On the traces of Sobolev functions on Lipschitz surfaces

Аннотация

Functions from the Sobolev spaces W_p¹(Q) are considered on a unit cube Q ⊂ Rⁿ, and the properties of their traces on Lipschitz surfaces are examined. The relation is found between the Hölder exponent α and the Hausdorff dimension of the family of poor k-dimensional planes Γ on which the traces do not belong to C^α(Γ). For the corresponding families of poor k-dimensional Lipschitz surfaces, estimates in terms of p-modules are obtained.