Knowlpedia

Proposition

Product estimate in a two-index analytic coefficient space

Coefficient convolution and the Leibniz rule are bounded by the factorial-binomial weights.

Statement

For the , coefficient multiplication (FG)α=∑i=0αFiGα−i(FG)_\alpha=\sum_{i=0}^\alpha F_iG_{\alpha-i} obeys

∥FG∥R,ρ≤256∥F∥R,ρ∥G∥R,ρ.\|FG\|_{R,\rho}\le256\|F\|_{R,\rho}\|G\|_{R,\rho}.

The constant is uniform in R,ρR,\rho; no optimality is claimed.

Convolution estimate

For every N≥0N\ge0, splitting the sum at N/2N/2 gives

∑i=0N(N+1)2(i+1)2(N−i+1)2≤8∑j=1∞j−2≤16.\sum_{i=0}^N\frac{(N+1)^2}{(i+1)^2(N-i+1)^2} \le8\sum_{j=1}^\infty j^{-2}\le16.

In the Leibniz formula for ∂ηβ(FiGα−i)\partial_\eta^\beta(F_iG_{\alpha-i}), the derivative binomial cancels the derivative factorials. The remaining binomial ratio is at most one because

(i+kk)(α−i+β−kβ−k)≤(α+ββ).\binom{i+k}{k}\binom{\alpha-i+\beta-k}{\beta-k} \le\binom{\alpha+\beta}{\beta}.

The left side counts a restricted collection of subsets counted on the right. Applying the convolution estimate to both indices gives 16216^2. Rescaling the norm by 256 makes multiplication submultiplicative, producing a Banach-algebra norm.