Vol. 2 No. 1 (2009)

An atomic decomposition of variable Besov and Triebel-Lizorkin spaces is given.

We consider Dirichlet type problem in upper half-plane for improperly elliptic equation $u_{z\bar{z}^2}=0$, with boundary functions from the class $L^1(\rho)$ $(\rho=(1+|x|)^{-\alpha}, \alpha\geq0)$. The solutions of the problem are obtained in explicit form.

In this note we derive the Monge-Ampere type equation in Euclidian coordinates describing the refraction phenomena of perfect lens. This simplies the regularity issues of the weak solutions on the problem.