tailieunhanh - Weighted boundedness for a rough homogeneous singular integral

A weighted norm inequality for a homogeneous singular integral with a kernel belonging to a certain block space is proved. Also, some applications of this inequality are obtained. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals. | Turk J Math 29 (2005) , 75 – 97. ¨ ITAK ˙ c TUB Weighted Boundedness for a Rough Homogeneous Singular Integral Hussain Al-Qassem Abstract A weighted norm inequality for a homogeneous singular integral with a kernel belonging to a certain block space is proved. Also, some applications of this inequality are obtained. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals. Key Words: Singular integral, rough kernel, block space, weighted norm inequality. 1. Introduction Let Sn−1 denote the unit sphere in Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ = dσ(·). Throughout this paper, p0 will denote the dual exponent to p, that is 1/p + 1/p0 = 1. Also, we shall let Ω be a homogeneous function of zero which satisfies Ω ∈ L1 (Sn−1 ) and Z Ω (u) dσ (u) = 0. () Sn−1 For γ > 1, let ∆γ (R+ ) denote the set of all measurable functions h on R+ such that 1 sup R>0 R ZR γ |h (t)| dt 0 |y|>ε () where y0 = y/ |y| ∈ Sn−1, and f ∈ S(Rn ), the space of Schwartz functions. ∗ ∗ by Th,Ω . Also, we denote TΓ,h,Ω If Γ(t) = t, we shall denote TΓ,h,Ω by Th,Ω and TΓ,h,Ω ∗ ∗ by TΩ and TΓ,h,Ω by TΩ when h = 1 and Γ(t) = t. The study of the Lp (1 1. Then Th,Ω is bounded on Lp (ω) if q 0 ≤ p 1 (see (0,0) for example, [1], [2], [4]). Also, it was proved in [3] that the condition Ω ∈ Bq (Sn−1 ) is the best possible for the Lp boundedness of TΩ to hold. Namely, the Lp boundedness (0,υ) of TΩ may fail for any p if it is replaced by a weaker condition Ω ∈ Bq (Sn−1) for any (κ,υ) Sn−1 will be recalled in −1 1. The definition of the block space Bq Section 2. The primary concern of this paper is studying the Lp (ω) boundedness of the operators (0,0) ∗ for ω ∈ A˜Ip/γ 0 (R+ ), Ω ∈ Bq (Sn−1 ) for some q > 1 and h ∈ ∆γ (R+ ) for Th,Ω and Th,Ω some γ > 1. The main results of this paper are the following: Theorem . Let h ∈ ∆γ (R+ ) with γ > 1. Let Γ be in C 2 ([0, ∞)), convex, and .