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We investigate local properties of the Green function of the complement of a compact set$E$υ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density of$E$at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..

Let$D$be a convex domain with smooth boundary in complex space and let$f$be a continuous function on the boundary of$D$. Suppose that$f$holomorphically extends to the extremal discs tangent to a convex subdomain of$D$. We prove that f holomorphically extends to$D$. The result partially answers a conjecture by Globevnik and Stout of 1991.

In this paper we consider operators of the form$H$=λ(-$i$∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in$L$^{2}(ℝ^{$n$}) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.