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Pseudodifferential operators with rough symbols
"... ABSTRACT. In this work, we develop L p boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(L p) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimat ..."
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ABSTRACT. In this work, we develop L p boundedness theory for pseudodifferential operators with rough (not even continuous in general) symbols in the x variable. Moreover, the B(L p) operator norms are estimated explicitly in terms of scale invariant quantities involving the symbols. All the estimates are shown to be sharp with respect to the required smoothness in the ξ variable. As a corollary, we obtain L p bounds for (smoothed out versions of) the maximal directional Hilbert transform and the Carleson operator. 1.
Vector valued inequalities for families of bilinear Hilbert transforms and applications to biparameter problems
, 2012
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Plinio: Logarithmic Lp Bounds for Maximal Directional Singular Integrals in the Plane
 J. Geom. Anal
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ON THE HILBERT TRANSFORM AND C 1+ǫ FAMILIES OF LINES
, 2004
"... Abstract. We study the operator ∫ 1 Hvf(x): = p.v. f(x − yv(x)) dy y −1 defined for smooth functions on the plane and measurable vector fields v from the plane into the unit circle. We prove that if v has 1+ǫ derivatives, then Hv extends to a bounded map from L2 (R2) into itself. What is noteworthy ..."
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Abstract. We study the operator ∫ 1 Hvf(x): = p.v. f(x − yv(x)) dy y −1 defined for smooth functions on the plane and measurable vector fields v from the plane into the unit circle. We prove that if v has 1+ǫ derivatives, then Hv extends to a bounded map from L2 (R2) into itself. What is noteworthy is that this result holds in the absence of some additional geometric condition imposed upon v, and that the smoothness condition is nearly optimal. Whereas Hv is a Radon transform for which there is an extensive theory, see e.g. [5], our methods of proof are necessarily those associated to Carleson’s theorem on Fourier series [3]. A previous paper of the authors [10], has shown how to adapt the approach of Lacey and Thiele [11] to the current setting; herein those ideas are combined with a crucial maximal function
Hilbert transform along measurable vector fields constant on Lipschitz curves
 L2 boundedness, arXiv:1401.2890
"... We prove the L2 boundedness of the Hilbert transform (without cutoff) along a family of measurable vector fields which are constant on Lipschitz curves with a global angle condition, see Theorem 1.4. A BesicovitchKakeya set counter example shows that, without this angle assumption, our operator ma ..."
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We prove the L2 boundedness of the Hilbert transform (without cutoff) along a family of measurable vector fields which are constant on Lipschitz curves with a global angle condition, see Theorem 1.4. A BesicovitchKakeya set counter example shows that, without this angle assumption, our operator may be unbounded in Lp for any p ∈ (1,∞). 1 Introduction and statement of the main result Consider a vector field assigning to every point x ∈ R2 a unit vector v(x). For some 0> 0 define the maximal operator