Anisotropic fractional Sobolev norms

Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev s  -seminorm of a function f∈W1,p(Rn)f∈W1,p(Rn) converges to the Sobolev seminorm of f   as s→1−s→1−. The anisotropic s-seminorms of f   defined by a norm on RnRn with unit ball K are shown to converge to the anisotropic Sobolev seminorm of f   defined by the norm with unit ball Zp⁎K, the polar LpLp moment body of K  . The limiting behavior for s→0+s→0+ is also determined (extending results by Mazʼya & Shaposhnikova).