A first-moment bound for the bath correlation convolution

Apply the Fourier continuity theorem to each frequency entry supplied by the kernel contract. Thus every time entry of $h$ is continuous, and $k(v)=\mathrm{\parallel }h(v){\mathrm{\parallel }}_{2,1}$ is a nonnegative continuous function with moments $G,H$. In particular $(u,v)\mapsto k(v-u)k(v)$ is continuous and is measurable for the product of the Borel sigma-algebras. All uses of Tonelli below therefore have product-measurable integrands and sigma-finite factors; no completed-product theorem is needed.

Translations preserve the Borel restriction $\lambda$ of Lebesgue measure. Reflection $R(v)=-v$ is a Borel bijection, so $\nu (E)=\lambda (R^{-1}E)$ is another Borel measure. For every bounded open interval $(a,b)$, its reflection $(-b,-a)$ is its translate by $-(a+b)$; hence $\nu ((a,b))=\lambda ((a,b))$. The bounded open intervals together with the empty set form a generating pi-system, with common finite-measure exhaustion $(-m,m)$. Measure uniqueness gives $\nu =\lambda$. Thus reflection, translations and their compositions preserve the Borel measure, and integral invariance justifies substitution by any of these maps, including for nonnegative integrals that may be infinite.

For finite compatible matrices, $\mathrm{\mid }(AB{)}_{ij}\mathrm{\mid }\le ({\sum }_a\mathrm{\mid }{A}_{ia}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}({\sum }_a\mathrm{\mid }{B}_{aj}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ by Cauchy–Schwarz. Summing over $i,j$ gives $\mathrm{\parallel }AB{\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}\le \mathrm{\parallel }{A}^{†}{\mathrm{\parallel }}_{2,1}\mathrm{\parallel }B{\mathrm{\parallel }}_{2,1}$. Put $q(u)={\int }_{\mathbb{R}}k(v-u)k(v)dv$. By $h(u-v{)}^{†}=h(v-u)$, the matrix product in the convolution has entry norm at most $k(v-u)k(v)$. The convolution premise and the entrywise integral triangle inequality therefore give $\mathrm{\parallel }C(u){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}\le q(u)$ almost everywhere. Tonelli and the substitution $w=v-u$, with $v$ fixed, give ${\int }_{\mathbb{R}}q(u)du={\int }_{\mathbb{R}}k(v){\int }_{\mathbb{R}}k(w)dwdv=G^2<\mathrm{\infty }$. Thus $q$ is measurable and finite almost everywhere; in particular the majorant also proves absolute convergence of the matrix convolution almost everywhere.

Substitution $z=v+u$ gives $q(-u)={\int }_{\mathbb{R}}k(z)k(z-u)dz=q(u)$. Hence reflection invariance identifies ${\int }_0^{\mathrm{\infty }}uq(u)du$ with half of ${\int }_{\mathbb{R}}\mathrm{\mid }u\mathrm{\mid }q(u)du$, including when these integrals are infinite. By the almost-everywhere majorant, ${\int }_0^{\mathrm{\infty }}u\mathrm{\parallel }C(u){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}du\le \frac{1}{2}{\int }_{\mathbb{R}}\mathrm{\mid }u\mathrm{\mid }q(u)du$.

Apply Tonelli and, at each fixed $v$, substitute $w=v-u$ using step 1.2. Since $\mathrm{\mid }v-w\mathrm{\mid }\le \mathrm{\mid }v\mathrm{\mid }+\mathrm{\mid }w\mathrm{\mid }$, this gives ${\int }_{\mathbb{R}}\mathrm{\mid }u\mathrm{\mid }q(u)du={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\mathrm{\mid }v-w\mathrm{\mid }k(w)k(v)dwdv\le HG+GH=2GH$. All integrands interchanged here are nonnegative and product-measurable; Tonelli factors the last two terms using the finite moments $G,H$. Together with step 3.1 this yields ${\int }_0^{\mathrm{\infty }}u\mathrm{\parallel }C(u){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}du\le GH$. Multiplication by $4\gamma$ proves finiteness of $M$ and $M\le 4\gamma GH=\mathrm{\Gamma }\tau$.