Function Classes on the Unit Disc: An Introduction (De Gruyter Studies in Mathematics, 52)
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In "Function Classes on the Unit Disc: An Introduction," Miroslav Pavlović explores various classes of functions within the unit disc, presenting new results and simplified proofs of established theorems. Key topics include the Fefferman-Stein theorem on subharmonic behavior, conjugate functions, fractional integration on Bergman spaces, and Fefferman's duality theorem. The book delves into applications of Hardy-Littlewood inequalities to maximal theorems and convergence issues, a study of BMOA through Green's formula as proposed by Knese, and examines singular inner functions within Besov and Hardy-Sobolev spaces. It also discusses g-functions for all \( p > 0 \), Calderón's area theorem, and presents a new proof of the Littlewood-Paley inequality for univalent functions by Astala and Koskela. Additional insights are provided on Lipschitz spaces, coefficient multipliers, and duality considerations. The monograph includes discussions on analytic functions and lacunary series in quasi-Banach spaces with practical applications to function spaces and composition operators. Sixteen open questions are posed throughout the text. This work is intended for readers with a strong background in Lebesgue integration, complex analysis, functional analysis, and Fourier series.
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