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Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ where $M$ is a positive constant number. Does this imply that $f$ is identically zero?

The motivations comes from section 3 page 42 of the following paper, "A Dynamical Approach to Quasi Analytic type Problems

In a similar way we generalize the question as follows:

Let $\Delta$ be the Laplace operator associated to a Riemannian manifold. Assume that $f:M\to \mathbb{R}$ is a smooth function which satisfies $|\Delta(f)(x)|\leq M |f(x)|,\quad \forall x\in M$ where $M$ is positive constant. Does this imply that $f$ is identically zero(at least locally around the point $p$)?

Ali Taghavi
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1 Answers1

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The answer to both questions is yes. For the first, assume only that $f \in C^2$, that $f(0) = f'(0) = 0$, and that $|f''| \leq M|f|$. Let $g(x) = f^2(x) + f'^2(x)$. Then $$g' = 2f'(f''+f) \leq 2(M+1)|ff'| \leq (M+1)g.$$ Hence $(e^{-(M+1)x}g)' \leq 0$, and since $g \geq 0$ and $g(0) = 0$ we conclude that $g = 0$.

This is a very simple form of Carleman estimates, which can be used to show that if $|\Delta f| \leq M|f|$ and $f$ vanishes to infinite order at a point, then $f$ vanishes in a neighborhood of this point. An excellent reference for this is a set of lecture notes by Carlos Kenig, which can be found here.

Connor Mooney
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  • Thank you very much for your very interesting answer. Regarding the first part i think you consider $f:=f(-x)$. Regarding the second part do you think that every arbitrary elliptic operator on a manifold satisfy this vanishing property(aroundflat point)?I did not read the details of the linked paper you provided yet. As another question, let'us consider non elliptic operator associated to derivational operator defined by a vector field. We consider the set of all points$ p\in M$ whose small neighborhoods fulfill the Carleman estimate you mentioned. Do you think that there are some dynamic. – Ali Taghavi May 09 '21 at 16:04
  • interpretation for this set? – Ali Taghavi May 09 '21 at 16:04
  • To be honnest, I was interested in this flat consideration(for derivational operator associated to a vector field since I was commenting on this MO answer https://mathoverflow.net/a/176509/36688 – Ali Taghavi May 09 '21 at 16:11
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    The "strong unique continuation" property holds for operators of the form $Lu := \text{div}(A(x)\nabla u) + b(x)\cdot \nabla u + V(x)u$, provided $A$ is uniformly elliptic and Lipschitz, and $b,,V$ are e.g. bounded. The Lipschitz regularity of $A$ cannot be dropped. See e.g. the work of Garofalo-Lin for an approach to this result based on a monotonicity formula (https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160400305) or the following paper of Koch and Tataru (https://math.berkeley.edu/~tataru/papers/esucp3.pdf) and the references therein for an approach based on Carleman estimates. – Connor Mooney May 09 '21 at 20:33
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    Regarding unique continuation for non-elliptic operators, I am not sure- however, the following expository paper of Tataru may be a useful starting point: https://math.berkeley.edu/~tataru/papers/shortucp.ps – Connor Mooney May 09 '21 at 20:36
  • Thank you again for your interesting answer. For a vector field $X$ on a manifold $M$ , we consider the set $\mathcal{G}(X)$ consisting of all $p \in M$ such that $D(f)=X.f$ is a G.S operator at p(or with your terminology, $D$ satisfies Carleman estimate at $P$. What can be said about topology-dynamical properties of $\mathcal{G}(X)$ ?is it compact when $M$ is compact? Is it closed? is it flow invariant? – Ali Taghavi May 13 '21 at 14:05