449 research outputs found
A modulation invariant Carleson embedding theorem outside local L 2
No full textYen Do and Christoph Thiele developed a theory of Carleson embeddings in outer Lp spaces for the wave packet transform Fφ(f)(u,t,η)=∫f(x)eiη(u−x)φ(u−xt)dxt,(u,t,η)∈R×(0,∞)×R of functions f ∈ Lp(R) in the range 2 ≤ p ≤ ∞, referred to as local L2. In this article, we formulate a suitable extension of this theory to exponents 1 < p < 2, answering a question posed by Do and Thiele. The proof of our main embedding theorem involves a refined multi-frequency Calderón-Zygmund decomposition in the vein of work by Di Plinio and Thiele and by Nazarov, Oberlin, and Thiele. We apply our embedding theorem to recover the full known range of Lp estimates for the bilinear Hilbert transforms without reducing to discrete model sums or appealing to generalized restricted weak-type interpolation
Positive sparse domination of variational Carleson operators
No full textDue to its nonlocal nature, the r-variation norm Carleson operator Cr does not yield to the sparse domination techniques of Lerner [15, 17], Di Plinio and Lerner [6], Lacey [14]. We overcome this difficulty and prove that the dual form of Cr can be dominated by a positive sparse form involving Lp averages. Our result strengthens the Lp-estimates by Oberlin et al. [18]. As a corollary, we obtain quantitative weighted norm inequalities improving the results in [8] by Do and Lacey. Our proof relies on the localized outer Lp-embeddings of Di Plinio and Ou [7] and Uraltsev [19]
A sharp estimate for the Hilbert transform along finite order lacunary sets of directions
No full textLet be a nonnegative integer and be a lacunary set of directions of order . We show that the norms, , of the maximal directional Hilbert transform in the plane are comparable to . For vector fields with range in a lacunary set of of order and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field , is -bounded for all . These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.20 pages, 2 figures. Submitted. Changes: clarified the definition of D-lacunary set and streamlined the notatio
Robust exponential attractors for the strongly damped waveequation with memory. I
No full textWe consider the singular limit of the semilinear strongly damped wave equation with memory ∂ ttu - γΔ ∂ t u - k (0)Δ u - ∫0∞ {k'} (s)Δ u(t - s)ds + φ (u) = f, in presence of an arbitrarily growing nonlinearity φ, as the memory kernel k(s)-k(∞) converges to the Dirac mass at zero. The existence of a robust family of regular exponential attractors is established, under a necessary and sufficient condition on k, along with quantitative estimates of the closeness of the equation with memory to the corresponding limit equation. © 2008 Pleiades Publishing, Ltd
Lacunary Fourier and Walsh-Fourier series near L1
No full textWe prove the following theorem: given a lacunary sequence of integers {nj}, the subsequences Fnj f and Wnj f of respectively the Fourier and the Walsh-Fourier series of f: T → C converge almost everywhere to (Formula presented.) Our integrability condition (1) is less stringent than the homologous assumption in the almost everywhere convergence theorems of Lie [14] (Fourier case) and Do and Lacey [6] (Walsh-Fourier case), where a triple-log term appears in place of the quadruple-log term of (1). Our proof of the Walsh-Fourier case is self-contained and, in antithesis to [6], avoids the use of Antonov's lemma [1, 19], relying instead on the novel weak-Lp bound for the lacunary Walsh-Carleson operator (Formula presented.). © 2013 Universitat de Barcelona
Weak-Lp bounds for the Carleson and Walsh-Carleson operators
No full textWe prove a weak-Lp bound for the Walsh-Carleson operator for p near 1, improving on a theorem of Sjölin. We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function f∈LlogL(T) converge for almost every x∈T. © 2014 Académie des sciences
GRISVARD'S SHIFT THEOREM NEAR L-infinity AND YUDOVICH THEORY ON POLYGONAL DOMAINS
No full textLet Omega subset of R-2 be a bounded, simply connected domain with boundary partial derivative Omega of class C-1,C-1 except at finitely many points S-j where partial derivative Omega is locally a corner of aperture alpha(j) <= pi/2. Improving on results of Grisvard [J. Monogr. Stud. Math. 24, Pitman, Boston, MA, 1985; J. Math. Pures Appl., 74 (1995), pp. 3-33], we show that the solution G(Omega)f to the Dirichlet problem on Omega with data f is an element of L-p(Omega) and homogeneous boundary conditions satisfies the estimates parallel to G(Omega)f parallel to(W2),(p(Omega)) <= Cp parallel to f parallel to L-p(Omega) for all 2 <= p < infinity, parallel to D(2)G(Omega)f parallel to(ExpL1(Omega)) <= C parallel to f parallel to L-infinity(Omega). The proof uses sharp L-p bounds for singular integrals on power weighted spaces inspired by the work of Buckley [Trans. Amer. Math. Soc., 340 (1993), pp. 253-272]. Our results lead to the extension of the Yudovich theory [V. I. Yudovich, Z. Vycisl. Mat. i Mat. Fiz., 3 (1963), pp. 1032-1066; Math. Res. Lett., 2 (1995), pp. 27-38] of existence, uniqueness, and regularity of weak solutions to the Euler equations on Omega x (0, T) to polygonal domains Omega as above
Logarithmic L p Bounds for Maximal Directional Singular Integrals in the Plane
No full textLet K be a Calderón-Zygmund convolution kernel on R. We discuss the L p -boundedness of the maximal directional singular integral T V f(x)= sup v ε V | ∫R f(x+t v) K(t)dt|where V is a finite set of N directions. Logarithmic bounds (for 2≤p<∞) are established for a set V of arbitrary structure. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L p almost orthogonality principle for Fourier restrictions to cones. © 2012 Mathematica Josephina, Inc
Endpoint bounds for the bilinear hilbert transform
No full textWe study the behavior of the bilinear Hilbert transform (BHT) at the boundary of the known boundedness region H. A sample of our results is the estimate (Formula Presented) valid for all tuples of sets Fj ⊂ R of finite measure and functions fj such that |fj| ≤ 1Fj, j = 1, 2, 3, with the additional restriction that f3 be supported on a major subset F′3 of F3 that depends on {Fj : j = 1, 2, 3}. The use of subindicator functions in this fashion is standard in the given context. The double logarithmic term improves over the single logarithmic term obtained by D. Bilyk and L. Grafakos. Whether the double logarithmic term can be removed entirely, as is the case for the quartile operator, remains open. We employ our endpoint results to describe the blow-up rate of weak-type and strong-type estimates for BHT as the tuple (Formula Presented) approaches the boundary of H. We also discuss bounds on Lorentz-Orlicz spaces near L2⁄3, improving on results of M. Carro et al. The main technical novelty in our article is an enhanced version of the multi-frequency Calder ́on-Zygmund decomposition
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