Skip to main content Accesibility Help
×
×
Home

A COMPACTNESS PRINCIPLE FOR MAXIMISING SMOOTH FUNCTIONS OVER TOROIDAL GEODESICS

  • STEFAN STEINERBERGER (a1)
Abstract

Let $f\in C^{2}(\mathbb{T}^{2})$ have mean value 0 and consider

$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D6FE}\,\text{closed geodesic}}\frac{1}{|\unicode[STIX]{x1D6FE}|}\biggl|\int _{\unicode[STIX]{x1D6FE}}f\,d{\mathcal{H}}^{1}\biggr|,\end{eqnarray}$$
where $\unicode[STIX]{x1D6FE}$ ranges over all closed geodesics $\unicode[STIX]{x1D6FE}:\mathbb{S}^{1}\rightarrow \mathbb{T}^{2}$ and $|\unicode[STIX]{x1D6FE}|$ denotes its length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $\unicode[STIX]{x1D6FE}$ attaining the supremum in terms of the smoothness of the function: for all $s\geq 2$ ,
$$\begin{eqnarray}|\unicode[STIX]{x1D6FE}|^{s}{\lesssim}_{s}\biggl(\max _{|\unicode[STIX]{x1D6FC}|=s}\Vert \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}f\Vert _{L^{1}(\mathbb{T}^{2})}\biggr)\Vert \unicode[STIX]{x1D6FB}f\Vert _{L^{2}}\Vert f\Vert _{L^{2}}^{-2}.\end{eqnarray}$$

Copyright
Footnotes
Hide All

This work is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.

Footnotes
References
Hide All
[1] Berger, M., A Panoramic View of Riemannian Geometry (Springer, Berlin, 2003).
[2] Grafakos, L., Classical Fourier Analysis, 2nd edn, Graduate Texts in Mathematics, 249 (Springer, Berlin–New York, 2008).
[3] Klingenberg, W., Lectures on Closed Geodesics, Grundlehren der mathematischen Wissenschaften, 230 (Springer, Berlin–New York, 1978).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed