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This might be very trivial, or not.

Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there exists $s\geq 0$ such that

\begin{equation} \int_{\mathbb{R}^n}|x|^s e^{p(x)}\, dx<\infty. \end{equation}

The question is if it is then possible to find $\varepsilon>0$ such that \begin{equation} \int_{\mathbb{R}^n}|x|^{s+\varepsilon} e^{p(x)}\, dx<\infty. \end{equation}

In other words: if the set of $s\geq 0$ such that $$ \int_{\mathbb{R}^n}|x|^s e^{p(x)}\, dx<\infty $$ is non empty, is it also open?

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  • $\begingroup$ The degree restriction seems totally useless because we can always add a few squares of dummy variables to $-p$ after which the whole game will still be played in the cylinder where the sum of squares of the dummy variables is less than $1$, say, i.e. in the original space. $\endgroup$
    – fedja
    Aug 25, 2016 at 1:34

1 Answer 1

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In hyperspherical coordinates, we carry out the angular integrations and are left with the radial integral, $$\int_{\mathbb{R}^n}|x|^{s+\varepsilon} e^{p(x)}\, dx=\int_0^\infty f(r) r^{n-1+s+\epsilon}\,dr,$$ with $f(r)$ the angular average of the positive function $e^{p(x)}$, so $f(r)\geq 0$ for all $r\geq 0$.

What I would now need is to establish that $f(r)$ for large $r$ decays as $r^{-\alpha} e^{-\beta r^\gamma}$, with $\alpha,\beta,\gamma \geq 0$ -- without any logarithmic factors. If $\beta$ and $\gamma$ are both $\neq 0$, the integral converges for any $s,\epsilon\geq 0$. Otherwise we have $\alpha>n+s$ if the integral converges for some $s\geq 0$, and then it will also converge for $\epsilon=(\alpha-n-s)/2>0$.

It seems natural that the angular average of $e^{p(x)}$ will not produce $\log r$ terms, but I'm not sure how to formalize this.

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