The Hollenbeck–Verbitsky (2010) conjecture is false

Let P₊ and P₋ be the analytic and co-analytic projections on the unit circle. Hollenbeck and Verbitsky (2010) conjectured that for 1 < p < 2 and 0 < s < sec²(π/(2p)) the best constant in ‖(|P₊f|s + |P₋f|s)1/s‖p ≤ C‖f‖p is 21/s/(2cos(π/(2p))). The conjecture was proved for 1 < p ≤ 4/3, and for 4/3 < p < 2 when s ≤ p/(p−1). We show that for every p ∈ (4/3,2) it fails for all s in an interval of positive length ending at sec²(π/(2p)). For example, it fails at p = 3/2, s = 3.98, inside the conjectured range 0 < s < 4. Thus, for 1 < p < 2, the conjecture holds on its whole range exactly when p ≤ 4/3. All earlier proofs for 1 < p < 2 rest on a pointwise inequality in two variables. Melentijević observed that it fails at s = sec²(π/(2p)) when p > 4/3, and Jaguzović found a point where it fails at p = 3/2, s = 4. We prove that for s ≤ sec²(π/(2p)) every failure of this pointwise inequality yields a function f that violates the conjectured bound, so that the conjectured constant is the best one if and only if s ≤ s0(p), the largest s for which the pointwise inequality holds. We give an explicit counterexample and certified bounds for s0(p) at six values of p. All these results except the explicit counterexample and the lower bounds for s0(p) have also been proved in the Lean proof assistant, assuming the cited results.