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...al polynomials (containing linear term), the problem is harder, especially for the full range p1, p2, r, since it involves the results of the uniform estimates for the bilinear Hilbert transform (see =-=[7]-=-, [13], [23]) and for the bilinear maximal transform (see [12]). The constraint of r > (d− 1)/d is superfluous if there is some additional convexity condition for the curve. For instance, for the mono...

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...sis In this section we prove (1.3) with 1/2 < r ≤ 1 for the paraproducts by time frequency analysis, which was used for establishing the L p (uniform) estimates for the bilinear Hilbert transforms in =-=[6, 10, 11, 12, 13, 14, 15, 21]-=-. Let F be a measurable set in R. X(F) denotes the set of all measurable functions supported on F such that the L ∞ norms of the functions are no more than 1. A function in X(F) can be considered esse...

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...as dist(Γ, ξo) . 2 k2 where ξo ∈ Q1 ×Q2 ×Q ′ 2. 7. trees The standard approach to prove the desired estimates for the forms Λ~P, ~Q is to organize our collections of tri-tiles ~P, ~Q into trees as in =-=[6]-=-. We may assume and shall do so for the rest of this article that ~P and ~Q are sparse of rank 1. We review standard definitions and comments for trees from [15]. Definition 7.1. For any 1 ≤ j ≤ 3 and...

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...ebrated results of Lacey and Thiele [27, 28, 29, 30] gave boundedness of Bm from L p × Lq to Lr′ with 1 < p, q < ∞ and 0 < 1r′ = 1 p + 1 q < 3 2 . These results were later extended by Grafakos and Li =-=[19, 32]-=- to cover the case where m is the characteristic function of a polygon. We refer the reader to [1], where the first author proved boundedness for particular square functions built on a covering of the...

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... some A ⊂ {1, 2, . . . , n} with |A| = m we have ¯ωi ⊆ 3 ¯ωi ′ for each i ∈ A, then there exist 1 ≤ i1 = i2 ≤ n such that ¯ωj � 3 ¯ωj ′ for each j ∈ {i1, i2}. We recall the following definition from =-=[4]-=-. Definition 2.2. A collection G of intervals in R d is called a central grid if • (G1) R, R ′ ∈ G and R ∩ R ′ = ∅ implies R ⊆ R ′ or R ′ ⊆ R • (G2) R, R ′ ∈ G and R ⊆ R ′ implies C2R ⊆ R ′ • (G3) R,...

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...tion of this strip and the convex hull of triangles a, b, d. The above theorem provides uniform estimates in the inner triangle c. This special case of Theorem 3.1 was previously shown by Grafakos/Li =-=[7]-=-, and Li [13] has shown uniform estimates in triangles a and b. These results together give uniform bounds in the convex hull of a, b, c. Uniform estimates near the points (1, 0, 0) and (0, 1, 0) rema...

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... n = 1 for a certain range of indices. In the bilinear case, the role of the directional Hilbert transforms is played by the bilinear Hilbert transforms and their uniform boundedness (Grafakos and Li =-=[10]-=-, Li [16]) is crucial in our approach. We recall that if the function Ω is smooth then the boundedness of the bilinear Calderón–Zygmund operator follows from the results of Coifman and Meyer [4], [5],...

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... pGENERALIZATIONS OF THE CARLESON-HUNT THEOREM I.THE CLASSICAL SINGULARITY CASE3 [17], [21] and will mark as “standard” any results that are well understood by now in this framework, as in [1], [3], =-=[5]-=-, [9], [11], [17], [21], [24], etc. The authors have been partially supported by NSF. The second author was also partially supported by an Alfred P. Sloan Research Fellowship. Both of the authors woul...

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..., I, J}. The dyadic intervals are a grid. A grid G is central iff for all I, J ∈ G, with I ⊄= J we have 100I ⊂ J. The reader can find the details on how to construct such a central grid structure in =-=[7]-=-. Let ρ be rotation on T by an angle of π/2. Coordinate axes for R 2 are a pair of unit orthogonal vectors (e, e⊥) with ρe = e⊥. 2.2. Definition. We say that ω ⊂ R 2 is a rectangle if it is a product ...