The critical logarithmic endpoint model

Definitions and mathematical statement

For $0<\varepsilon<1/2$, real $\lambda$ and initially smooth compactly supported $f:{\mathbb{R}}\to{\mathbb{C}}$, use the reconstructed symmetric principal value \[T_{\varepsilon,\lambda}f(x)=\operatorname{p.v.}\int_{\mathbb{R}} f(x-t)e^{i\lambda|t|^\varepsilon}\frac{{\,\mathrm{d}} t}{t},\qquad M_\varepsilon f(x)=\sup_{\lambda\in{\mathbb{R}}}|T_{\varepsilon,\lambda}f(x)|.\] Original Question 12 (Guo). Is there a universal constant $C>0$, independent of $\varepsilon$, such that \[\|M_\varepsilon f\|_{L^2({\mathbb{R}})}\le C\|f\|_{L^2({\mathbb{R}})}\quad\text{for every }0<\varepsilon<1/2?\] This is a uniform family of maximal-in-$\lambda$ bounds, not a single maximal operator also taking a supremum in $\varepsilon$. The exponent and principal-value/absolute-value symbols are reconstructed from damaged extraction; the registry marks the statement reconstructed_unverified.

Attached unreviewed endpoint reduction. For fixed real $\mu$, choose the allowed scaling $\lambda=\mu/\varepsilon$ and remove the scalar phase. The aid claims pointwise convergence on $C_c^1({\mathbb{R}})$, \[e^{-i\mu/\varepsilon}T_{\varepsilon,\mu/\varepsilon}f(x)\longrightarrow S_\mu f(x):=\operatorname{p.v.}\int_{\mathbb{R}} f(x-t)|t|^{i\mu}\frac{{\,\mathrm{d}} t}{t}.\] Consequently the universal bound would imply, with the same constant, \[\left\|\sup_{\mu\in{\mathbb{R}}}|S_\mu f|\right\|_2\le C\|f\|_2.\] The aid obtains this through finite parameter maxima and Fatou's lemma, then the appropriate parameter-continuity/density argument; it does not exchange an unproved infinite supremum with a limit. The logarithmic model is necessary, not claimed sufficient for the original estimate.

With Fourier convention $\widehat f(\xi)=\int e^{-ix\xi}f(x){\,\mathrm{d}} x$, the reported exact multiplier for $\xi\ne0$ is \[\widehat{S_\mu f}(\xi)=2\Gamma(i\mu)\sinh(\pi\mu/2)\operatorname{sgn}(\xi)|\xi|^{-i\mu}\widehat f(\xi).\] At $\mu=0$ its removable value is $-i\pi\operatorname{sgn}\xi$, the unnormalized Hilbert-transform multiplier. The gamma expression is interpreted by continuation at that parameter.

For every fixed $0<a<b<\infty$ and $M<\infty$, define annular convolution kernels \[K_{\varepsilon,\mu}(t)=\mathbf1_{\{a\le|t|\le b\}}\frac{\exp\left(i\mu(|t|^\varepsilon-1)/\varepsilon\right)}t,\qquad K_{0,\mu}(t)=\mathbf1_{\{a\le|t|\le b\}}\frac{|t|^{i\mu}}t.\] The aid asserts $\sup_{|\mu|\le M}\|K_{\varepsilon,\mu}*-K_{0,\mu}*\|_{L^2\to L^2}=O_{a,b,M}(\varepsilon)$. This is fixed-annulus, bounded-$\mu$ control, not full-kernel operator-norm convergence uniformly over all parameters. The aid reports the universal AIM estimate as open; all reduction and multiplier assertions require independent review.

Short English statement

Guo asks for an $L^2$ bound uniform as the power exponent tends to zero. The aid identifies a necessary logarithmic-modulation maximal problem by scaling the phase coefficient like $1/\varepsilon$, gives its exact multiplier, and quantifies convergence only on fixed annuli with bounded logarithmic parameter. It does not prove the requested universal bound.

Sources

• Local UnsolvedMath registry, permanent ID 20000539, source code catalogue entry; aim-analysis-notes.json, notes[70], tag question.
• AIM workshop Carleson theorems and multilinear operators, Question 12 (Guo), equation (0.17): https://aimath.org/pastworkshops/multilinopsproblems.pdf. Supplied verification describes comparison with the PDF and Guo's operator definition, but gives no separate bibliographic locator. No PDF was consulted here.
• Attached machine-generated research attempt 1: result reduction, problem status at run open, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation. Original literature field empty.