On the Hausdorff Dimension of $n\times m$ Concentration Sets

Michael Bildhauer

Abstract

The Hausdorff dimension of $n\times m$ concentration sets is studied. Uniform dimension estimates at each cross section in ${\ifmmode{I\hskip -3pt R} \else{\hbox{$I\hskip -3pt R\;$}}\fi }^{n}$ and Hölder continuity in ${\ifmmode{I\hskip -3pt R} \else{\hbox{$I\hskip -3pt R\;$}}\fi }^{m}$ are used to deduce dimension estimates in ${\ifmmode{I\hskip -3pt R} \else{\hbox{$I\hskip -3pt R\;$}}\fi }^{n+m}$.

AMS Subject Classification: 35D99, 76C99.

• Introduction
• Hausdorff Measures
• Space-Time Concentration Sets
• On the Assumptions of the Theorem
• Two-Dimensional Fluid Flow
• Examples