Weighted Bounds for Multilinear Square Functions

Abstract

Abstract. Let P = (p1, . . . , pm) with 1 < p1, . . . , pm < ∞, 1/p1 + · · · + 1/pm = 1/p and w = (w1, . . . , wm) ∈ A P . In this paper, we investigate the weighted bounds with dependence on aperture α for multilinear square functions S α,ψ ( f ). We show thatThis result extends the result in the linear case which was obtained by Lerner in 2014. Our proof is based on the local mean oscillation technique presented firstly to find the weighted bounds for Calderón-Zygmund operators. This method helps us avoiding intrinsic square functions in the proof of our main result.

New bounds for bilinear Calderón–Zygmund operators and applications

“…|f j | Q 0 by definition. Thus, the cubes Q i satisfy Lacey's conditions (2) and (3) and to complete the proof, we only need to show that with a suitable choice of C 0 T the cubes also satisfy property (1). Let us split the set E 0 into two parts:…”

Weak and strong type estimates for the multilinear pseudo-differential operators

“…, 3 n . These results are explicitly proved in [2]. By (3.6) and (3.7), the inequalities (1.3) and (1.4) follow from the following…”

“…For the sake of completeness we here give the details. Together with Lemma 2.2, the hypothesis (1) or (2) indicates that u ∈ A 1 and v 1 m ∈ A ∞ . The former implies that…”

“…where 0 < δ 0 < min{δ, 1 2 }. Thus, together with (3.27) and (3.28), the estimate (2.2) applied to Q 0 = Q and f = S α ( f ) 2 gives that…”

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