Noncommutative Bourgain multi-frequency maximal inequality

Determine whether the operator-valued estimate for shrinking frequency neighborhoods holds: for rational or arbitrary frequencies λ_1,…,λ_K∈𝕋 and the sets R_j={ξ∈𝕋:min_{1≤r≤K}d_𝕋(ξ,λ_r)≤2^{-j}}, prove or disprove the bound ||(𝒯_{1_{R_j}}f)_j||_{L_2(𝒜;ℓ_∞)}≲(log(K+1))^2||f||_2.

Background

The scalar proof of Bourgain’s polynomial maximal estimate uses a logarithmic maximal inequality for Fourier projections onto shrinking neighborhoods of finitely many frequencies. The paper needs an operator-valued substitute for this estimate but avoids the issue by combining an alternative decomposition with a noncommutative sampling principle.

The unresolved estimate concerns the maximal L2(ℓ∞)-norm of Fourier multipliers whose symbols are indicators of the sets R_j. Establishing it would provide a direct noncommutative counterpart of Bourgain’s scalar multi-frequency maximal inequality and could simplify or strengthen the treatment of major-arc approximations.

References

It is presently unclear whether eq:nc-Bourgain-Lemma41 holds, and this appears to be an interesting problem in its own right.

eq:nc-Bourgain-Lemma41:

∥(T1Rj(f))j∥L2(;ℓ∞)≲(log⁡(K+1))2∥f∥2.\left\| \left( \mathcal T_{\mathbf1_{R_j}}(f) \right)_j \right\|_{L_2(;\ell_\infty)} \lesssim \bigl(\log(K+1)\bigr)^2\|f\|_2.

— Noncommutative maximal inequalities for polynomial ergodic averages  (2610.06455 - Hong et al., 5 Oct 2026) in Remark 5.5, Section 5, subsection “L_2-approximation estimates”