Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

Abstract

Abstract. Minimal regularity conditions on the kernels of bilinear operators are identified and shown to be sufficient for the existence of end-point estimates within the context of the bilinear Calderón-Zygmund theory.

An Endpoint Weak-Type Estimate for Multilinear Calderón–Zygmund Operators

“…Referring to the contents of [20], the current proof shows that the sets E i,j can be constructed as cubes, that the regularity of Lemma 1 is only required for collections of pairwise disjoint cubes, and that Theorem 2 is not necessary for the weak-type estimate. Section 2 describes the definitions and preliminary results, including a weighted version of the regularity condition first described for bilinear kernels in [16]. Section 3 contains two proofs of the main result, Theorem 4.…”

“…As in the classical theory, their proof uses the Calderón-Zygmund decomposition. Other proofs, also using the Calderón-Zygmund decomposition, were later given in the bilinear setting by Pérez and Torres in [16], and by Maldonado and Naibo in [12]. Another proof was given by the author and Wick in [20] using a variation of the Nazarov-Treil-Volberg method.…”

Estimates for the maximal bilinear singular integral operators

“…As it was pointed out in [12] that, if T is a bilinear singular integral operator of type ω with ω satisfies (1.3), then T also satisfies (1.4)-(1.6). The purpose of this paper is to consider the behavior on product of the Lebesgue spaces for the maximal operator associated with the bilinear singular integral operator whose kernel satisfies the conditions (1.4)-(1.6).…”

“…Fairly recently, Pérez and Torres [12] considered this question and obtained the following result. Theorem 1.1 Let T be a bilinear singular integral operator with kernel K in the sense of (1.1), and is bounded from L q 1 (R n ) × L q 2 (R n ) to L q, ∞ (R n ) for some q 1 , q 2 ∈ [1, ∞], and q ∈ ( 1 2 , ∞) with 1 q = 1 q 1 + 1 q 2 .…”

“…When K is a bilinear Calderón-Zygmund kernel, Grafakos and Torres [4] considered the endpoint estimate for T on the space of type L 1 (R n ) × L 1 (R n ), and established a T 1 type theorem for such an operator T . Lerner et al [9] introduced a new maximal operator and established the weighted estimates with multiple weights for the multilinear Calderón-Zygmund singular integral operators; see also [12] for more results on the multilinear Calderón-Zygmund operator. Maldonado and Naibo [11] generalized the bilinear Calderón-Zygmund operators, and introduced the bilinear singular integral operator of type ω.…”