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I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value of the polynomial calculated on the surface.

More precisely: suppose $p$ a polynomial of $d$ real variables of fixed total degree $n$, $M$ a compact surface in $\mathbb{R}^{d}$ (as smooth and good as we need), i'm lookin for something near to $${\| \partial_i p\|}_{M+B(0,\epsilon)} \leq \text{Something}(n,d, \epsilon) \|p \|_M $$ where $M+B(0,\epsilon)$ means an $\epsilon$-neighborhood of $M$ and $\|f\|_K$ is the supnorm of $f$ on $K$.

There is a tangential version of Markov inequality... seems to me to not suffice...

Thank you for any suggestions!

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  • $\begingroup$ So $M$ is compact and $M+B(0,\epsilon)$ is some $\epsilon$-neighborhood of $M$? $\endgroup$
    – J.C. Ottem
    Jan 11, 2011 at 0:12
  • $\begingroup$ I think the OP meant to use the word estimate, not extimate, in the title. Perhaps someone with editing clout could fix it? $\endgroup$
    – drbobmeister
    Jan 11, 2011 at 0:57
  • $\begingroup$ Federico: I've taken the liberty of editing your question to include your updates, and in passing to correct some spelling. Hope you don't mind. $\endgroup$
    – Yemon Choi
    Jan 11, 2011 at 20:04
  • $\begingroup$ May be this question is somehow related: mathoverflow.net/questions/12298/… $\endgroup$
    – Nurdin Takenov
    Jan 12, 2011 at 0:11

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Ok, my original question was not so precisely posed..

$Something(n,d,\epsilon)$ depends of course on M too!

I know (the existence of) that article because before asking here i try to research some statements in google and mathscinet but without any positive result.

The problem is not the verse of inequality: it is the classical one, think to Markov Bernstein and Cauchy's inequalities. The (big?) problem is dimensional: for (good) full dimensional compact subset of $ \mathbb{R}^n $ and $\mathbb{C}^n$ too one can use Markov multivariate Inequality or Bernstein one. For a smooth compact manifold there is an equivalent statement, but it is only for partial derivatives with respect to tangential directions. Moving along tangential direction you obviously can't obtain a full neiborhood of manifold, hence i can't provide inequality for all the points that i need.

Perhaps a domination of all derivatives by value of polynomial on the manifold is impossible because of the nature of the problem...could be?

PS now that we focused on another error in my original question, how the right behaviour is like? Should i edit my first post and fix or not?

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    $\begingroup$ can you make your statement more precise please? do you require p to be homogenuous? what is wrong with $p(x_1,\dots,x_d)=x_1^2+…+x_d^2−1$ and $M=\mathbb{S}^{d-1}$? $\endgroup$
    – Olivier Bégassat
    Mar 24, 2011 at 4:02

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