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90
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 (6 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
UNIMODULAR FOURIER MULTIPLIERS FOR MODULATION SPACES
"... Abstract. We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiξα, where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lpspaces. As a consequence, the phases ..."
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Cited by 17 (2 self)
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Abstract. We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiξα, where α ∈ [0, 2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lpspaces. As a consequence, the phasespace concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers ξ  −δ sin(ξ  α) for 0 ≤ δ ≤ α. 1.
Filters, mollifiers and the computation of the Gibbs phenomenon
, 2007
"... We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accu ..."
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Cited by 15 (5 self)
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We are concerned here with processing discontinuous functions from their spectral information. We focus on two main aspects of processing such piecewise smooth data: detecting the edges of a piecewise smooth f, namely, the location and amplitudes of its discontinuities; and recovering with high accuracy the underlying function in between those edges. If f is a smooth function, say analytic, then classical Fourier projections recover f with exponential accuracy. However, if f contains one or more discontinuities, its global Fourier projections produce spurious Gibbs oscillations which spread throughout the smooth regions, enforcing local loss of resolution and global loss of accuracy. Our aim in the computation of the Gibbs phenomenon, is to detect edges and to reconstruct piecewise smooth f’s, while regaining the high accuracy encoded in the spectral data. To detect edges, we utilize a general family of edge detectors based on concentration kernels. Each kernel forms an approximate derivative of the delta function, which detects edges by separation of scales. We show how such
Extrapolation and sharp norm estimates for classical operators on weighted Lebegue spaces
 Publ. Math
"... We obtain sharp weighted Lp estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < ∞ the norm of a sublinear operator on Lr (w) is bounded by a function of the Ar characteristic constant of the weight w, the ..."
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Cited by 9 (5 self)
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We obtain sharp weighted Lp estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 < r < ∞ the norm of a sublinear operator on Lr (w) is bounded by a function of the Ar characteristic constant of the weight w, then for p> r it is bounded on Lp (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on Lp (v) by the same increasing function of the r−1 p−1 power of the Ap characteristic constant of v. For some operators these bounds are sharp, but not always. In particular, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Grenoble 57 no 6
, 2007
"... The paper concerns the magnetic Schrödinger operator H(a, V) = ..."
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Cited by 9 (3 self)
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The paper concerns the magnetic Schrödinger operator H(a, V) =
ON THE H 1 –L 1 BOUNDEDNESS OF OPERATORS
"... Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from ..."
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Cited by 9 (0 self)
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Abstract. We prove that if q is in (1, ∞), Y is a Banach space, and T is a linear operator defined on the space of finite linear combinations of (1,q)atoms in Rn with the property that sup{‖Ta‖Y: a is a (1,q)atom} < ∞, then T admits a (unique) continuous extension to a bounded linear operator from H1 (Rn)toY. We show that the same is true if we replace (1,q)atoms by continuous (1, ∞)atoms. This is known to be false for (1, ∞)atoms. 1.
THE REGULARITY AND NEUMANN PROBLEM FOR NONSYMMETRIC ELLIPTIC OPERATORS
, 2006
"... We will consider the Dirichlet problem Lu=0, in Ω ..."
On Mott’s formula for the acconductivity in the Anderson model
"... Olivier Lenoble, and Peter Müller* We study the acconductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the acconductivity is bounded from above by Cν 2 ( ..."
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Cited by 7 (2 self)
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Olivier Lenoble, and Peter Müller* We study the acconductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the acconductivity is bounded from above by Cν 2 (log 1 ν)d+2 at small frequencies ν. This is to be compared to Mott’s formula, which predicts the leading term to be Cν 2 (log 1 ν)d+1.
Strichartz estimates for the magnetic Schrödinger equation
, 2005
"... We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n ≥ 6. ..."
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Cited by 5 (1 self)
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We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n ≥ 6.
Decay estimates for a class of wave equations
, 802
"... Abstract In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by e itφ( √ −∆), where φ: R → R is smooth away from the origin. Especially, the decay estimates for the solutions of the KleinGordon equation and the beam equation are simplified and sl ..."
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Cited by 5 (3 self)
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Abstract In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by e itφ( √ −∆), where φ: R → R is smooth away from the origin. Especially, the decay estimates for the solutions of the KleinGordon equation and the beam equation are simplified and slightly improved.