  In this paper, we prove a new generalized Mikhlin multiplier theorem whose
conditions are given with respect to fractional derivatives in integral forms
with two different integration intervals. We also discuss the connection
between fractional derivatives and stable processes and prove a version of
Mikhlin theorem under a condition given in terms of the infinitesimal generator
of symmetric stable process. The classical Mikhlin theorem is shown to be a
corollary of this new generalized version in this paper.
