Bellman function technique for multilinear estimates and an application to generalized paraproducts (Q2909231)

Let \(I\) be a dyadic interval, let \(\varphi^D_I:=|I|^{1/2}\chi_I\) be the Haar scaling function, and let \(\psi^D_I:=|I|^{1/2}(\chi_{I_{\mathrm{left}}}-\chi_{I_{\mathrm{right}}})\) be the Haar wavelet, where \(\chi_I\) denotes the characteristic function of \(I\) and \(I_{\mathrm{left}}\), \(I_{\mathrm{right}}\) respectively denote the left and right halves of \(I\). Let \(\mathcal{C}\) denote the collection of all dyadic squares in \(\mathbb{R}^2\), NEWLINE\[NEWLINE\mathcal{C}:=\{[2^k\ell_1,2^k(\ell_1+1))\times[2^k\ell_2,2^k(\ell_2+1)):\;k,\ell_1, \ell_2\in \mathbb{Z}\}.NEWLINE\]NEWLINE Let \(m,n\) be positive integers and choose \(E\subseteq\{1,\dots,m\}\times\{1,\dots,n\}\), \(S\subseteq\{1,\dots,m\}\) and \(T\subseteq\{1,\dots,n\}\). The author represents \(E\) as the set of edges of a simple bipartite undirected graph with vertices \(\{x_1,\dots,x_m\}\) and \(\{y_1,\ldots,y_n\}\), where \(x_j\) and \(y_j\) are connected by an edge if and only if \((i,j)\in E\). The author introduces a two-dimensional multilineal form \(\Lambda\) by: NEWLINE\[NEWLINE\begin{multlined} \Lambda((F_{i,j})_{(i,j)\in E}):=\sum_{I\times J\in\mathcal{C}}|I|^{2-(m+n)/2} \int_{\mathbb{R}^{m+n}}\left[\prod_{(i,j)\in E}F_{i,j}(x_i,y_j)\right]\left[\prod_{i\in S}\psi^D_I(x_i)\right]\\ \times\left[\prod_{i\in S^\complement}\varphi^D_I(x_i)\right] \left[\prod_{i\in T}\psi^D_I(y_i)\right]\left[\prod_{i\in T^\complement}\varphi^D_I(y_i)\right]\,dx_1\cdots dx_mdy_1\cdots dy_n,\end{multlined}NEWLINE\]NEWLINE where \(F_{i,j}\) is a measurable, bounded and compactly supported function, \(S^\complement:=\{1,\dots,m\}\setminus S\) and \(T^\complement:=\{1,\dots,n\}\setminus T\). The bipartite graph determined by \(E\) splits into connected components. Let \(d_{i,j}\) denote the larger size of the two bipartition classes of the connected component containing an edge \((x_i,y_j)\). Then the author proves that the series \(\sum_{I\times J\in\mathcal{C}}\) defining \(\Lambda\) converges absolutely, and the form \(\Lambda\) satisfies the estimate NEWLINE\[NEWLINE|\Lambda((F_{i,j})_{(i,j)\in E})|\lesssim_{m,n,(p_{i,j})} \prod_{(i,j)\in E}\|F_{i,j}\|_{\mathrm{L}^{p_{i,j}}(\mathbb{R}^2)},NEWLINE\]NEWLINE whenever the exponents \((p_{i,j})_{(i,j)\in E}\) are such that \(\sum_{(i,j)\in E}1/p_{i,j}=1\) and \(d_{i,j}<p_{i,j}<\infty\) for each \((i,j)\in E\). To aid understanding the author also explains the key steps on a concrete example.