Contractive Hardy--Littlewood inequalities in the Dirichlet range

2025/10/16 by Brevig, Ole Fredrik, Kulikov, Aleksei, Seip, Kristian +1
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2510.14333

Abstract

The class A_α^p consists of those analytic functions f in the unit disc such that ‖f‖_α,p^p := |f(0)|^p+∫₀¹ ((d)/(dr) M_p^p(r,f)) (1-r²)^α-1 dr lt; ∞, where M_p^p(r,f) is the radial integral mean of |f|^p and 0<α, p <∞. For α>1, A_α^p is the standard weighted Bergman space, and A₁^p=H^p. We consider A_α^p for 0<α<1 and show that (weighted) isometric conformal invariance extends to this range, and we also clarify the relation between A_α^p and the classical Besov spaces. Our main result is the contractive inequality ‖f‖_β,q ≤ ‖f‖_α,p, valid when 0<α<β<∞ and α/p=β/q. We also identify the functions for which equality is attained. We thus extend recent results of the second-named author (1≤ α<β) and Llinares (β=1 and p=2). The extension of results from the classical range 1≤ α< ∞ to the Dirichlet range 0<α<1 uses arguments relying on analytic continuation.

Citations

• Functionals with extrema at reproducing kernels
• Bergman projection induced by radial weight
• Majorant Series

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