Bilinear Littlewood-Paley for circle and transference

Abstract

In this paper we have obtained the boundedness of bilinear Littlewood-Paley operators on the circle group T by using appropriate transference techniques. In particular, bilinear analogue of Carleson's Littlewood-Paley result for all possible indices has been obtained. Also, we prove some bilinear analogues of de Leeuw's results concerning multipliers of R n .

A note on bi-linear multipliers

“…But, we already know that p, q ≥ 2 is a necessary condition for the boundedness of the bi-linear Littlewood-Paley operator S (see the paper by P. Mohanty and S. Shrivastava [8] for a proof of this assertion) and hence we get a contradiction if either of p and q is less than 2.…”

On bilinear Littlewood-Paley square functions

“…In this section we study bilinear square functions on T d . As mentioned previously, Mohanty and Shrivastava [9] proved that the bilinear Carleson's Littlewood-Paley operator maps L p (T)×L q (T) into L r (T) for exponents p, q, r satisfying 2 ≤ p, q ≤ ∞ and the Hölder condition 1 p + 1 q = 1 r . The authors used vector-valued transference methods to prove their result.…”

“…As far as our knowledge is concerned, not much is known about bilinear square functions. We refer the interested reader to [1,2,3,8,9] for the work done so far (to our knowledge) on bilinear square functions.…”

“…The study of bilinear square functions on T was initiated by Mohanty and Shrivastava [9]. They proved the bilinear analogue of Carleson's Littlewood-Paley theorem on T. Unlike the linear case, the bilinear Carleson's Littlewood-Paley theorem on T is not straightforward (for more details see [9]). The authors used suitable transference techniques in order to prove their result.…”

A homomorphism theorem for bilinear multipliers

“…where m (n, m) = m(π (n), π (m)), for n, m ∈ Z d . The previous result allows us to obtain a generalization of Diestel and Grafakos's [16] Proposition 2 for p < 1 (and also its weak type counterpart), on the restriction to a lower dimension of a bilinear multiplier (see also the result of Mohanty and Shrivastava [26,Theorem 4.1] for multipliers of strong type).…”