Weighted monomial integrals and monomial norms for the disc, ball and polydisc

Given: An integer $m\ge 1$, the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, and a multi-index $\alpha =({\alpha }_0,\dots ,{\alpha }_{m-1})\in {\mathbb{N}}^m$. Write $\alpha !:={\prod }_{j<m}{\alpha }_j!$, $\mathrm{\mid }\alpha \mathrm{\mid }:={\sum }_{j<m}{\alpha }_j$, and $z^{\alpha }:={\prod }_{j<m}{z}_j^{{\alpha }_j}$ using the conventions of $C^k$ maps and multi-index derivative notation in Euclidean space and Integer powers in the complex field.

For $d\ge 1$, $\beta \in {\mathbb{N}}^d$ and $t\in \mathbb{N}$, write $I_d(\beta ,t):={\int }_{{\mathbb{B}}^d}\mathrm{\mid }{z}^{\beta }{\mathrm{\mid }}^2(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{t}d{\lambda }_{2d}(z)$.

The only choice principle assumed is ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). It is required by the polar-coordinate, chart/polar surface, sigma-finiteness and product-Lebesgue suppliers below; no full Axiom of Choice or arbitrary-index selection is used.

The complex-real identification and Euclidean norm are those of Complex $m$-space and its real coordinate dictionary, and the open unit ball ${\mathbb{B}}^m$ and open unit polydisc ${\mathbb{D}}^m$ are those of Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$.

For every nonnegative Borel function on ${\mathbb{R}}^n$, polar coordinates give $\int fd{\lambda }_n={\int }_0^{\mathrm{\infty }}{\int }_{S^{n-1}}f(r\omega )r^{n-1}d\sigma (\omega )dr$ (The polar surface set function on the unit sphere, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

On each angular chart $X(\theta )=(\cos \theta ,\sin \theta )$ of $S^1$, the derivative has norm $1$, so the surface density is $1$; summing a partition of unity over one full turn gives chart surface measure ${\int }_0^{2\pi }1d\theta =2\pi$. The chart surface measure equals the polar measure $\sigma$ (Surface integration on compact C1 hypersurfaces, Agreement with the existing polar sphere measure).

For $p,q>0$, the Euler Beta integral is $B(p,q)={\int }_0^1{t}^{p-1}(1-t{)}^{q-1}dt$ and converges (Euler's real Beta integral, Euler's Beta integral converges exactly for two positive parameters); change of variables is valid on the improper interval after compact truncation (Change of variable in an improper integral).

The Beta-Gamma identity is $B(p,q)=\mathrm{\Gamma }(p)\mathrm{\Gamma }(q)\mathrm{/}\mathrm{\Gamma }(p+q)$, and $\mathrm{\Gamma }(n+1)=n!$ for every $n\in \mathbb{N}$ (The real Beta--Gamma identity, $\mathrm{\Gamma }(n+1)=n!$ for every natural number $n$, The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$).

Nonnegative product-measurable functions satisfy Tonelli's iterated-integral identity on sigma-finite measure spaces; under ${\mathrm{A}\mathrm{C}}_{\omega }$, product Lebesgue measure agrees with Euclidean Lebesgue measure on Borel sets, and equality of these measures gives equality of nonnegative integrals by simple approximation and monotone convergence (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure, Every nonnegative measurable function admits an explicit increasing sequence of simple approximations, Monotone convergence for the integral).

In real coordinates the displayed integrands are nonnegative polynomials times indicators of open balls or polydiscs. Finite sums and products of coordinate polynomials are continuous by the $C^k$ algebra theorem; the preimage criterion makes continuous maps Borel measurable, and products of measurable functions remain measurable ($C^k$ Euclidean maps and diffeomorphisms, $C^k$ Euclidean maps are closed under componentwise algebra and composition, The Borel sigma-algebra of a topological space, A measurable function between measurable spaces, A continuous map has Borel preimages of Borel sets, Arithmetic and lattice operations preserve measurability whenever they are defined, Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$).

A nonnegative real has a unique nonnegative square root (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$).

Induction on the positive integer dimension is valid (The principle of mathematical induction).

Multi-indices, their factorials and lengths, and complex nonnegative integer powers have the conventions used in the statement ($C^k$ maps and multi-index derivative notation in Euclidean space, The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, Integer powers in the complex field).