First take an integer \(k\geq 0\) and \(a>0\). Put \(c=a/4\), choose an integer \(M\geq k/2\), and set \(C=\max (1,M!c^{-M})\). For \(|x|\geq 1\), the exponential series gives \(e^{cx^2}\geq (cx^2)^M/M!\), hence \[|x|^k e^{-a x^2/2}\leq M!c^{-M} e^{-a x^2/4}.\] For \(|x|\leq 1\), the left side is at most \(e^{-a x^2/2}\), which is at most \(e^{-a x^2/4}\). The global bound therefore holds with \(C\) and includes \(k=M=0\). Its right side is integrable by the scalar lemma with parameter \(a/2\).

In an orthonormal eigenchart, write \[\begin {aligned} P(x)&=\sum _\alpha c_\alpha \prod _i y_i^{\alpha _i},\\ w_Q(x)&=\prod _i e^{-q_i y_i^2/2}. \end {aligned}\] There are finitely many nonnegative-integer multiindices \(\alpha \). Nonnegative Tonelli factors the weighted absolute integral of each monomial into finite scalar integrals just proved. The triangle inequality bounds the polynomial integral by the finite sum of these integrals multiplied by \(|c_\alpha |\). Products and powers remain polynomials, so the same argument applies. In dimension zero each function is constant and its integral is its absolute value at zero.