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Geometric majorants for analytic germs

Statement

For any finite family of analytic germs $f_{j}$ in $n \geq 1$ variables at zero there are $M, r > 0$ such that $∣ [z^{α}] f_{j} ∣ \leq M r^{- ∣ α ∣}$ for every $j, α$ . Consequently $f_{j} ≪ M / (1 - (z_{1} + \dots + z_{n}) / r)$ , and $f_{j} - f_{j} (0) ≪ M / (1 - (z_{1} + \dots + z_{n}) / r) - M$ .

Facts & Assumptions

Given: A finite family of analytic germs $f_{j}$ in $n \geq 1$ variables at the origin.

[F1]

Real analytic germs have holomorphic complexifications on a positive polydisc. (Real analytic germs in several variables).

[F2]

Cauchy estimates bound every derivative by its factorial times the boundary supremum and inverse polyradius powers. (Cauchy estimates for mixed derivatives on a polydisc).

Proof

1.1

If the family is empty take $M = r = 1$ . Otherwise F1 gives a common complex polydisc of positive polyradius $R$ ; take $r = \frac{1}{2} \min_{i} R_{i}$ . The finitely many complexifications are continuous on the compact distinguished boundary of the $r$ -polydisc. Let $M$ be the larger of $1$ and their finitely many boundary suprema. F2 and $[z^{α}] f_{j} = D^{α} f_{j} (0) / α!$ give the asserted bound.

givenF1F2

2.1

The coefficient of $z^{α}$ in $M \sum_{q \geq 0} ((z_{1} + \dots + z_{n}) / r)^{q}$ is $M r^{- ∣ α ∣} ∣ α ∣! / α!$ . The multinomial factor counts arrangements of a multiset and is at least one, including $α = 0$ . Hence it dominates the bound in step 1.1. Subtracting the respective constants leaves coefficient zero at degree zero and the same inequality in positive degrees, proving the second comparison.

step 1.1algebra∎

Source notes

Gantumur, §3 equations (29)–(30), printed pp. 7–8; local proof uses polydisc Cauchy estimates.

Depends on

Used by

A concrete geometric majorant Example
Analytic ODE systems from majorants Lemma
Positive majorants dominate the Cauchy recursion Lemma

Dependency tree · two levels

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Sources

Gantumur, Math 580 Lecture Notes 2: The Cauchy-Kovalevskaya Theorem (standard reference, not scraped)