Approximation of Subharmonic Functions in the Unit Disk

Анотація

We prove that if $u$ is a subharmonic function in $\mathbb{D}=\{|z|<1\}$, then there exists an absolute constant $C$ and an analytic function $f$ in  $\mathbb{D}$  such that $\int_{\mathbb{D}} |u(z)-\log|f(z)||dm(z)<C$, where $m$ denotes the plane Lebesgue measure. We also (following the arguments of Lyubarskii and Malinnikova) answer Sodin's question, namely, we show that the logarithmic potential of measure $\mu$ supported in a square $Q$, with $\mu(Q)$ being an integer $N$, admits approximations by the subharmonic function $\log|P(z)|$, where $P$ is a polynomial with $\int_{Q} |\mathcal{U}_{\mu}(z)-\log|P(z)||dxdy=O(1)$, independent of $N$ and $\mu$. We also consider uniform approximations.

Mathematics Subject Classification: 31A05, 30E10.