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
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