bibkey: chenwang2012weighted authors: Xin Chen and Jian Wang year: 2012 title: Weighted Poincare Inequalities for Nonlocal Dirichlet Forms doi: null url: https://arxiv.org/abs/1207.7140v1 claim: Theorem 5.1 controls a weaker-weight variance for the exact theta Gamma jump energy. A separate paper deduction gives zero Gamma-only gap in the pole measure and leading prime/Gamma compensation on the same normalized far-tail tests; the global joint Weil inequality remains unproved. strata_touched: [] license: bibliographic-reference-only triage: anchor

A reusable weighted estimate for the Gamma jump energy

Primary statement and hypotheses

The inspected source is Chen–Wang, arXiv:1207.7140v1, submitted 31 July 2012, with its versioned primary PDF. Example 2.3, equation (2.30), printed p.14, defines

$$D_{\psi ,V}(h,h)=\frac{1}{2}\iint |h(y)-h(x){|}^2\psi (|x-y|)e^{-V(x)}{e}^{-V(y)}dxdy.$$

Here $V$ is locally bounded, $e^{-V}$ is bounded and integrable, and ${\int }_0^{\infty }(1\wedge r^2)r^{d-1}\psi (r)dr<\infty$. The ambient probability measure is

$$d{\mu }_{2V}=\frac{e^{-2V(x)}dx}{\int e^{-2V(x)}dx}.$$

Theorem 5.1, printed pp.32–33, assumes $\psi :(0,\infty )\to (0,\infty )$ is continuous. With $\gamma (r)={\inf }_{0<s\le r+1}\psi (s)$, its additional hypotheses are

$${\int }_0^1{r}^{d-1}\psi (r{)}^{-1}dr<\infty ,{\int }_{|x|\ge 1}\frac{e^{-3V(x)}}{\gamma (|x|)}dx<\infty ,$$

and, for some $0<{\alpha }_0<1$,

$${\int }_1^{\infty }{r}^{d+{\alpha }_0-1}\psi (r)dr<\infty ,\underset{|x|\to \infty }{\mathrm{limsup}}\frac{{\sup }_{|z|\ge |x|}{e}^{-V(z)}}{\gamma (|x|)|x{|}^{{\alpha }_0}}=0.$$

The conclusion is that there exists $C_1>0$ such that, for every real $h\in C_b^{\infty }({\mathbb{R}}^d)$,

$$\int |h-{\mu }_{2V}(h){|}^2{e}^{V(x)}\gamma (|x|)d{\mu }_{2V}(x)\le C_1{D}_{\psi ,V}(h,h).\text{\hspace{1em}(1)}$$

The theorem does not specify a numerical value for $C_1$. Applying (1) separately to real and imaginary parts gives the same estimate for complex tests.

Parameter map to the theta kernel

Use the original positive smooth even $\Phi$ identified in Romik’s theta representation, and set

$$d=1,V=-\log \Phi ,\psi (t)=\frac{e^{t/2}}{2\sinh t}=\frac{e^{-t/2}}{1-e^{-2t}}.$$

This is the density already used by the project’s archimedean jump decomposition. Symmetry of the double integral gives the exact parameter identification

$$D_{\psi ,V}(h,h)=E_{\Gamma }(h):={\int }_0^{\infty }\psi (t)\int \Phi (x)\Phi (x+t)|h(x+t)-h(x){|}^{2}dxdt.\text{\hspace{1em}(2)}$$

No normalization factor is inserted into this energy. Put

$$A=\int \Phi (x{)}^{2}dx,m_2(h)=A^{-1}\int h(x)\Phi (x{)}^{2}dx.$$

The function $\psi$ decreases strictly, so $\gamma (r)=\psi (r+1)$. Near zero, $\psi (t)\sim (2t{)}^{-1}$; at infinity, $\psi (t)\sim e^{-t/2}$. These estimates give both the form-integrability condition and the inverse-kernel integral in (1), and give the positive-order tail moment for any fixed ${\alpha }_0\in (0,1)$.

The theta derivative tail bound, obtained from the original normally convergent series and evenness, gives

$$\Phi (x)\le C{e}^{c|x|}\exp (-\frac{\pi }{2}{e}^{2|x|}).$$

It implies boundedness and integrability of $\Phi$, integrability of ${\Phi }^3/\gamma$, and the stated radial-supremum limit. Positivity and smoothness make $V$ locally bounded. Thus (1) applies and yields the unconditional paper-level source application

$$G(h):=\int |h(x)-m_2(h){|}^2\Phi (x)\gamma (|x|)dx\le C_{1}A{E}_{\Gamma }(h).\text{\hspace{1em}(3)}$$

In particular (3) holds for the even complex compact smooth tests of the theta-weighted Weil interface.

The weight and constant that remain unpaid

The full even Weil interface uses

$$d\nu (x)=2\Phi (x)\cosh (x/2)dx,E_{\Gamma }(h)+E_{\mathrm{prime}}(h)\ge \frac{1}{2}{\mathrm{Var}}_{\nu }(h).$$

Its center is $\nu (h)$, whereas (3) uses $m_2(h)$. More fundamentally, the ratio between the target density and the controlled density is

$$\frac{2\Phi (x)\cosh (x/2)}{\Phi (x)\gamma (|x|)}=\frac{2\cosh (|x|/2)}{\psi (|x|+1)}\sim e^{|x|+1/2}.\text{\hspace{1em}(4)}$$

It is unbounded. Changing the center alone cannot give a uniform domination of the target variance by this weighted integral. For example, even smooth bumps supported near $\pm R$, normalized in $L^2(\nu )$, have ${\mathrm{Var}}_{\nu }$ tending to one while $G$ tends to zero: the ratio of controlled to target densities tends to zero on those supports, and their ${\mu }_{2V}$ means tend to zero by Cauchy–Schwarz and the theta tail. This is a weight-comparison observation, not a counterexample to (3) or to the full energy bound; it does not estimate the energies of these bumps.

The source therefore supplies a genuine weaker-weight coercive estimate for exactly $E_{\Gamma }$. It supplies neither the numerical constant nor the common-test prime/Gamma comparison required at one-half. Replacing the ${\Phi }^2$ speed measure by $\nu$ also changes the generator contract. Neither an unchanged spectral gap nor a generator realization on $L^2(\nu )$ follows from Example 2.3.

The source application, tail checks and weight comparison here are paper deductions, not new Lean proofs or an originality claim. The source PDF has 42 pages and SHA-256 82e717b0b790b169103b9cf1f9c39bd8612aaaf534723fad6199f6d6ecc64ca5. No theorem from a later journal version is used without its own source check.

The Gamma-only target gap is zero

Use the original theta kernel and the pole probability measure from the even Weil interface: $d\nu =2\Phi (x)\cosh (x/2)dx$, with $I=\int \Phi (x)e^{x/2}dx=1/2$. The following is a paper deduction from the published theta inputs and the actual jump energy, rather than an additional theorem attributed to Chen–Wang.

Fix a nonzero real $u\in C_c^{\infty }((0,1))$, extended by zero. For $R\ge 2$ put ${\delta }_R=e^{-2R}$ and

$$h_R(x)=a_R\left[u\left(\frac{x-R}{{\delta }_R}\right)+u\left(\frac{-x-R}{{\delta }_R}\right)\right],\int |h_R{|}^{2}d\nu =1,$$

with $a_R>0$. These are actual even compact smooth tests supported in $A_R=[R,R+{\delta }_R]\cup [-R-{\delta }_R,-R]$. Romik’s Lemma 2.3, printed p.10, equations (2.8)–(2.9), supplies the upper tail and first-term remainder. Its two-sided consequence is

$$cg(r)\le \Phi (r)\le Cg(r),g(r)=\exp (9r/2-\pi e^{2r}),r\ge 1.$$

The local-scale identity recorded there gives $\Phi (r)\asymp \Phi (R)$ for $R-{\delta }_R\le r\le R+2{\delta }_R$. This is comparison by constants, not a ratio tending to one. Thus

$$a_R^{-2}{\asymp }_u{\delta }_R\Phi (R)e^{R/2},N_R:=\int h_R^{2}dx\le C_u{e}^{-R/2}/\Phi (R),$$

$$\Vert h_R^{\prime }{\Vert }_2^2={\delta }_R^{-2}\frac{\Vert u^{\prime }{\Vert }_2^2}{\Vert u{\Vert }_2^2}{N}_R,W_R:=\int \Phi h_R^{2}dx\le e^{-R/2}.$$

Also $|\nu(h_R)|^2\le\nu(A_R) =O(\delta_R\Phi(R)e^{R/2})$,so$\operatorname{Var}_\nu(h_R)\to1$. All comparison constants here and below are independent of $R$; those indexed by $u$ may depend on that fixed bump.

Let $S(r)={\sup }_{|x|\ge r}\Phi (x)$ and $J_R(t)=\int \Phi (x)\Phi (x+t)|h_R(x+t)-h_R(x){|}^{2}dx$. For $0<t<{\delta }_R$, the contributing pairs lie in the local comparison region. The translation derivative bound gives $J_R(t)\le C\Phi (R{)}^2{t}^2\Vert h_R^{\prime }{\Vert }_2^2$. Since $\psi (t)\le C/t$ for $t<1$,

$${\int }_0^{{\delta }_R}\psi (t)J_R(t)dt\le C_u\Phi (R)e^{-R/2}.$$

This pays the derivative cost of the shrinking bump. For ${\delta }_R\le t<1$, the bound $|a-b{|}^2\le 2(|a{|}^2+|b{|}^2)$ gives $J_R(t)\le 4S(R-1)W_R$, hence

$${\int }_{{\delta }_R}^1\psi (t)J_R(t)dt\le C_u(1+R)S(R-1)e^{-R/2}.$$

For $t\ge 1$, use $\psi (t)\le C{e}^{-t/2}$ and

$$\int \Phi (y)e^{-|x-y|/2}dy\le I{e}^{-|x|/2}.$$

For $x\ge 0$ this follows from $e^{-|x-y|/2}\le e^{-x/2}{e}^{y/2}$; evenness supplies $x<0$. It gives ${\int }_1^{\infty }\psi (t)J_R(t)dt\le C{e}^{-R}$. Together,

$$E_{\Gamma }(h_R)\le C_u\left[e^{-R}+(1+R)S(R-1)e^{-R/2}+\Phi (R)e^{-R/2}\right]=O_u(e^{-R}).\text{\hspace{1em}(5)}$$

The theta tail makes the last two terms $o(e^{-R})$. Nonnegativity and the normalized variance give the paper-level conclusion

$$\underset{\begin{array}{c}h\in C_c^{\infty }(\mathbb{R};\mathbb{C})\mathrm{even}\\ {\mathrm{Var}}_{\nu }(h)>0\end{array}}{\inf }\frac{E_{\Gamma }(h)}{{\mathrm{Var}}_{\nu }(h)}=0.\text{\hspace{1em}(6)}$$

Thus Gamma alone cannot control the target variance by any positive uniform constant on this compact test class. This strengthens the weight-comparison observation to an estimate on the actual energy and does not conflict with (3), whose variance density is weaker.

Exact leading Gamma compensation on the same tests

Put $f_R=\Phi h_R$ and ${\mathrm{Corr}}_f(t)=\int f(x+t)\overline{f(x)}dx$. For the shifts at least one, expanding the square gives the exact identity

$$E_{\Gamma ,\ge 1}(h_R)=\int |h_R(x){|}^2{D}_{\Gamma ,\mathrm{far}}(x)d\nu (x)-2\Re {\int }_1^{\infty }\psi (t){\mathrm{Corr}}_{f_R}(t)dt,$$

$$D_{\Gamma ,\mathrm{far}}(x)=\frac{{\int }_{|x-y|\ge 1}\Phi (y)\psi (|x-y|)dy}{2\cosh (x/2)}.$$

For each fixed $y$, as $x\to +\infty$,

$$\frac{e^x}{2\cosh (x/2)}{1}_{\{|x-y|\ge 1\}}\psi (|x-y|)⟶e^{y/2}.$$

The integrand without $\Phi (y)$ is bounded by $C{e}^{y/2}$, using $\psi (s)\le C{e}^{-s/2}$ for $s\ge 1$ and $|x-y|\ge x-y$. Dominated convergence and evenness therefore give $e^{|x|}{D}_{\Gamma ,\mathrm{far}}(x)\to I$ at both tails. On the support of $h_R$, $|x|\in [R,R+{\delta }_R]$, so normalization makes the diagonal contribution $I{e}^{-R}+o(e^{-R})$.

The far correlation is supported in $[2R,2R+2{\delta }_R]$. Writing $M_R=\Vert f_R{\Vert }_2^2=O(\Phi (R)e^{-R/2})$, Cauchy–Schwarz gives

$$\left|{\int }_1^{\infty }\psi (t){\mathrm{Corr}}_{f_R}(t)dt\right|\le C{\delta }_R{e}^{-R}{M}_R=o(e^{-R}).$$

The two smaller-shift estimates in (5) are $o(e^{-R})$. Consequently

$$E_{\Gamma }(h_R)=\frac{1}{2}{e}^{-R}+o(e^{-R}).\text{\hspace{1em}(7)}$$

Compact full-form estimate without a prime number theorem

Reuse the existing compact full-energy identity and prime jump decomposition. For an even compact smooth $f$ the entire finite prime diagonal mass cancels, leaving

$$Q(f)=P(f)+G(f)+c_{\Gamma }\Vert f{\Vert }_2^2-2\sum _{n\ge 2}\frac{\Lambda (n)}{\sqrt{n}}\Re {\mathrm{Corr}}_f(\log n),\text{\hspace{1em}(8)}$$

where

$$G(f)={\int }_0^{\infty }\psi (t)\Vert f(\cdot +t)-f{\Vert }_2^{2}dt,P(f)=2{\left|\int e^{x/2}f(x)dx\right|}^2,$$

and $c_{\Gamma }=\Re \mathrm{digamma}(1/4)-\log \pi$. Equation (8) is a reuse of the compact formula, not a new criterion. For support in $A_R$, a prime correlation is possible only when $\log n\in [2R,2R+2{\delta }_R]$: the same-interval differences are at most ${\delta }_R<\log 2$. The corresponding integer interval has length at most

$$e^{2R}(e^{2{\delta }_R}-1)\le 2e^2.$$

There are therefore uniformly boundedly many possible integer terms. Since $\Lambda (n)\le \log n$, their complete coefficient sum is $O((R+1)e^{-R})$. Every prime power is included; no short-interval prime number theorem is needed.

The differentiated theta series estimate $|{\Phi }^{\prime }(r)/\Phi (r)|\le C{e}^{2r}$, together with local comparison and bump derivative scaling, gives $\Vert f_R^{\prime }{\Vert }_2^2\le C_u{\delta }_R^{-2}{M}_R$. Splitting $G$ at ${\delta }_R$, the translation derivative bound on the first range and $\Vert f(\cdot +t)-f{\Vert }_2^2\le 4\Vert f{\Vert }_2^2$ on the second give

$$G(f_R)\le C_u(1+R)M_R,P(f_R)\le C{\delta }_R{e}^R{M}_R.$$

Cauchy–Schwarz bounds each correlation by $M_R$. Thus

$$|Q(f_R)|\le C_u(1+R)M_R=o(e^{-R}).\text{\hspace{1em}(9)}$$

Also $|\nu (h_R){|}^2\le \nu (A_R)=o(e^{-R})$, so ${\mathrm{Var}}_{\nu }(h_R)=1+o(e^{-R})$. Apply the full weighted identity from the even Weil interface to these same tests, retaining all weighted prime diagonals:

$$E_{\mathrm{prime}}(h_R)=\frac{1}{2}{\mathrm{Var}}_{\nu }(h_R)+Q(f_R)-E_{\Gamma }(h_R)=\frac{1}{2}-\frac{1}{2}{e}^{-R}+o(e^{-R}).\text{\hspace{1em}(10)}$$

Positivity on the restricted disconnected support class

For every even complex compact smooth $f$ supported in $A_R$, ${\mathrm{Corr}}_f(t)=0$ for ${\delta }_R<t<1$. Hence $\Vert f(\cdot +t)-f{\Vert }_2^2=2\Vert f{\Vert }_2^2$ on that range. From (8), $P(f)\ge 0$ and the complete prime coefficient bound give

$$Q(f)\ge \left[c_{\Gamma }+2{\int }_{{\delta }_R}^1\psi (t)dt-C(R+1)e^{-R}\right]\Vert f{\Vert }_2^2.\text{\hspace{1em}(11)}$$

Since $\psi (t)=(2t{)}^{-1}+O(1)$ near zero, the bracket is $2R+O(1)$. It is positive for all sufficiently large $R$, uniformly over this support class. In particular $Q(f_R)>0$ there. Equations (7), (9) and (10) describe how the same normalized tests approach the target energy ratio one-half from a positive full-form side.

This class does not exhaust all even compact tests. The global same-test joint lower bound remains unproved, as do RH and Robin’s full inequality. These deductions establish no operator realization, essential-spectrum statement or continuous transport of FIB operations. They are independently checked paper derivations, without new Lean certification or an originality claim; the published inputs and existing compact identities retain their own provenance.

A five-mode FIB address selects an exact prime-correlation cut

The sparse support can be centered using the project’s existing FIB quantity observation, without changing either the theta kernel or $Q$. Use $M=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)$, $S=M^3=\left(\begin{array}{cc}1& 2\\ 2& 3\end{array}\right)$ and $q(a,b)=2a+3b$. The increments for [null,2,3,2 5,5] are respectively $(0,0),(1,0),(0,1),(2,1),(1,1)$. For the low-to-high address [null]^L followed by one nonempty window $\sigma$, its readout is $H_L=q(S^L{d}_{\sigma })$. Unit1 and End are separate; no unit offset is added here.

These are applications of the Fibonacci recurrence and the classical Lucas/Fibonacci trace identity. Each sequence satisfies $H_{L+2}=4H_{L+1}+H_L$, since $S^2=4S+I$. Parity is already visible in the composition cut: $S\equiv I(\mathrm{mod}2)$ and $q(a,b)\equiv b(\mathrm{mod}2)$. The entirely null address has $H=0$ and is not a logarithmic support center.

For any integer $H\ge 2$, set $R=\log H$, $\delta =H^{-2}$ and use $A_R=[R,R+\delta ]\cup [-R-\delta ,-R]$. Same-interval differences are at most $\delta <\log 2$. Cross-interval correlations can be nonzero only for

$$2\log H<\log n<2\log H+2H^{-2}.$$

At either endpoint the overlap has measure zero, so $n=H^2$ contributes nothing even when it is a prime power. Also $H^2{e}^{2/H^2}<H^2+3$. Indeed, for $t=2/H^2\le 1/2$, $e^t\le 1+t+e^{1/2}{t}^2/2$, whence $H^2(e^t-1)\le 2+2e^{1/2}/H^2<3$. Thus only $H^2+1$ and $H^2+2$ can survive in the correlation sum (8).

For even $H$, $H^2+2\equiv 2(\mathrm{mod}4)$ and $H^2+2>2$, so it is not a prime power. For odd $H$, the same argument excludes $H^2+1$. For $H=F_j$, $j\ge 4$, the other exclusion is supplied by the classical Cassini and Catalan identities:

$$F_j^2+1=\{\begin{array}{ll}{F}_{j-1}{F}_{j+1},& j\text{ even},\\ F_{j-2}{F}_{j+2},& j\text{ odd}.\end{array}$$

The existing Vajda identity supplies these specializations. Pinned mathlib also supplies Int.fib_succ_mul_fib_pred_sub_fib_sq, Int.fib_add_sq_sub_fib_mul_fib_add_two_mul and Nat.fib_gcd. For even $j$, $\gcd (j-1,j+1)=1$; for odd $j$, $\gcd (j-2,j+2)=\gcd (j-2,4)=1$. Fibonacci strong divisibility therefore makes the two factors coprime. Both exceed one in the stated range, so their product is not a prime power.

For [null]^L[2], $L\ge 1$, these two exclusions eliminate every von-Mangoldt-weighted correlation. Equation (8) consequently reduces, for every even complex compact smooth $f$ supported in its $A_R$, to

$$Q(f)=P(f)+G(f)+c_{\Gamma }\Vert f{\Vert }_2^2.\text{\hspace{1em}(12)}$$

The $L=0$ exception is essential: $H=2$ gives the possible correlation $H^2+1=5$. For the other three nonempty modes, only $H^2+2$ remains a candidate; it is not asserted composite or excluded. For example, the [5] seed $H=5$ gives the prime power $27$, so restricting the sum to primes would lose a real candidate.

This is a support calculation and a use of existing recurrence and divisibility results, not new Fibonacci mathematics or an originality claim. It covers only these one-nonempty-window addresses. No density or exhaustion of the full test space follows. For nonzero tests the weighted $E_{\mathrm{prime}}$ still has its positive diagonal terms, including large prime powers whose compact correlations vanish. Equation (12) does not make that energy zero and does not supply the global joint bound needed for RH or Robin. The address-to-support bridge is a paper derivation, without an exact Lean proof of its analytic conclusion.