of Bellman functions rewrites an inequality as an extremal problem, whose solution is often given by the solution of a differential equation. For these situations, it was not always possible to find the extremal, but super-solutions of the equation were found which were shown to be adequate for the problem at hand. (It should be noted that Bellman functions are not the only way to prove these inequalities.) In this paper, the authors characterize the extremals and use extremals to prove several well known inequalities for \(\mathrm{BMO}\).NEWLINENEWLINEFor a function \(\phi\) defined on \(\mathbb{R}\), set \( <\phi >_J = \frac{1}{{|J| }} \int_J \phi \). The authors start with the \(L_2\) characterization of \(\mathrm{BMO}(I)\), NEWLINE\[NEWLINE \mathrm{BMO}(I) \equiv \{ \phi | \;|| \phi ||_{\mathrm{BMO}(I)}^2 = \sup_{J \subset I} < | \phi - <\phi^2>_J |^2_J < \infty. NEWLINE\]NEWLINE \(\mathrm{BMO}_{\epsilon}(I)\) denotes a ball of radius \(\epsilon\) in this space. The authors develop three well-known inequalities for \(\mathrm{BMO}\) using their technique. The first is the equivalence of any \(p\)-norm to the \(2\)-norm, NEWLINE\[NEWLINE c_p || \phi ||_{\mathrm{BMO}(I)} \leq ( \sup_{J \subset I} < | \phi - <\phi>_J |^p_J > )^{1/p} \leq C_p || \phi ||_{\mathrm{BMO}(I)} NEWLINE\]NEWLINE The second is the weak form of the John-Nirenberg inequality, which asserts that for any \(\lambda>0\), NEWLINE\[NEWLINE \frac{1}{|I|} | \{ t \in I | |\phi(t) - <\phi>_I | > \lambda \} \leq c_1 e^{-c_2 \lambda/ || \phi ||_{\mathrm{BMO}(I)}}.NEWLINE\]NEWLINE The third is the integral form of this inequality, which says that there exists an \(\epsilon_0 > 0\) such that for every \(0 < \epsilon < \epsilon_0\) and every \(\phi \in \mathrm{BMO}_{\epsilon}(I)\), NEWLINE\[NEWLINE <e^{\phi}>_I \leq C_{\epsilon} e^{<\phi>_I}. NEWLINE\]NEWLINENEWLINENEWLINE\noindent The authors introduce a Bellman function NEWLINE\[NEWLINE\mathcal{B}_{\epsilon}(x_1, x_2, f) = \sup_{ \phi \in \mathrm{BMO}_{\epsilon}(I)} \{ < f\;\circ \phi>_I | <\phi>_I = x_1, <\phi^2>_I = x_2 \} NEWLINE\]NEWLINE and point out that a proof of sharp constants \(c_p, C_p\) of the first inequality above was found by finding super-solutions for \(f(u) = |u|^p\), for the second by doing the same for \(f(u) = \chi_{(-\infty, - \lambda] \cup [\lambda, \infty) }\) and for the integral form by choosing \(f(u) = e^u\).NEWLINENEWLINEThe authors show that under mild conditions on \(f\), the extremal functions of the Bellman functional satisfy a homogeneous Monge-Ampere boundary value problem on a parabolic domain. They give a geometric algorithm for solving it. The algorithm produces the exact Bellman function, as well as optimizers for the inequalities being considered.