The sharp radius in Korenblum's maximum principle for the Fock space

Abstract

Korenblum's maximum principle states that if $|f| \le |g|$ near the boundary, then $\| f \| \le \| g \|$. The optimal size of the region of domination has been studied in Bergman spaces since 1991 and in Fock spaces since 2006, but only bounds have been obtained. We determine the optimal size for the Fock space $F^2$ of entire functions that are square integrable with respect to $e^{-|z|^2}\,dA(z)$: if $f, g\in F^2$ and $|f(z) \le |g(z)|$ for all $|z|>1$, then $\| f \| \le \| g \|$. The radius $1$ is optimal and we determine all pairs for which equality holds. To our knowledge, this is the first space of Bergman or Fock type in which the optimal radius in Korenblum's principle has been determined. The proof combines a Schwarz-Pick estimate centred at infinity with a moment-duality argument, and requires no numerical computation. The same method gives a moment criterion for weighted Fock-type spaces. For every non-increasing radial weight, the optimal radius equals the upper bound given by the pairs $f\equiv c$, $g(z)=z$. In particular, the upper bounds of Wee and Le for such weighted Fock spaces are sharp in the Hilbert space case. We also show that the optimal radius is $\sqrt{(β+1)/α}$ for the weight $|z|^{2β}e^{-α|z|^2}$, whenever $-1<β\le5.44$.