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17
Some remarks on multilinear maps and interpolation
"... Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasinormed rearrangementinvariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integra ..."
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Cited by 21 (15 self)
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Abstract. A multilinear version of the Boyd interpolation theorem is proved in the context of quasinormed rearrangementinvariant spaces. A multilinear Marcinkiewicz interpolation theorem is obtained as a corollary. Several applications are given, including estimates for bilinear fractional integrals. 1.
On Xray transforms for rigid line complexes and integrals over curves
 in R , Proc
, 1999
"... Abstract. Endpoint estimates are proved for model cases of restricted Xray transforms and singular fractional integral operators in R 4. 1. ..."
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Cited by 9 (5 self)
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Abstract. Endpoint estimates are proved for model cases of restricted Xray transforms and singular fractional integral operators in R 4. 1.
Two endpoint bounds for generalized Radon transforms
"... The purpose of this note is to prove L p → L q inequalities for averaging operators in the plane (also known as generalized Radon transforms). To describe our setup let ΩL and ΩR be open sets in R 2 and let M be a submanifold in ΩL × ΩR which will contain the singular support of the kernel of our op ..."
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Cited by 8 (2 self)
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The purpose of this note is to prove L p → L q inequalities for averaging operators in the plane (also known as generalized Radon transforms). To describe our setup let ΩL and ΩR be open sets in R 2 and let M be a submanifold in ΩL × ΩR which will contain the singular support of the kernel of our operator. We assume that the projections M → ΩL and M → ΩR have surjective differential;
Restriction of Fourier transforms to curves and related oscillatory integrals
, 2006
"... Abstract. We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in R d, d ≥ 3, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in R d we obtain sharp uniform L p → L q bounds with respect to affin ..."
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Cited by 6 (3 self)
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Abstract. We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in R d, d ≥ 3, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in R d we obtain sharp uniform L p → L q bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall. 1.
SHARP RATE OF AVERAGE DECAY OF THE FOURIER TRANSFORM OF A BOUNDED SET
, 2003
"... Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that ..."
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Cited by 5 (3 self)
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Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L²average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in R d is
Fourier restriction to polynomial curves I: a geometric inequality, preprint
"... Abstract. We prove a Fourier restriction result for general polynomial curves in R d. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishin ..."
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Cited by 4 (1 self)
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Abstract. We prove a Fourier restriction result for general polynomial curves in R d. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal estimate for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. Applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established. 1. Introduction. Recently
THE MULTILINEAR MARCINKIEWICZ INTERPOLATION THEOREM REVISITED: THE BEHAVIOR OF THE CONSTANT
"... Abstract. We provide a selfcontained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise. The advantage of this proof is that it yields a ..."
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Cited by 3 (3 self)
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Abstract. We provide a selfcontained proof of the multilinear extension of the Marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates, considering possible degenerate situations that may arise. The advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted weak type bounds; it also provides an explicit estimate in terms of the exponents of the initial estimates: the constant blows up like a negative power of the distance from the intermediate point to the boundary of the convex hull of the initial points. In memory of Nigel Kalton 1.
Restriction of Fourier transforms to curves, II: Some classes with vanishing torsion
 J. Austr. Math. Soc
"... Abstract. We consider the Fourier restriction operators associated to certain degenerate curves in R d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform feat ..."
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Cited by 2 (1 self)
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Abstract. We consider the Fourier restriction operators associated to certain degenerate curves in R d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves. 1.
Average decay estimates for Fourier transforms of measures supported on curves
 J. Geom. Anal
"... Abstract. We consider Fourier transforms ̂µ of densities supported on curves in R d. We obtain sharp lower and close to sharp upper bounds for the decay rates of ‖̂µ(R·) ‖ L q (S d−1), as R → ∞. 1. ..."
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Cited by 2 (1 self)
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Abstract. We consider Fourier transforms ̂µ of densities supported on curves in R d. We obtain sharp lower and close to sharp upper bounds for the decay rates of ‖̂µ(R·) ‖ L q (S d−1), as R → ∞. 1.
A SELBERG INTEGRAL FORMULA AND APPLICATIONS
 PACIFIC JOURNAL OF MATHEMATICS VOL. 191, NO. 1, 1999
, 1999
"... We obtain a 3fold Selberg integral formula. As a consequence we are able to compute the explicit value of the sharp constant in a trilinear fractional integral inequality due to Beckner. ..."
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Cited by 2 (0 self)
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We obtain a 3fold Selberg integral formula. As a consequence we are able to compute the explicit value of the sharp constant in a trilinear fractional integral inequality due to Beckner.