No abstract uploaded!

No abstract uploaded!

A function$G$in a Bergman space$A$^{$p$}, 0<$p$<∞, in the unit disk$D$is called$A$^{$p$}-inner if |$G$|^{$p$}−1 annihilates all bounded harmonic functions in$D$. Extending a recent result by Hedenmalm for$p$=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ_{$D$}denotes the characteristic function of$D$and$G$is an arbitrary$A$^{$p$}-inner function, is continuous in$C$, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman space$A$^{2}to a general$A$^{$p$}-setting.

Let$f$be an entire function of order at least 1/2,$M(r)$=max_{|}$z$|=$r$|$f(z)$|, and$n(r, a)$the number of zeros of$f(z)-a$in |$z$|≤$r$. It is shown that lim sup_{$r$→∞}$n(r, a)$/log$M (r)$≥1/2π for all except possibly one$a$∈C.

We establish several conditions, sufficient for a set to be (quasi)conformally removable, a property important in holomorphic dynamics. This is accomplished by proving removability theorems for Sobolev spaces in$R$^{$n$}. The resulting conditions are close to optimal.