Motivated by a theorem of Gordon and Hedenmalm in 1999, the study composition operators acting on various scales function spaces Dirchlet series has arisen intensive interest. In this paper, we characterize boundedness induced specific Dirichlet symbols from Bergman space to Hardy series.

We prove the existence of solutions to Dirichlet problems associated with the p(x)-quasilinear elliptic equation Au = − div a(x, u,∇u) = f(x, u,∇u). These solutions are obtained in Sobolev spaces with variable exponents.

Let ? be a positive finite Borel measure on the unit circle. The associated Dirichlet space D(?) consists of holomorphic functions disc whose derivatives are square integrable when weighted against Poisson integral ?. We give sufficient condition subset E circle which ensures that is uniqueness set for D(?). also some examples measures and sets