Cowling’s Lp-integrability conjecture

Let $G$ be a Kunze--Stein locally compact group, let $\lambda_G$ denote its left regular representation, and let $\pi$ be a unitary representation of $G$ such that $\pi\prec\lambda_G$ . Then, for every $p>2$ , the representation $\pi$ is $L^p$ -integrable: there exists a dense subspace $\mathcal H_0$ of its Hilbert space such that $g\mapsto\langle\pi(g)\xi,\eta\rangle$ belongs to $L^p(G)$ for all $\xi,\eta\in\mathcal H_0$ .

References

Primary source

arXiv

Additional references

On Cowling's L^p-integrability conjecture for Kunze--Stein groups — arXiv — Siwei Liang

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new preprint proves the conjectured integrability implication for a broad class of groups and disproves a related cyclic version, but the full conjecture remains open.

Cowling’s conjecture concerns an integrability implication for matrix coefficients of representations of Kunze–Stein groups. The latest claim covers a broad admissible class but does not assert the result for every Kunze–Stein group.

September 15, 2026 development

Siwei Liang’s preprint claims the matrix-coefficient implication for an admissible class containing all $S$ -algebraic groups and certain tree-automorphism groups, and gives a counterexample to the related cyclic-vector formulation. These are claimed results and have not been independently verified in the retrieved material.

Current status (as of September 2026): The conjecture is claimed for a substantial class of groups, and its cyclic-vector analogue is claimed false, but the full statement for all Kunze–Stein groups remains open and the new claims are unverified.

Sources

No solutions have been posted yet.