Gauged Sobolev inequalities

Abstract

We present two-scale Morrey–Sobolev inequalities for measure-valued Lagrangeans on quasi-metric balls, scaled according to refined power laws. The fine tuning is given by suitable gauge functions, typically of logarithmic type. Fractal examples with fluctuating geometry are described.

†Dedicated to Ebba Swantje

Keywords:

• Sobolev inequalities
• spaces of homogeneous type
• quasimetric spaces
• fractals

Keywords:

• AMS: Primary 46E35
• 46E30
• Secondary 43A85
• 43A15

Acknowledgement

The author wishes to thank Professor Maria Agostina Vivaldi for helpful discussions and useful comments on this work.

Notes

†Dedicated to Ebba Swantje