Holomorphic L2 mean-value estimate

Statement

Let $U\subseteq\mathbb C^d$ be open and $f:U\to\mathbb C$ holomorphic. If $B_r(z)\subset U$ , then

|f(z)|\le C_d r^{-d}\|f\|_{L^2(B_r(z))}.

Proof

The function $|f|^2$ is plurisubharmonic, hence subharmonic as a function on $\mathbb R^{2d}$ . Its ball mean-value inequality gives

|f(z)|^2\le |B_r|^{-1}\int_{B_r(z)}|f(w)|^2\,dw.

Since $|B_r|\asymp_d r^{2d}$ , taking square roots proves the estimate.

Use

The estimate converts the weighted $L^2$ control supplied by the Hörmander theorem into pointwise upper bounds and local nonvanishing bounds.

References

Lars Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990.