bibkey: mantoiupurice2007hardy
authors: Marius Măntoiu and Radu Purice
year: 2007
title: Hardy type inequalities with exponential weights for a class of convolution operators
doi: 10.1007/s11512-006-0035-4
url: https://projecteuclid.org/journals/arkiv-for-matematik/volume-45/issue-1/Hardy-type-inequalities-with-exponential-weights-for-a-class-of/10.1007/s11512-006-0035-4
claim: Lemma 3.1 supplies exponential integrability of inverse Fourier transforms of sufficiently many derivatives of strip-analytic symbols. Applied with one derivative to the exact Gamma symbol, it pays the bounded capped-weight commutator used in the same-source weighted null estimate. The paper does not supply that actual theta/Möbius null-source theorem, positivity, RH or Robin.
strata_touched: []
license: citation-only
triage: anchor

Exponential convolution-kernel supplier for the actual Gamma weight

Măntoiu–Purice, Hardy type inequalities with exponential weights for a class of convolution operators, Arkiv för Matematik 45 (2007), 83–103, DOI 10.1007/s11512-006-0035-4. The published primary, Notation 2.1 and Lemma 3.1, printed pp.87 and 89, are the supplier.

Published statement and the required symbol class

For a strip-analytic symbol $\rho$, Notation 2.1’s ${\mathcal{O}}_0^s({\mathbb{C}}_{\delta }^n)$ requires, uniformly on each smaller closed strip, $$|{\partial }^{\alpha }\rho (x+iy)|\le C_{\alpha }\min \{⟨x{⟩}^{s-|\alpha |},⟨\rho (x+iy)⟩\}.$$ For $s>0$ and an integer $m>s$, Lemma 3.1 gives $$\underset{|\alpha |=m}{\max }\underset{0\le t\le 1}{\sup }{\int }_{{\mathbb{R}}^n}{e}^{t\gamma |h|}|({\partial }^{\alpha }\rho {)}^{\vee }(h)|dh<\infty (0<\gamma <\delta ).$$ This is a literature-attested kernel lemma. Its use does not require the regular-energy hypothesis of Theorem 2.4. No new general kernel or Agmon theorem is asserted.

Actual-symbol map and remaining source obligations

The weighted null application, VHN5–VHN6, uses $n=1$, $\delta =5/4$, $0<s<1$ and $m=1$. Digamma recurrence shifts its first surviving pole from the apparent quarter-width to $5/4$, retaining the original rational factor. The resulting Gamma symbol has logarithmic growth and derivatives of order $O(⟨\Re z{⟩}^{-j})$ on smaller closed strips, so it satisfies the published symbol-class conditions. The lemma gives $e^{\gamma |h|}(p^{\prime }{)}^{\vee }(h)\in L^1$; the commutator difference cancels the off-diagonal factor $1/h$ in the kernel of $p(D)$.

The same application separately proves actual theta-image domain membership, all-dilation mixed-moment coercivity and the complete mixed arithmetic debit bound. These are source-relative interfaces, not conclusions of the published lemma. In particular the original moment is nonlocal in the Gamma coordinate, so no scalar-potential eigenfunction-decay theorem is substituted. The resulting estimate has no Lean/kernel certification, literature-priority claim or positivity conclusion for the complete energy.