Wave equation and multiplier estimates on ax + b groups

Detlef Müller; Christoph Thiele

Studia Mathematica (2007)

  • Volume: 179, Issue: 2, page 117-148
  • ISSN: 0039-3223

Abstract

top
Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e i t √ L ψ ( √ L / λ ) for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev estimates for solutions to the wave equation associated to L. There appears no dispersive effect with respect to the L ∞ -norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.

How to cite

top

Detlef Müller, and Christoph Thiele. "Wave equation and multiplier estimates on ax + b groups." Studia Mathematica 179.2 (2007): 117-148. <http://eudml.org/doc/284937>.

@article{DetlefMüller2007,
abstract = {Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $e^\{it√L\} ψ(√L/λ)$ for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev estimates for solutions to the wave equation associated to L. There appears no dispersive effect with respect to the $L^\{∞\}$-norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.},
author = {Detlef Müller, Christoph Thiele},
journal = {Studia Mathematica},
keywords = {affine group; wave equation; spectral multiplier},
language = {eng},
number = {2},
pages = {117-148},
title = {Wave equation and multiplier estimates on ax + b groups},
url = {http://eudml.org/doc/284937},
volume = {179},
year = {2007},
}

TY - JOUR
AU - Detlef Müller
AU - Christoph Thiele
TI - Wave equation and multiplier estimates on ax + b groups
JO - Studia Mathematica
PY - 2007
VL - 179
IS - 2
SP - 117
EP - 148
AB - Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form $e^{it√L} ψ(√L/λ)$ for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev estimates for solutions to the wave equation associated to L. There appears no dispersive effect with respect to the $L^{∞}$-norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.
LA - eng
KW - affine group; wave equation; spectral multiplier
UR - http://eudml.org/doc/284937
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.