Sobolev Trace Theorem and the Dirichlet Problem in the Unit Disk

We give a direct and elementary proof for the trace theorem in L _p -based Sobolev spaces, when the domain is the unit disk. We also consider the Dirichlet boundary problem for the Laplace equation, where the boundary value is a function in the Besov space. The Poisson kernel enables us to solve this problem in the unit disk more easily than in a general domain.