Citation Context

...invariant under translations in direction of Γ ′ , the operator itself has a modulation symmetry. Hence the title of this paper. Such forms and operators have been discussed in [20], [10] (also [25], =-=[23]-=-, [22], [9], [11]). Given a multiplier m, the main interest in the subject is to obtain L p estimates for Λm of the form |Λm(f1, . . . , fn)| ≤ C n∏ i=1 ‖fi‖pi (3) when 1 ≤ pi ≤ ∞. By duality this is ...

Citation Context

... � ∑ 2 2d/3 2 2d/3 2 −2Nd/3 � 1 d≥0 P ∈Pd d≥0 as wanted and this finishes the proof of the one-parameter case. The reader should compare this Proposition 3.6 with the corresponding Proposition 6.5 in =-=[16]-=-. Our present “lacunary setting” allows for an L1-type definition of the “energies” (instead of L2-type as in [16]) and this is why we can obtain the full range of estimates this time.BI-PARAMETER PA...

Citation Context

...ounterexamples rely strongly on a certain degeneracy in the phase function ∑ j (−1)jxj on the boundary of the simplex x1 < . . . < xn. If one replaced this phase by ∑ j xj, then we have shown in [9], =-=[10]-=- that the bound (2) in fact holds when p = 2 and n = 3. Indeed this statement is true for generic phases of the form ∑ j αjxj. A similar statement holds for (1) when p = 2 and n = 2 and will appear el...

Citation Context

... Li show in[15, 23] some uniform (with respect to the parameters λ1 and λ2) continuities. These proofs use a technical time frequency analysis, which was proven by C. Muscalu, T. Tao and C. Thiele in =-=[25, 26, 27]-=- and independently by J. Gilbert and A. Nahmod in [13, 14]. They also get a very important result in the study of bilinear operators : continuities in Lebesgue spaces for more singular operators than ...

Citation Context

... introduction. Thus in order to carry out the program of [2] in the endpoint case p = 2 it is necessary 6 (among other things) to bound T on L 2 . This motivates the work of this paper and the sequel =-=[15]-=-. 10. Appendix II: Twalsh,P,Q and T In this rather informal section we briefly explain why Twalsh,P,Q is the natural Walsh model of T+. Here T+ is defined like T but with an integration 0 < ξ1 < ξ2 < ...

Citation Context

...ruct the multipliers m ′ N , m′′ N and m ′′′ N we plan to carve these two symbols carefully into smaller pieces. To take care of the first symbol χξ1<ξ2, we first need to recall some definitions from =-=[10]-=-. Definition 4.1. For n ≥ 1 and σ ∈ {0, 1 2 , 3 3 }n. We define the shifted n-dyadic mesh D = Dn σ to be the collection of cubes of the form D n α := {2j (k + (0, 1) n + (−1) j σ)|j ∈ Z, k ∈ Z n }. We...

Citation Context

...)‖G‖ r Lq(LQ). This completes the proof of the Theorem 4.5. 5. Discretization In this section we introduce the discrete model sums for the operator BP . First we follow the notation and comments from =-=[8]-=- and introduce rank 1 collections of tri-tiles, which are used in the decomposition of the bilinear Hilbert transform [5], [8]. We use rank 1 collections in the decomposition of BP in the first coordi...