Riesz summability of multiple Hermite series inL p spaces

Karadzhov, Georgi E.

doi:10.1007/bf02572353

Cited by 29 publications

(48 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This leads to an alternative proof of the eigenfunction bounds of Karadzhov [7] and Thangavelu [13]. Our approach has the advantage that is robust enough so that it allows us to obtain the same results with x 2 replaced by potentials in a very large class.…”

Section: Introductionmentioning

confidence: 79%

See 1 more Smart Citation

L p eigenfunction bounds for the Hermite operator

Koch¹,

Tataru²

2005

Duke Math. J.

104

122

View full text Add to dashboard Cite

Abstract. We obtain L p eigenfunction bounds for the harmonic oscillator H = −∆+x 2 in R n and for other related operators, improving earlier results of Thangavelu and Karadzhov. We also construct suitable counterexamples which show that our estimates are sharp. IntroductionThe question of obtaining L p eigenfunction bounds for elliptic operators on compact manifolds has been considered in Sogge's work, for which we refer the reader to his book [11]. The L p eigenfunction bounds in [11] are sharp, and turn out to be related to a variable coefficient version of the restriction theorem, and further to a phase curvature condition for Fourier integral operators. In this analysis a special role is played by the Laplace-Beltrami operator for the sphere, which is the worst case because it has many highly concentrated eigenfunctions. This is connected to the fact that it has a periodic Hamilton flow.In this article we consider the problem of obtaining L p eigenfunction bounds for the Hermite operator H = −∆ + x 2 in R n and also for a larger class of related operators of the form H V = −∆ + V . Within this class the Hermite operator plays a role similar to that of the spherical Laplacian, in that it has a periodic Hamilton flow and many highly concentrated eigenfunctions.This question has received considerable interest in the context of Riesz summability for the harmonic oscillator in the work of Thangavelu [12], [14], [13] and Karadzhov [7].Our interest in it has a different source, namely the strong unique continuation problem for parabolic equations. In this context the work of Thangavelu and Karadzhov has already found applications in Escauriaza [2] and EscauriazaVega [3]. Further applications are contained in a forthcoming paper of the authors. We note in passing that the related strong unique continuation problem for second order elliptic operators is related to the eigenfunction bounds for the spherical harmonics. This was first observed in work of Jerison [6]; see also the authors paper [10] and further references therein.2000 Mathematics Subject Classification. 35S05, 35B60.

show abstract

Section: Introductionmentioning

confidence: 79%

“…This question has received considerable interest in the context of Riesz summability for the harmonic oscillator in the work of Thangavelu [12], [14], [13] and Karadzhov [7].…”

Section: Introductionmentioning

confidence: 99%

L p eigenfunction bounds for the Hermite operator

Koch¹,

Tataru²

2005

Duke Math. J.

104

122

View full text Add to dashboard Cite

show abstract

“…In the simplest form, these have the form t −τ − The L p spectral projection bounds for the Hermite operator were independently obtained by Thangavelu [28] and Kharazdhov [16]; see also the simplified proof in the authors's paper [19]. These bounds were essential in the proof of L p Carleman inequalities for the heat operator of Escauriaza [8] and Escauriaza and Vega [10] which yield SUCP(I) when g = I n , W = 0 and…”

Section: Sucp(ii)mentioning

confidence: 98%

Carleman Estimates and Unique Continuation for Second Order Parabolic Equations with Nonsmooth Coefficients

Koch

Tataru

2009

Communications in Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…The main tool used in one of the end-points of the analytic interpolation is the following restriction theorem for the projections associated to Hermite functions that was first proved by Karadzhov [21] (see also [38, the remark to Proposition 1]).…”

Section: Proof Of the Estimatesmentioning

confidence: 99%

Carleman inequalities and the heat operator

Escauriaza¹

2000

Duke Math. J.

107

View full text Add to dashboard Cite

1. Introduction. The unique continuation property is best understood for secondorder elliptic operators. The classic paper by Carleman [8] established the strong unique continuation theorem for second-order elliptic operators that need not have analytic coefficients. The powerful technique he used, the so-called "Carleman weighted inequality," has played a central role in later developments. In the 1950s, Aronszajn [3] and Cordes [11] generalized Carleman's result to higher dimensions. In recent years, this subject attracted attention from a great number of people. Many efforts have been made to relax the smoothness hypothesis on the coefficients (see [7], [15], [31], [2], and [16]). Using this method, Jerison and Kenig [19] obtained the strong unique continuation property for operators of the form + V with V ∈ L

show abstract

Riesz summability of multiple Hermite series inL p spaces

Cited by 29 publications

References 8 publications

L p eigenfunction bounds for the Hermite operator

L p eigenfunction bounds for the Hermite operator

Carleman Estimates and Unique Continuation for Second Order Parabolic Equations with Nonsmooth Coefficients

Carleman inequalities and the heat operator

Contact Info

Product

Resources

About