Best Constants in Inequalities Involving Analytic and Co-Analytic Projections and Riesz’S Theorem in Various Function Spaces

Abstract

Let P+ be the Riesz’s projection operator and let P− = I − P+. We consider estimates of the expression ∥(|P+f|s+|P−f|s)1s∥Lp(T) in terms of Lebesgue p-norm of the function f ∈ Lp(T). We find the accurate estimates for p ≥ 2 and 0 < s ≤ p, thus significantly improving the Kalaj result who treated this problem for s = 2 and 1 < p< ∞. Interestingly, for this range of s the appropriate vector-valued inequality holds with the same constant. Additionally, we obtain the right asymptotic of the constants for large s. This proves the conjecture of Hollenbeck and Verbitsky on the Riesz projection operator in some cases.