Discrete decompositions for bilinear operators and almost diagonal conditions (Q2759059)

The authors establish a general sufficient condition for boundedness of a bilinear singular integral operator based on extensions of the matrix-boundedness methods of Frazier and Jawerth for linear singular integrals to the case of bilinear operators. One of the key ingredients is the notion of a partial transpose: if \(B\) is a bilinear operator on a pair of function spaces \(X_{1},X_{2}\) then \(B_{1}^{\ast }\) is the transpose of the linear operator \(f\mapsto B(f,g)\) whenever \(g\) is fixed and \(B_{2}^{\ast }\) is the transpose of \(g\mapsto B(f,g)\) whenever \(f\) is fixed. The kernels of these transposes suitably permute the variables of the kernel of \(B\). Frazier and Jawerth devised a systematic method for dealing with linear singular integral operators by showing that the operators were almost diagonal when restricted to a suitable family of molecules that could be regarded as a generalized wavelet family. Namely, if \(\varphi _{\nu ,k}\) denotes a function having a certain number of vanishing moments and is suitably localized near the dyadic cube whose lower vertex is \(k/2^{\nu }\) then \(T\) is bounded on certain Triebel-Lizorkin norms that reflect the cancellation and localization properties of the \(\varphi _{\nu ,k}\) provided the matrix \(\langle T(\varphi _{\nu ,k}),\varphi _{\mu ,l}\rangle \) decays in an appropriate sense when \(k\) is far from \(l\) and/or \(\mu \) is far from \(\nu \). The authors extend the methods of Frazier and Jawerth by showing that if the entries of the tensor \(|\langle B(\varphi _{\nu ,k},\varphi _{\mu ,l}),\varphi _{\lambda ,m}\rangle |\) indexed by vertices of dyadic cubes decay in a sufficient manner away from the diagonals -- where two of the three indices agree -- then \(B\) is bounded on corresponding Triebel-Lizorkin spaces. Among the tools used are an extension of basic convolution estimates for pairs of molecules to estimates for integrals of products of three molecules which in turn motives a definition of bilinear molecules which, instead of being thought of as bumps localized near a single dyadic cubes, can be thought of as a superposition of such bumps. As in the work of Frazier and Jawerth, the Fefferman-Stein vector-valued maximal theorem is a key ingredient in the proof.