Certain Multi(sub)linear Square Functions

Abstract

Let d ≥ 1, ℓ ∈ Z d , m ∈ Z + and θ i , i = 1, . . . , m are fixed, distinct and nonzero real numbers. We show that the m-(sub)linear version below of the Ratnakumar and Shrivastava [11] Littlewood-Paley square functionwhen 2 ≤ p i < ∞ satisfy 1/p = 1/p 1 + · · · + 1/p m and 1 ≤ p < ∞. Our proof is based on a modification of an inequality of Guliyev and Nazirova [6] concerning multilinear convolutions.2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B25.

Multilinear square functions and multiple weights

“…This gives us a complete analogue of the corresponding classical result about Littlewood-Paley operators in the context of multilinear multiplier operators. The proof of our main result (Theorem 3.1) is motivated from the ideas presented in [15] (see also [6]) for the bilinear square functions and [16] for the classical Littlewood-Paley operators.…”

“…The bilinear Littlewood-Paley operators associated with such bilinear multipliers may be defined in a similar fashion. We refer the interested reader to [1,3,5,6,8,14,15] and the references therein for some relevant background on this.…”