Monotonic rearrangements of functions with small mean oscillation (Q2804302)

This paper deals with the monotonic rearrangement operator in some spaces of functions, specifically in the BMO space and the Muckenhoupt classes \(A_{p}\). In one variable it is known that this operator has norm equal to one. For \(f\) in one of these spaces, let \(f^{*}\) its monotonic rearrangement, that is, \(f^{*}\) is a monotone function with the same distribution function as \(f\). Then one has NEWLINE\[NEWLINE \| f^{*}\|_{\mathrm{BMO}}\leq \| f\|_{\mathrm{BMO}},\quad [f^{*}]_{A_{p}}\leq [f]_{A_{p}}. NEWLINE\]NEWLINE These equalities do not hold in higher dimensions and the problem to calculate the norm of the monotonic rearrangement seems to be a very difficult one. As a step to the solution of this problem the authors consider the action of the monotonic rearrangement operator on the dyadic BMO and Muckenhoupt classes.NEWLINENEWLINELet us recall the definition of these two dyadic classes of functions:NEWLINENEWLINELet \(n\in\mathbb{N}\) and let \(\mathcal{D}\) be the set of all dyadic subcubes of \([0,1]^{n}\). The dyadic BMO space on \([0,1]^{n}\) is defined as NEWLINE\[NEWLINE\mathrm{BMO}^{d}([0,1]^{n})=\Bigl\{\varphi\in L^{1}([0,1]^{n}): \sup_{I\in \mathcal{D}} (\langle\varphi^{2}\rangle_{I}-\langle \varphi\rangle^{2}_{I})<\infty\Bigr\}, NEWLINE\]NEWLINE where \(\langle\varphi\rangle_{I}\) is the average of \(\varphi\) over \(I\). If the supremum is taken over all intervals \(I\), one gets the ordinary BMO space.NEWLINENEWLINEFor \(\varphi \in\mathrm{BMO}^{d}([0,1]^{n})\), its monotonic rearrangement is a monotone function \(\varphi^{*}\) on \([0,1]\) with the same distribution function as \(\varphi\). The following result is proved:NEWLINENEWLINEThe monotonic rearrangement operator acts from the space \(\text{BMO}^{d}([0,1]^{n})\) to \(\mathrm{BMO}([0,1])\) with norm \((1+2^{n})/(2^{1+n/2})\).NEWLINENEWLINENext, consider the dyadic Muckenhoupt class \(A_{p}^{d}\) on \([0,1]^{n}\). For each constant \(Q\), define NEWLINE\[NEWLINE A^{d}_{2,Q}([0,1]^{n})=\Bigl\{\varphi\in L^{1}([0,1]^{n}):\sup_{I\in\mathcal{D}} (\langle \varphi\rangle_{I}\cdot \langle \varphi^{-1}\rangle_{I})\leq Q\Bigr\}. NEWLINE\]NEWLINE If one takes the supremum over all intervals \(I\) one gets the ordinary class \(A_{2,Q}([0,1]^{n})\). The following result is now proved:NEWLINENEWLINEThe monotonic rearrangement operator acts from \(A^{d}_{2,Q}([0,1]^{n})\) to\linebreak \(A_{2,Q'}([0,1])\) if and only if NEWLINE\[NEWLINE Q'\geq \frac{Q(2^{n}+1)^{2}-(2^{n}-1)^{2}}{2^{n+2}}. NEWLINE\]NEWLINENEWLINENEWLINEThese results are obtained as a consequence of a general theorem established in an abstract geometric context, involving a martingale embedding theorem.NEWLINENEWLINETo every pair of unbounded open strictly convex domains \(\Omega_{0}, \Omega_{1}\subset \mathbb{R}^{2}\) such that \(\overline{\Omega}_{1}\subset \Omega_{0}\) and any ray lying inside~\(\Omega_{0}\) can be shifted to lie inside \(\Omega_{1}\), one associates the space \(\mathbf{A}_{\Omega}\), where \(\Omega=(\overline{\Omega_{0}\backslash\Omega_{1}})\). For \(J\subset\mathbb{R}\) an interval and \(\varphi: J\to \partial \Omega_{0}\) a summable function one says that \(\varphi\in \mathbf{A}_{\Omega}\) if \(\langle \varphi\rangle_{I}\in\Omega\) for every interval \(I\subset J\).NEWLINENEWLINEIt is well known that BMO spaces and Muckenhoupt classes can be obtained as the \(\mathbf{A}_{\Omega}\) spaces for convenient pairs of domains \(\Omega_{0}\), \(\Omega_{1}\).NEWLINENEWLINEOn the other hand, let \((X,\mathfrak{U},\mu)\) be a probability space and let \(\mathcal{F}=\{\mathcal{F}_{n}\}_{n\geq 0}\) be an increasing discrete time filtration. The class~\(\mathbf{A}^{\mathcal{F}}_{\Omega}\) consists of all functions~\(\varphi: X\to \partial \Omega_{0}\) such that \(\langle \varphi\rangle_{\omega}\in \Omega\) for every atom of \(\mathcal{F}_{n}\), \(n\geq 0\).NEWLINENEWLINEA pair \(\tilde{\Omega}_{0}\), \(\tilde{\Omega}_{1}\) is called an \(\alpha\)-extension of the pair \(\Omega_{0}\), \(\Omega_{1}\) if every straight line segment contained in \(\Omega\) (i.e.\ \((\overline{\Omega_{0}\backslash \Omega_{1}})\)) is also contained in \(\tilde{\Omega}\) (i.e. \((\overline{\tilde{\Omega}_{0}\backslash \tilde{\Omega}_{1})}\)). Then the abstract general theorem to which we referred above stays that, under appropriate conditions, the pair \(\tilde{\Omega}_{0}\), \(\tilde{\Omega}_{1}\) is an \(\alpha\)-extension of the pair \(\Omega_{0}\),~\(\Omega_{1}\) if and only if the space~\(\mathbf{A}_{\tilde{\Omega}}\) contains the monotonic rearrangement~\(\varphi^{*}\) of each function \(\varphi\in \mathbf{A}^{\mathcal{F}}_{\Omega}\) such that the martingale generated by \(\mathcal{F}\) and \(\varphi\) satisfies the so-called \(\alpha\)-condition.