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63
A bilinear approach to the restriction and Kakeya conjectures
 J. Amer. Math. Soc
, 1998
"... Abstract. Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the (L p, L p) spherical restriction theorem of Wolff [27] ..."
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Cited by 54 (22 self)
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Abstract. Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the (L p, L p) spherical restriction theorem of Wolff [27] from p> 42/11 to p> 34/9, and also obtain a sharp (L p, L q) spherical
A sharp bilinear restriction estimate for paraboloids, Geom
 Func. Anal
"... Abstract. Recently Wolff [28] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces ” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon ..."
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Cited by 45 (8 self)
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Abstract. Recently Wolff [28] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces ” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.
Restriction and Kakeya phenomena for finite fields
 DUKE MATH. J
, 2004
"... The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. I ..."
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Cited by 27 (0 self)
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The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there
The BochnerRiesz Conjecture Implies The Restriction Conjecture
 363 – 375. MR 1666558 52
, 1999
"... this paper. The author also thanks the reviewer for some helpful remarks. ..."
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Cited by 26 (10 self)
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this paper. The author also thanks the reviewer for some helpful remarks.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (9 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Riesz transform, Gaussian bounds and the method of wave equation
 Math. Z
"... Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We ..."
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Cited by 22 (1 self)
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Abstract. For an abstract selfadjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the LaplaceBeltrami operator on Riemannian manifolds for p> 2. 1.
Improved bounds for BochnerRiesz and maximal BochnerRiesz operators
 DUKE MATH. J
, 2004
"... In this note we improve the known L pbounds for BochnerRiesz operators and their maximal operators. ..."
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Cited by 16 (3 self)
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In this note we improve the known L pbounds for BochnerRiesz operators and their maximal operators.
Nonlinear Schrödinger equations at critical regularity
 CLAY LECTURE NOTES
, 2009
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Singular integral operators with rough convolution kernels
 J. Amer. Math. Soc
, 1996
"... The purpose of this paper is to investigate the behavior on L1 (Rd), d ≥ 2, of a class of singular convolution operators which are not within the scope of the standard CalderónZygmund theory. An important special case occurs if the convolution kernel K is homogeneous of ..."
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Cited by 14 (1 self)
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The purpose of this paper is to investigate the behavior on L1 (Rd), d ≥ 2, of a class of singular convolution operators which are not within the scope of the standard CalderónZygmund theory. An important special case occurs if the convolution kernel K is homogeneous of