A T(1) theorem for entangled multilinear dyadic Calderón-Zygmund operators
The paper under review deals with multilinear forms defined by NEWLINE\[NEWLINE \Lambda_{E}(F_{1},\dots,F_{|E|}) = \int \limits _{\mathbb{R}^{N}}K(x_{1},\dots,x_{N})\prod \limits _{(i,j) \in E} F_{(i,j)}(x_{i},x_{j})dx_{1}\dots dx_{N}, NEWLINE\]NEWLINE where \((\{1,\dots,N\},E)\) is a graph, and \(K\) is a kernel satisfying appropriate Calderón-Zygmund conditions. The case where \(N=2n\) and \(E=\{(i,n+i) \;;\; i=1,\dots,n\}\) corresponds to the influential multilinear theory developed by \textit{L. Grafakos} and \textit{R. H. Torres} [Adv. Math. 165, No. 1, 124--164 (2002; Zbl 1032.42020)]. Generalising this theory to more complicated graphs is important to study operators that are invariant under certain modulations. For instance, operators such as the two dimensional bilinear Hilbert transforms from [\textit{C. Demeter} and \textit{C. Thiele}, Am. J. Math. 132, No. 1, 201--256 (2010; Zbl 1206.42010)], correspond to the case where \(N=4\), \(E=\{(1,3), (1,4), (2,3)\}\) and \(K\) is of the form \((x_{1},...,x_{4}) \mapsto k(x_{1}-x_{2},x_{3}-x_{4})\). Obtaining \(L^p\) bounds for such operators is a fundamental question in harmonic analysis with important applications to ergodic theory. A key open question in this direction is whether or not the trilinear form associated with the triangular Hilbert transform NEWLINE\[NEWLINE \Lambda(F_{1},F_{2},F_{3}) = \int \limits _{\mathbb{R}^{3}}F_{1}(x,y)F_{2}(y,z)F_{3}(z,x)\frac{1}{x+y+z}dxdydz, NEWLINE\]NEWLINE is bounded on \(L^3\times L^3 \times L^3\). This corresponds to the case \(N=3\), \(E = \{(1,2),(2,3),(3,1)\}\).NEWLINENEWLINEThe paper under review is part of the development of a theory that aims to solve such open questions. At this stage, the triangular Hilbert transform remains out of reach, but simpler forms that can be thought of as building blocks for \(\Lambda\) can be treated. These forms arise from graphs \((\{1,\dots,N\},E)\) that have a bipartite structure: \(N=n+m\) and \(E \subset \{1,\dots,m\}\times\{1,\dots,n\}\), and kernels that are perfect dyadic Calderón-Zygmund kernels (i.e. constant on \(m+n\) dimensional dyadic cubes that do not intersect the diagonal \(D=\{(x_{1},\dots,x_{m},y_{1},\dots,y_{n})\in \mathbb{R}^{m+n} \;;\; x_{1}=\dots=x_{m} \;\text{and} \; y_{1}=\dots=y_{n}\}\)). The authors establish a \(T(1)\) theorem for such forms, proving that natural testing conditions imply boundedness in \(\prod \limits _{(i,j) \in E} L^{p_{(i,j)}}\) when \(\sum \limits _{(i,j)\in E} \frac{1}{p_{(i,j)}}=1\) and, for all \((i,j) \in E\), \(p_{(i,j)}>d_{(i,j)}\) for some finite sequence of natural numbers \(d_{(i,j)}\) satisfying \(\sum \limits _{(i,j)\in E} \frac{1}{d_{(i,j)}}>1\).NEWLINENEWLINEThe proof very much builds on the ideas, techniques, and results of \textit{V. Kovač} [Indiana Univ. Math. J. 60, No. 3, 813--846 (2011; Zbl 1256.42021)] where an \(L^p\) boundedness result for special kernels and special bipartite graphs (corresponding to certain cancellative paraproducts) is proved. The paper under review uses a clever series of decompositions based on graph theoretic ideas to essentially reduce the problem to these special graphs.
scientific article; zbMATH DE number 6369190