In this dissertation some new results on the boundedness of Calderón-Zygmund operators on Sobolev spaces on domains in Rd. First a T(P)-theorem is obtained which is valid for Wn,p (U), where U is a bounded uniform domain of Rd, n is a given natural number and p>d. Essentially, the result obtained states that a convolution Calderón-Zygmund operator is bounded on this function space if and only if T(P) belongs to Wn,p (U) for every polynomial P of degree smaller than n restricted to the domain. For indices p less or equal than d, a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of n=1 and p<=d, this Carleson condition is shown to be necessary in fact. The case where n is not integer and 0<n<1 is also studied, and analogous results to the former are obtained for a larger family of function spaces, the so-called Triebel-Lizorkin spaces. The thesis also contains some optimal conditions to establish when the Beurling transform of a polynomial restricted to a domain is contained in the Sobolev space Wn,p(U), where U is a bounded planar Lipschitz domain, in terms of the Besov regularity of the boundary of U. This result, in combination with the results mentioned above, provides an optimal condition to determine wether the Beurling transform is bounded on Wn,p(U) or not in terms of the regularity of the boundary for p>2. Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity Wn,p(U) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain U with boundary parameterizations in a certain Besov space and p>2, implies that the mapping itself is in Wn+1,p(U).
Singular Integral Operators on Sobolev Spaces on Domains and Quasiconformal Mappings
Prats, M. (Author). 16 Oct 2015
Student thesis: Doctoral thesis
Student thesis: Doctoral thesis