Maximal inequalities and Riesz transform estimates on L^p spaces for Schrödinger operators with nonnegative potentials

Auscher, Pascal; Ali, Besma Ben

doi:10.5802/aif.2320

Cited by 96 publications

(144 citation statements)

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“…We note for instance the boundedness of Riesz transform on L p for p ∈ (1, 2] if V ≥ 0 and V ∈ L 1 loc (R n ) (see E.M. Ouhabaz [19]), which means again a quite strong stability for low range of p. A similar result is obtained by Coulhon-Duong [6] for the Laplacian on a class of manifolds. P. Auscher and A. Ben Ali [2] improved a result of Z. Shen [20] on higher ranges of p, this is when the potential is non-negative and satisfy the q reverse Hölder inequality for some q ∈ (1, +∞], indeed they prove in this setting that there is ǫ > 0 such that T ∈ L(L p ) if p ∈ (1, q+ǫ) (Z. Shen's restriction was n/2 ≤ q < n). We remark that powers |z| −α satisfy the reverse Hölder inequality if α ∈ (−∞, n/q).…”

Section: Introductionmentioning

confidence: 99%

“…Let ∆ g be the positive Laplacian on M with respect to g. Then the Schrödinger operator P = ∆ g + V is self-adjoint on L 2 (M, dg) and its spectrum is σ(P ) = [0, ∞) ∪ σ pp (P ) where σ pp (P ) = {−k 2 1 ≥ · · · ≥ −k 2 N } is a finite set of negative eigenvalues (by convention k i > 0). Note that 0 can be an L 2 -eigenvalue but the half-line (0, ∞) is only continuous spectrum.…”

Section: Introductionmentioning

confidence: 99%

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Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

2008

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Abstract. Let M • be a complete noncompact manifold and g an asymptotically conic manifold on M • , in the sense that M • compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M . A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S n−1 ; such manifolds have an end that can be identified with R n \ B(R, 0) in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k 2 ) −1 where P = ∆g + V is the sum of the positive Laplacian associated to g and a real potential function V that is smooth on M and vanishes to third order at the boundary (i.e. decays to third order at infinity on M • ). We show that on a blown up version of M 2 × [0, k 0 ] the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on L p (M • ) for 1 < p < n, and that this range is optimal if V ≡ 0 or if M has more than one end. The result with V ≡ 0 is new even when M • = R n , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 < p < pmax where pmax > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary ∂M .Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In a follow-up paper we shall analyze the same situation in the presence of zero modes and zero-resonances.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

2008

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“…The proof is completely analogous to the one in R n ( see [22], [2]): Lemma 3.1. (Fefferman-Phong inequality).…”

Section: Principal Toolsmentioning

confidence: 99%

“…(see section 11 in [2], [19]) Let V be a non-negative measurable function. Then the following properties are equivalent:…”

Section: Reverse Hölder Classesmentioning

confidence: 99%

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Real Interpolation of Sobolev Spaces Associated to a Weight

Badr

2009

Potential Anal

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Hardy spaces for Bessel–Schrödinger operators

Kania

Preisner

2018

Mathematische Nachrichten

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Consider the Bessel operator with a potential on L2false((0,∞),xα0.16emdxfalse), namely Lffalse(xfalse)=−f′′false(xfalse)−αxf′false(xfalse)+Vfalse(xfalse)ffalse(xfalse).We assume that α>0 and V∈Lloc1false((0,∞),xα0.16emdxfalse) is a nonnegative function. By definition, a function f∈L1false((0,∞),xα0.16emdxfalse) belongs to the Hardy space scriptH1false(boldLfalse) if trueprefixsupt>0e−tboldLf∈L1false((0,∞),xα0.16emdxfalse).Under certain assumptions on V we characterize the space scriptH1false(boldLfalse) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to boldL for α∈(0,1) with no additional assumptions on the potential V.

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Maximal inequalities and Riesz transform estimates on L^p spaces for Schrödinger operators with nonnegative potentials

Cited by 96 publications

References 28 publications

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

Real Interpolation of Sobolev Spaces Associated to a Weight

Hardy spaces for Bessel–Schrödinger operators

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