Bernstein's inequality and the resolution of spaces of analytic functions
In Ann. Inst. Fourier 44, No. 4, 1091-1179 (1994; Zbl 0811.14046) the authors proved an extension theorem with bounds for polynomial functions in algebraic subvarieties of \(\mathbb{R}^n\) to rational functions in a neighborhood of the variety. Correspondingly, one obtains a priori bounds not just on the values of the extensions but also on their derivatives. For instance, one wants \[ \sup_{B (x, \rho)} |DF |\leq {C \over \rho} \sup_{B (x, \rho)} |F |\tag{*} \] where \(B(x, \rho)\) is the open ball of center \(x\) and radius \(\rho\) in \(\mathbb{R}^n\). When \(F\) is a polynomial in \(\mathbb{R}^n\), \(C\) depends only on the degree of \(F\), as shown by a classical theorem of S. Bernstein. Moreover, the algebraicity of the variety is tied to the existence of such bounds [\textit{L. Bos}, \textit{N. Levenberg}, \textit{P. Milman} and \textit{B. A. Taylor}, Indiana Univ. Math. J. 44, No. 1, 115-138 (1995; Zbl 0824.41015)]. The difficulty consists in obtaining some control of the constant \(C\) when \(F\) is only locally defined. In this paper the authors prove the following: Theorem. Let \(F_{1, \lambda}, \dots, F_{N, \lambda}\) be holomorphic functions in the complex ball \(B_c (0,1 + \varepsilon)\) of \(\mathbb{C}^n\), \(\varepsilon > 0\), depending real-analytically on \(\lambda \in U\), \(U\) open subset of \(\mathbb{R}^m\). Let \(V_\lambda = \text{span} \{F_{i, \lambda} : 1 \leq i \leq N\}\). Then, for any compact \(K \subseteq U\) there is a constant \(C\) such that for any \(\lambda \in K\), \(f \in V_\lambda\) and \(B(x, \rho) \subseteq B (0,1)\), the estimate (*) holds. The key technical lemma is a statement that appears naturally when one tries to relate the concept of multiplicity in the algebraic sense to that of counting roots of a system of equations [e.g. \textit{J. W. Milnor}, `Singular points of complex hypersurfaces' (1968; Zbl 0184.48405) and \textit{E. Bierstone} and \textit{P. Milman}, Publ. Math., Inst. Hautes Etud. Sci. 67, 5-42 (1988; Zbl 0674.32002)], namely; given a family \(F_{i, \lambda}\) as above, find a similar family \(G_{j, \lambda}\), \(1 \leq j \leq M\), such that (a) for every \(\lambda \in U\), \(\dim \text{span} \{G_{j, \lambda}\} = M\); (b) \(\text{span} \{F_{i, \lambda} \subseteq \text{span} \{G_{j, \lambda}\} \forall \lambda\;in U\); (c) there is a nowhere dense analytic subset \(A\) of \(U\) such that for all \(\lambda \in U \backslash A\), \(\text{span} \{F_{i, \lambda}\} = \text{span} \{G_{j, \lambda}\}\).
scientific article; zbMATH DE number 880072