Analysis & Applied Math

In this talk, we discuss the recent work on the operator norm of paraproducts on bi-parameter Hardy spaces. A paraproduct is a bilinear form arising from the product of two functions, both expanded in either a wavelet basis, such as the Haar, or in Littlewood-Paley pieces. In the one-parameter theory, the frequency interactions in the product of two functions are divided into either low-low, low-high, or high-low interactions, and each gives rise to a bilinear form called a one-parameter paraproduct. Some of these forms behave much better than the product itself, and for them, H¨older’s inequality holds for the full range of exponents, provided that the Lebesgue spaces are replaced by Hardy spaces and the space of bounded functions is replaced by functions of bounded mean oscillation. Similar results hold for any number of parameters as well.

In our recent work, it is shown that for all positive values of p, q, and r with 1/q = 1/p + 1/r , the operator norm of the dyadic paraproduct π_g from the bi-parameter dyadic Hardy space H_p^d to H_q^d is comparable to ∥g∥_{H_r^d}. In addition, similar results are obtained for bi-parameter Fourier paraproducts of the same form.