Multilinear paraproducts revisited (Q748338)

This 8-page note provides a simple new proof of the \(L^{p_0}\times L^{p_1}\times\cdots\times L^{p_m}\to L^{p,\infty}\) boundedness (for \(p_i\in[1,\infty)\) and \(p=(\frac{1}{p_0}+\frac{1}{p_1}+\ldots+\frac{1}{p_m})^{-1}\in[\frac{1}{m+1},\infty)\)) of multilinear paraproducts \[ P^{(k)}_{m+1}\vec{f}=\sum_j \Delta_j^{(0)}(f_0)\cdots\Delta_j^{(k)}(f_k)S_j^{(k+1)}(f_{k+1})\cdots S_j^{(m)} (f_m) \] built from Littlewood-Paley resolution operators \(\Delta_j^{(i)}\) and their partial sums \(S_j^{(i)}\). If \(k\geq 1\), so that there are at least two resolution operators, one gets an easy pointwise bound by a product of Littlewood-Paley square functions and Hardy-Littlewood maximal functions of the various \(f_i\). The \(L^{p,\infty}\) norm of this product is then controlled by a weak-type Hölder inequality and the well-known \(L^{p_i}\to L^{p_i,\infty}\) bounds for the two classical operators. In the main case that \(k=0\), each \(S^{(i)}_j\) is written as a sum of a finite number \(r_i\) of resolution operators (reducing to the above easy case) and a residual \(\tilde S^{(i)}_j=S^{(i)}_{j-r_i}\), so that the \(j\)-th term in the residual paraproduct \(g:=\tilde P^{(0)}_{m+1}(\vec{f})\) has Fourier support in an annulus \(|\xi|\sim 2^j\). Thus the Littlewood-Paley square function of \(g\) is controlled by a similar product as in the easy case. But the \(L^{p,\infty}\) norm of \(g\) itself is bounded by the \(L^{p,\infty}\) norm of its square function, thanks to a lemma of the second author [J. Fourier Anal. Appl. 20, No. 5, 1083--1110 (2014; Zbl 1309.42027)], which concludes the proof.