1
$\begingroup$

I have posted this problem in MSE long ago: https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this problem not so easy. I would like to describe it again.

Consider the polynomial function in $\mathbb{R}^n$: $$f(x)=\sum_{|\alpha|=1}^{m}c_\alpha x^{\alpha}$$ where $\alpha=(\alpha_1,\cdots,\alpha_n)\in \mathbb{N}^n$ is the multi-index with non-negative integers and $|\alpha|=\alpha_1+\cdots+\alpha_n$.

Problem: For given $p>0$ such that $|\alpha|p<1$ for any $\alpha$, show that in an open bounded domain $D\subset \mathbb{R}^n$ $$\int_{D}\frac{dx}{|f(x)|^p}<+\infty.$$ Example to understand:

(1) $$\int_{(-1,1)^2}\frac{dxdy}{|x^3-xy^3+y^2|^{1/5}}$$

(2) For any $0<\varepsilon<1$ $$\int_{(-1,1)^3}\frac{dxdydz}{|xyz+z^4-y^3x+y^2z|^{(1-\varepsilon)/4}}. $$

Attempt: When the case is special I can show this. For example, if $f(x)=x_1+\sum\limits_{~~~~~~|\alpha|=1\\ \text{no $x_1$ appears}}^{m}c_\alpha x^{\alpha}$, then I can take $f(x)=u_1$ and $x_i=u_i$. The Jacobi is $1$ and we can prove it. But in general it doesn't work.

On the other hand, I have tried the special version of Resolution of Singularities (see, e.g. page 147 in Atiyah's https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.3160230202). But this thoery only makes $f$ become monomial locally with abstract power $\beta_i$ ($g$ is the blow-down map): $$f\circ g(u)=c(u)u_1^{\beta_1}\cdots u_n^{\beta_n}.$$ Since the power $\beta_i$'s are abstract, the condition $|\alpha|p<1$ doesn't work.

I also tried to apply some theorem in real analysis. For example:

Suppose $f\geq 0$ and is finite almost everywhere. Let $E_k=\{x:f(x)>2^k\}$, then $f$ is integrable if and only if $$\sum_{k=-\infty}^{\infty}2^km(E_k)<\infty. $$ One can find this proposition (as an exercise) in Stein and Shakarchi's Real Analysis.

Back to this problem. $E_k=\{x:\frac{1}{|f(x)|^p}>2^k\}=\{x:|f(x)|<2^{-\frac{k}{p}}\}$. But I'm stuck in estimating the Lebesgue measure of $E_k$. Is there any way of reference to finish it?

Or you can give some new ways to show this problem. Thank you.

$\endgroup$

1 Answer 1

1
$\begingroup$

I'm questioner. Now I have found a reference:

Volume estimates of sublevel sets of real polynomials

Theorem 4.1 in it can solve this proplem.

This paper has been published at Annales Polonici Mathematici

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.