  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 -(d-1)) on H\"older 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 H\"older space of order alpha on the
boundary of D. Finally, a limiting case of the above results dealing with VMO
and Semmes-Kenig-Toro domains is considered.
