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 2^1/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.