Weighted Estimates for Bilinear Operators

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 797956, 10 pages http://dx.doi.org/10.1155/2014/797956 Research Article Weighted Estimates for Bilinear Operators Hua Zhu1 and Heping Liu2 1 2 Beijing International Studies University, Beijing 100024, China LMAM, School of Mathematical Science, Peking University, Beijing 100871, China Correspondence should be addressed to Hua Zhu; Received 31 May 2013; Accepted 26 November 2013; Published 6 February 2014 Academic Editor: Dashan Fan Copyright © 2014 H. Zhu and H. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator. 1. Introduction Bilinear (or multilinear) operators have attracted many researchers’ attention, due to their relations closely connected to the Cauchy integral along with Lipschitz curves, Calderón commutators, and compensated compactness. In [1–3] and references therein, we can see an extensive study on the Hardy space estimate of bilinear operators. In [4], we can see the bilinear operators related to a Schrödinger operator L and estimates of them with respect to the Hardy type space associated with the Schrödinger operator L under some general conditions. In our paper, we study weighted estimates for bilinear operators which have the same expression as the operators in [4]. In this paper, 𝑁 and 𝐾 will denote fixed integers ≥2. Given a matrix of convolution Calderón-Zygmund kernels 𝑗 𝑗 𝐾 𝑑 {𝐾𝑖 }𝑁 𝑖=1,𝑗=1 on R , we define {T𝑖 }𝑖,𝑗 as the associated CalderónZygmund operators. We denote by 𝐿(𝑓1 , . . . , 𝑓𝐾 ) the 𝐾-linear operator: Theorem 1. Assume that 𝑝1 , . . . , 𝑝𝐾 > 1 are given and let −1 −1 ) be their harmonic mean. We also 𝑟 = (𝑝1−1 + ⋅ ⋅ ⋅ + 𝑝𝐾 assume that the harmonic mean of any proper subset of the 𝑝𝑗 ’s is greater than 1. If, for all (𝑓1 , . . . , 𝑓𝐾 ) ∈ (𝐶0∞ )𝐾 , the 𝐾-linear operator 𝐿 satisfies ∫ 𝐿 (𝑓1 , . . . , 𝑓𝐾 ) 𝑑𝑥 = 0. (2) Then, for 𝜔 ∈ 𝐴 1 (R𝑑 ), one has following conclusions. (1) If 𝑟 > 1, 𝐿 maps 𝐿𝑝1 (𝜔) × ⋅ ⋅ ⋅ × 𝐿𝑝𝐾 (𝜔) → 𝐿𝑟 (𝜔). 𝑁 𝐿 (𝑓1 , . . . , 𝑓𝐾 ) = ∑ (𝑇𝑖1 𝑓1 ) ⋅ ⋅ ⋅ (𝑇𝑖𝐾 𝑓𝐾 ) , 𝑖=1 where 𝜙𝑡 (𝑥) = (1/𝑡𝑛 )𝜙(𝑥/𝑡) and 𝜙 is smooth, nonzero, and compactly supported, and we also denote by 𝐻𝑝,∞ (𝜔) the weak 𝐻𝑝 as defined in [7], that is, the set of all distributions 𝑓 on R𝑑 for which the maximal function sup𝑡>0 |𝜙𝑡 ∗ 𝑓(𝑥)| is in weak 𝐿𝑝 (𝜔). If 𝜔(𝑥) ≡ 1, the weak Hardy space 𝐻1,∞ (R𝑑 ) = 𝐻1,∞ (𝜔) was introduced by Feerman and Soria in [8]. 𝐻𝑝,∞ (R𝑑 ) first appeared in [9] (see also [10]). (1) (2) If 1 ≥ 𝑟 > 𝑑/(𝑑 + 1), 𝐿 maps 𝐿𝑝1 (𝜔) × ⋅ ⋅ ⋅ × 𝐿𝑝𝐾 (𝜔) → 𝐻𝑟 (𝜔). which originally is defined for smooth compactly supported functions 𝑓1 , . . . , 𝑓𝐾 . For 𝑝 ≤ 1, 𝜔 ∈ 𝐴 1 (R𝑑 ), we denote by 𝐻𝑝 (𝜔) the usual weighted Hardy space as defined in [5, 6], that is, the set of all distributions 𝑓 on R𝑑 for which the maximal function sup𝑡>0 |𝜙𝑡 ∗ 𝑓(𝑥)| is in 𝐿𝑝 (𝜔), (3) If 𝑟 = 𝑑/(𝑑 + 1), 𝐿 maps 𝐿𝑝1 (𝜔) × ⋅ ⋅ ⋅ × 𝐿𝑝𝐾 (𝜔) → 𝐿𝑟,∞ (𝜔). Let L = −Δ + 𝑉 be a Schrödinger operator on R𝑑 , 𝑑 ≥ 3, where 𝑉 ≢ 0 is a fixed nonnegative potential. We will assume 2 Journal of Function Spaces that 𝑉 belongs to reverse Hölder class RH𝑠 (R𝑑 ) for some 𝑠 ≥ 𝑑/2; that is, there exists 𝐶 = 𝐶(𝑠, 𝑉) > 0 such that ( 1/𝑠 1 1 ∫ 𝑉 (𝑥) 𝑑𝑥) , ∫ 𝑉(𝑥)𝑠 𝑑𝑥) ≤ 𝐶 ( |𝐵| 𝐵 |𝐵| 𝐵 (3) for every ball 𝐵 ⊂ R𝑑 . In what follows, 𝐵(𝑥, 𝑟) denotes the ball centered at 𝑥 and of the radius 𝑟. Trivially, RH𝑞 (R𝑑 ) ⊂ RH𝑝 (R𝑑 ) provided 1 < 𝑝 ≤ 𝑞 < ∞. It is well known that if 𝑉 ∈ RH𝑞 (R𝑑 ) for some 𝑞 > 1, then there exists 𝜀 > 0, which depends only on 𝑑 and the constant 𝐶 in (3), such that 𝑉 ∈ RH𝑞+𝜀 (R𝑑 ) (see [11]). Throughout this paper, we always assume that 0 ≢ 𝑉 ∈ RH𝑑/2 . Thus, 𝑉 ∈ RH𝑞0 for some 𝑞0 > 𝑑/2. Let {𝑇𝑡L }𝑡>0 be the semigroup of linear operators generated by L and let 𝑇𝑡L (𝑥, 𝑦) be their kernels; that is, 𝑇𝑡L 𝑓 (𝑥) = 𝑒−𝑡L 𝑓 (𝑥) = ∫ 𝑇𝑡L (𝑥, 𝑦) 𝑓 (𝑦) 𝑑𝑦, R𝑑 (4) 󵄨2 󵄨󵄨 󵄨𝑥 − 𝑦󵄨󵄨󵄨 0 ≤ 𝑇𝑡L (𝑥, 𝑦) ≤ 𝐻𝑡 (𝑥, 𝑦) = (4𝜋𝑡)−𝑑/2 exp (− 󵄨 ). 4𝑡 (5) The maximal function with respect to the semigroup {𝑇𝑡L }𝑡>0 is defined by 󵄨 󵄨 T∗ 𝑓 (𝑥) = sup 󵄨󵄨󵄨󵄨𝑇𝑡L 𝑓 (𝑥)󵄨󵄨󵄨󵄨 . (6) 𝑡>0 The weighted Hardy-type space related to L is naturally defined by (see [13]) (7) L Following [14], we define the auxiliary function 𝜌(𝑥, 𝑉) = 𝜌(𝑥) by 𝜌 (𝑥) = 𝜌 (𝑥, 𝑉) = sup {𝑟 > 0 : ̃ ± (𝑓, 𝑔) (𝑥) = (𝑇 ̃1 𝑓) (𝑥) (𝑇 ̃2 𝑔) (𝑥) 𝑇 ̃1 𝑔) (𝑥) ̃2 𝑓) (𝑥) (𝑇 ± (𝑇 1 𝑉 (𝑦) 𝑑𝑦 ≤ 1} . ∫ 𝑟𝑑−2 𝐵(𝑥,𝑟) (8) The auxiliary function 𝜌(𝑥) plays an important role in studying the boundedness of singular integral operators related to the Schrödinger operator L as well as the atomic 1 1 decomposition of 𝐻L and 𝐻L (𝜔) (see [13–15]). In our paper, we also consider the following bilinear operators: 𝑇± (𝑓, 𝑔) (𝑥) = (𝑇1 𝑓) (𝑥) (𝑇2 𝑔) (𝑥) ± (𝑇2 𝑓) (𝑥) (𝑇1 𝑔) (𝑥) , (9) where 𝑓 ∈ 𝐿𝑝 (𝜔), 𝑔 ∈ 𝐿𝑞 (𝜔) with 1 < 𝑝, 𝑞 < ∞ and 1/𝑝 + 1/𝑞 = 1, 𝑇𝑖 (𝑖 = 1, 2) are Calderón-Zygmund operators related to L and satisfy the following two conditions. (11) ̃− ̃ + or 𝑇 has the vanishing moment; that is, either 𝑇 satisfies R𝑑 By the Trotter product formula (cf. [12]), 󵄩 󵄩 󵄩 󵄩 with 󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐻1 (𝜔) = 󵄩󵄩󵄩T∗ 𝑓󵄩󵄩󵄩𝐿1 (𝜔) . (ii) One of the parallel bilinear operators ̃ ± (𝑓, 𝑔) (𝑥) 𝑑𝑥 = 0 ∀𝑓, 𝑔 ∈ 𝐶∞ (R𝑑 ) . ∫ 𝑇 𝑐 for 𝑡 > 0, 𝑓 ∈ 𝐿2 (R𝑑 ) . 1 𝐻L (𝜔) = {𝑓 ∈ 𝐿1 (𝜔) : T∗ 𝑓 (𝑥) ∈ 𝐿1 (𝜔)} , ̃𝑖 (i) There exist parallel Calderón-Zygmund operators 𝑇 related to the Laplacian Δ and a constant 𝛿 > 0 such that 𝐶 󵄨󵄨 ̃𝑖 (𝑥, 𝑦)󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨𝑇𝑖 (𝑥, 𝑦) − 𝑇 (10) 𝛿 󵄨󵄨 󵄨 󵄨 󵄨𝑑−𝛿 , 𝑥 ≠ 𝑦, 𝜌(𝑦) 󵄨󵄨𝑥 − 𝑦󵄨󵄨󵄨 ̃𝑖 (𝑥, 𝑦) denote the kernels of 𝑇𝑖 where 𝑇𝑖 (𝑥, 𝑦) and 𝑇 ̃𝑖 , respectively. and 𝑇 (12) We will show that either 𝑇+ or 𝑇− is bounded from 𝐿𝑝 (𝜔) × 1 (𝜔). 𝐿𝑞 (𝜔) to 𝐻L Theorem 2. Suppose that the bilinear operators 𝑇± are defined as above. Let 1 < 𝑝, 𝑞 < ∞ and 1/𝑝 + 1/𝑞 = 1. Then either 𝑇+ or 𝑇− (but not both), which corresponds to the parallel bilinear 1 (𝜔) and operator satisfying (12), maps 𝐿𝑝 (𝜔) × 𝐿𝑞 (𝜔) into 𝐻L there exists a constant 𝐶 > 0 such that 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩 ± 󵄩󵄩𝑇 (𝑓, 𝑔)󵄩󵄩󵄩𝐻1 (𝜔) ≤ 𝐶󵄩󵄩󵄩𝑓󵄩󵄩󵄩𝐿𝑝 (𝜔) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐿𝑞 (𝜔) . (13) L This paper is organized as follows. In Section 2, we give some notation and preliminary estimates on 𝜌(𝑥) and the kernel 𝑇𝑡 (𝑥, 𝑦) which have been proved in [14–16]. In Section 3, we prove Theorem 1, and in Section 4 we prove Theorem 2. Throughout this paper, we will use 𝐶 to denote a positive constant, which is not necessarily the same at each occurrence. By 𝐴 ∼ 𝐵, we mean that there exists a constant 𝐶 > 1, such that 𝐶−1 ≤ 𝐴/𝐵 ≤ 𝐶. For a given ball 𝐵, we denote by 𝐵∗ the concentric ball with twice radius, and 𝐵∗∗ = (𝐵∗ )∗ . 2. Preliminaries Throughout this paper, we will denote 𝜔(𝐸) := ∫𝐸 𝜔(𝑥)𝑑𝑥 for any set 𝐸 ⊂ R𝑑 . For 1 ≤ 𝑝 ≤ ∞, denote by 𝑝󸀠 the adjoint number of 𝑝; that is, 1/𝑝 + 1/𝑝󸀠 = (...truncated)