Characterizing regularity of domains via Riesz transforms on their boundaries
Abstract
Given a domain D in R^d with mild geometric measure theoretic assumptions on its boundary, we show that boundedness of the principal value Riesz tranforms (witn kernel of homogeneity (d1)) on Hölder spaces of order alpha on the boundary of D is equivalent to D being a Lyapunov domain of order alpha (i.e., the boundary of D is an hypersurface of class 1+alpha). Another equivalent condition involving Riesz transforms on D is discussed. We also prove that on Lyapunov domains of order alpha the higher order Riesz transforms associated with an odd polynomial are bounded on the Hölder space of order alpha on the boundary of D. Finally, a limiting case of the above results dealing with VMO and SemmesKenigToro domains is considered.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2444
 Bibcode:
 2014arXiv1410.2444M
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Primary 42B20;
 Secondary 15A66;
 42B37;
 35J15
 EPrint:
 55 pages, exposition improved, references added