Prove the integral of multi-variable rational fraction is convergent

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.

1 Answer

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