zbMATH — the first resource for mathematics

A local Bernstein inequality on real algebraic varieties. (English) Zbl 0911.32011
Let \(S\) be the set of singular points of a \(d\)-dimensional real algebraic subset \(X\) of \(\mathbb{R}^N\), and let \(U\) be an open and relatively compact subset of \(X-S\). Let \(R\) be a positive real number, and let \(B_R\) be the open ball of radius \(R\) centered at the origin of \(\mathbb{R}^d\). Let \(\varphi: U\times B_r\to X\) be a real analytic map. For each point \(x\in U\), let \(\varphi_x: B_R\to X\) be defined as \(\varphi_x(y)=\varphi(x,y)\). Assume that \(\varphi_x(0)=0\) and \(\varphi_x\) is an analytic isomorphism of \(B_R\) onto its image \(B_R(x)\), which is assumed to be an open subset of \(X-S\).
The aim of the paper under review is to prove the following inequalities, known as the doubling and the Bernstein inequalities: Given a compact subset \(K\) of \(U\), there exist real positive numbers \(C\) and \(r_0\) such that if \(x\in K\) and \(F\) is a polynomial \(F\in\mathbb{R}[X_1,\dots,X_N]\) of degree \(D\), then for every \(0<r<r_0\), \[ \sup\biggl\{\bigl| F\varphi_x(y) \bigr|\mid y\in B_{2r}(x)\biggr\}\leq e^{CD}\sup\biggl\{\bigl| F\varphi_x(y)\bigr |\mid y\in B_r(x) \biggr\}, \tag{1} \] \[ \sup\biggl\{\bigl|\nabla F\varphi_x(y)\bigr|\mid y\in B_r(x) \biggr\}\leq{CD^2\over r}\sup \biggl\{\bigl| F\varphi_x(y)\bigr|\mid y\in B_r(x)\biggr\}.\tag{2} \] It must be pointed out that these inequalities are uniform for \(x\in K\) and optimal in their dependence on \(D\) and \(r\), except for the dependence of \(C\) and \(r_0\) on the set \(X\).
The proof is extremely technical, and so it is a great merit to have written it so transparently. It consists in first proving an analogue of the doubling inequality for complex algebraic varieties and combine it with two clever lemmas on functions of one complex variable.

32C07 Real-analytic sets, complex Nash functions
14P05 Real algebraic sets
32B20 Semi-analytic sets, subanalytic sets, and generalizations
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI EuDML
[1] L. Bos, N. Levenberg, P. Milman and B.A. Taylor. Tangential Markov inequalities characterize algebraic submanifolds of \(\mathbb{R}\) N . Indiana Univ. Math. J.44 (1995), 115–137. · Zbl 0824.41015 · doi:10.1512/iumj.1995.44.1980
[2] L. Bos, P. Milman. Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains. Geometric and Functional Analysis5 (1995), 853–923. · Zbl 0848.46022 · doi:10.1007/BF01902214
[3] H. Donnelly, C. Fefferman. Nodal sets of eigenfunctions on Riemannian manifolds. Inventiones Math.93 (1988), 161–183. · Zbl 0659.58047 · doi:10.1007/BF01393691
[4] C. Fefferman, R. Narasimhan. On the polynomial-like behavior of certain algebraic functions. Annales Inst. Fourier44 (1994), 1091–1179. · Zbl 0811.14046
[5] C. Fefferman, R. Narasimhan. Bernstein’s inequality and the resolution of spaces of analytic functions. Duke Math. Journal81 (1995), 77–98. · Zbl 0854.32006 · doi:10.1215/S0012-7094-95-08108-3
[6] A. Sadullaev. An estimate for polynomials on analytic sets. Math. USSR. Izv.20 (1983), 493–502. · Zbl 0582.32023 · doi:10.1070/IM1983v020n03ABEH001612
[7] G. Szegö. Über einen Satz von A. Markoff. Math. Zeit.23 (1925), 45–61. · JFM 51.0097.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.