Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-67gxp Total loading time: 0.278 Render date: 2021-03-06T06:30:02.254Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Spectral Theory of the Hermite Operator on Lp(Rn)

Published online by Cambridge University Press:  17 July 2014

Corresponding
E-mail address:
Get access

Abstract

We prove that the minimal operator and the maximal operator of the Hermite operator are the same on Lp(ℝn) , 4 / 3 <p< 4. The domain and the spectrum of the minimal operator (=maximal operator) of the Hermite operator on Lp(ℝn), 4/3 <p< 4, are computed. In addition, we can give an estimate for the Lp-norm of the solution to the initial value problem for the heat equation governed by the minimal (maximal) operator for 4/3<p<4.

Type
Research Article
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below.

References

Askey, R. A., Wainger, S., Mean convergence of expansions in Laguerre and Hermite series. Amer. J. Math., 87 (1965), 695708. CrossRefGoogle Scholar
Catană, V.. The heat equation for the generalized Hermite and the generalized Landau operators. Integral Equations Operator Theory, 66 (2010), 4152. CrossRefGoogle Scholar
V. Catană. The heat kernel and Green function of the generalized Hermite operator, and the abstract Cauchy problem for the abstract Hermite operator, in Pseudo-Differential Operators: Analysis, Applications and Computations,. Operator Theory: Advances and Applications 213 (2011), 155–171.
X. Duan. The heat kernel and Green function of the sub-Laplacian on the Heisenberg group, in Pseudo-Differential Operators, Generalized Functions and Asymptotics, 231 (2013), 55–75.
M. Reed, B. Simon. Fourier Analysis, Self-Adjointness. Academic Press, 1975.
Simon, B.. Distributions and their Hermite expansions. J. Math. Phys., 12 (1970), 140148. CrossRefGoogle Scholar
E. M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, 1993.
Taylor, A. E.. Spectral theory of the closed distributive operators. Acta Math., 84 (1951), 189224. CrossRefGoogle Scholar
A. E. Taylor, D. Lay. Introduction to Functional Analysis. Second Edition, Wiley, 1980.
Wong, M. W.. The heat equation for the Hermite operator on the Heisenberg group. Hokkaido Math. J., 34 (2005), 393404. CrossRefGoogle Scholar
Wong, M. W.. Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator. Ann. Global Anal. Geom., 28 (2005), 271283. CrossRefGoogle Scholar
M. W. Wong. An Introduction to Pseudo-Differential Operators. Second Edition, World Scientific, 1999.
M. W. Wong. Partial Differential Equations: Topics in Fourier Analysis. CRC Press, 2014.

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 16 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 6th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Spectral Theory of the Hermite Operator on Lp(Rn)
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Spectral Theory of the Hermite Operator on Lp(Rn)
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Spectral Theory of the Hermite Operator on Lp(Rn)
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *