A transfer principle and the full Banach range for lacunary rotation-coupled circle averages

Definitions and mathematical statement

Let $\sigma$ be probability arclength measure on $S^1\subset{\mathbb{R}}^2$, and let $R_\theta$ be rotation through angle $\theta$. For a nonempty radius set $E\subset(0,\infty)$, define, initially for smooth compactly supported functions, \[ \mathcal M_{E,\theta}(f,g)(x)=\sup_{t\in E} \left|\int_{S^1}f(x-tw)g(x-tR_\theta w)\,{\,\mathrm{d}}\sigma(w)\right|. \] Krause's Question 7 concerns $E=(0,\infty)$ and $\theta=\pi/3$: find exponent triples for which there is a finite constant independent of $f,g$ such that \[ \|\mathcal M_{(0,\infty),\pi/3}(f,g)\|_{L^r({\mathbb{R}}^2)} \leq C_{p,q,r}\|f\|_{L^p({\mathbb{R}}^2)}\|g\|_{L^q({\mathbb{R}}^2)}. \] The source says only "for certain $p,q,r$," without an endpoint or full-range conjecture. Finite positive exponents use their usual $L^p$ norms or quasi-norms, with the essential-supremum convention at $\infty$. For any scale-invariant continuous-radius bound, the natural scaling relation is $1/r=1/p+1/q$, with $1/\infty=0$.

The workshop text asserts that this question is equivalent to the corresponding estimate for \[ \mathcal M_{(0,\infty),\pi}(f,g)(x)=\sup_{t>0} \left|\int_{S^1}f(x-tw)g(x+tw)\,{\,\mathrm{d}}\sigma(w)\right|. \] The equivalence is retained as a source assertion; no transformation, constants, or precise endpoint scope is supplied. It is not an asserted identity between the two operators.

The attached aid describes a linear-maximal transfer. Put \[ \mathcal C_E h(x)=\sup_{t\in E}\int_{S^1}|h(x-tw)|\,{\,\mathrm{d}}\sigma(w). \] For conjugate exponents $a,a'>1$, $1/a+1/a'=1$, it records the pointwise estimate \[ \mathcal M_{E,\theta}(f,g) \leq[\mathcal C_E(|f|^a)]^{1/a} [\mathcal C_E(|g|^{a'})]^{1/a'}. \] Using the appropriate linear circular maximal bounds, the aid reports the known continuous-radius estimates on the Hölder line with $r>2$. For the dyadic radius set $E=\{2^j:j\in{\mathbb{Z}}\}$, it reports strong bounds for all \[ 1<p,q\leq\infty,\quad1<r\leq\infty,\qquad \frac1r=\frac1p+\frac1q<1, \] uniformly in every rotation $\theta$, including $\pi/3$ and $\pi$. The dyadic result does not replace the continuous supremum in the original problem. The aid identifies the continuous-radius region $p,q>2$, $1<r\leq2$, on the same Hölder line as unresolved in the literature considered. It makes no claim of novelty for the elementary pointwise Hölder inequality itself. These are supplied unreviewed partial-result and status claims, not a new sharp-range theorem.

Short English statement

Find $L^p\times L^q\to L^r$ estimates for a continuous-radius bilinear circular maximal operator with a sixty-degree rotation. The source also asserts an antipodal equivalence. The attached aid gives a transfer from linear circular maximal bounds and a larger dyadic-radius range, which leaves the stated continuous-radius question distinct.

Sources

• Krause, Question 7, equations (0.7) and (0.8), AIM workshop Carleson theorems and multilinear operators: https://aimath.org/pastworkshops/multilinopsproblems.pdf. The supplied verification says the damaged displays were checked against the official AIM TeX source as well as the PDF; no separate TeX URL is supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000534$, exact source code catalogue entry; canonical aim-analysis-notes.json, notes[65].
• The attached research attempt supplies the radius-set transfer and dyadic result but no full external references for the linear maximal inputs. Attempt 1 is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.