{"product_id":"function-classes-on-the-unit-disc-an-introduction-9783110281231","title":"Function Classes on the Unit Disc: An Introduction","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThis monograph contains a study on various function classes, a number of new results and new or easy proofs of old results (Fefferman-Stein theorem on subharmonic behavior, theorems on conjugate functions and fractional integration on Bergman spaces, Fefferman's duality theorem), which are interesting for specialists; applications of the Hardy-Littlewood inequalities on Taylor coefficients to (C, α)-maximal theorems and (C, α)-convergence; a study of BMOA, due to Knese, based only on Green's formula; the problem of membership of singular inner functions in Besov and Hardy-Sobolev spaces; a full discussion of g-function (all p \u0026gt; 0) and Calderón's area theorem;  a new proof, due to Astala and Koskela, of the Littlewood-Paley inequality for univalent functions; and new results and proofs on Lipschitz spaces, coefficient multipliers and duality, including compact multipliers and multipliers on spaces with non-normal weights.   It also contains a discussion of analytic functions and lacunary series with values in quasi-Banach spaces with applications to function spaces and composition operators. Sixteen open questions are posed.   The reader is assumed to have a good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series.   Further information can be found at the author's website at http:\/\/poincare.matf.bg.ac.rs\/~pavlovic.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003ePreface   1 Quasi-Banach spaces     1.1 Quasinorm and p-norm   1.2 Linear operators    1.3 The closed graph theorem     The open mapping theorem    The uniform boundedness principle    The closed graph theorem    1.4 F-spaces    1.5 The spaces lp    1.6 Spaces of analytic functions    1.7 The Abel dual of a space of analytic functions     1.7a Homogeneous spaces    2 Interpolation and maximal functions     2.1 The Riesz\/Thorin theorem    2.2 Weak Lp-spaces and Marcinkiewicz’s theorem   2.3 The maximal function and Lebesgue points   2.4 The Rademacher functions and Khintchine’s inequality    2.5 Nikishin’s theorem   2.6 Nikishin and Stein’s theorem    2.7 Banach’s principle, the theorem on a.e. convergence, and Sawier’s theorems    2.8 Addendum: Vector-valued maximal theorem    3 Poisson integral     3.1 Harmonic functions     3.1a Green’s formulas    3.1b The Poisson integral    3.2 Borel measures and the space h1   3.3 Positive harmonic functions    3.4 Radial and non-tangential limits of the Poisson integral     3.4a Convolution of harmonic functions    3.5 The spaces hp and Lp(T)    3.6 A theorem of Littlewood and Paley    3.7 Harmonic Schwarz lemma   4 Subharmonic functions     4.1 Basic properties    4.1a The maximum principle   4.1b Approximation by smooth functions    4.2 Properties of the mean values    4.3 Integral means of univalent functions    Prawitz’ theorem    Distortion theorems   4.4 The subordination principle    4.5 The Riesz measure     Green’s formula   The Riesz measure of | f |p (f  є H(D)) and | u |p (u є hp)    5 Classical Hardy spaces     5.1 Basic properties    The decomposition lemma of Hardy and Littlewood   5.1a Radial limits   The Poisson integral of log | f* |    5.2 The space H1   5.3 Blaschke products     Riesz’ factorization theorem   5.4 Some inequalities   5.5 Inner and outer functions     5.5a Beurling’s approximation theorem   5.6 Composition with inner functions. Stephenson’s theorems     5.6a Approximation by inner functions   6 Conjugate functions     6.1 Harmonic conjugates    6.1a The Privalov\/Plessner theorem and the Hilbert operator    6.2 Riesz projection theorem    6.2a The Hardy\/Stein identity   6.2b Proof of Riesz’ theorems   6.3 Applications of the projection theorem    6.4 Aleksandrov’s theorem: Lp(T) = Hp(T) + \\overline{\u003c!-- --\u003eHp}(T)    6.5 Strong convergence in H1   6.6 Quasiconformal harmonic homeomorphisms and the Hilbert transformation   7 Maximal functions, interpolation, and coefficients    7.1 Maximal theorems     7.1a Hardy\/Littlewood\/Sobolev theorem   7.2 Maximal characterization of Hp (Burkholder, Gundy and Silverstein)    7.3 “Smooth” Cesàro means     σα-maximal theorem   The “W-maximal” theorem    7.4 Interpolation of operators on Hardy spaces     7.4a Application to Taylor coefficients and mean growth    7.4b On the Hardy\/Littlewood inequality    7.4c The case of monotone coefficients   7.5 Lacunary series   7.6 A proof of the σα-maximal theorem    8 Bergman spaces: Atomic decomposition    8.1 Bergman spaces   8.2 Reproducing kernels   8.3 The Coifman\/Rochberg theorem     q-envelops of Hardy spaces   8.4 Coefficients of vector-valued functions. Kalton’s theorems     8.4a Inequalities for a Hadamard product    8.4b Applications to spaces of scalar valued functions   9 Lipschitz spaces     9.1 Lipschitz spaces of first order   9.2 Conjugate functions    9.3 Lipschitz condition for the modulus. Dyakonov’s theorems with simple proofs by Pavlovic   9.4 Lipschitz spaces of higher order   9.5 Lipschitz spaces as duals of Hp, p \u0026lt; 1   10 Generalized Bergman spaces and Besov spaces     10.1 Decomposition of mixed norm spaces: case 1 \u0026lt; p \u0026lt; ∞    10.1a Besov spaces   10.2 Decomposition of mixed norm spaces: case 0 \u0026lt; p ≤ ∞    10.2a Radial limits of Hardy\/Bloch functions     10.2b Fractional integration and differentiation   10.3 Möbius invariant Besov spaces    10.4 Mean Lipschitz spaces    10.4a Lacunary series in mixed norm spaces    10.5 Duality in the case 0 \u0026lt; p ≤ ∞   10.6 Appendix: Characterizations of Besov spaces   11 BMOA, Bloch space     11.1 The dual of H1 and the Carleson measures    Proof of Fefferman’s theorem   11.2 Vanishing mean osillation    11.3 BMOA and mean Lipschitz spaces   11.4 Coefficients of BMOA-functions     11.4a Lacunary series    11.5 The Bloch space    11.5a On the predual of B    Functions with decreasing coefficients   12 Subharmonic behavior    12.1 Subharmonic behavior and Bergman spaces     Two simple proofs of Hardy\/Littlewood\/Fefferman\/Stein theorem    12.2 The space hp, p \u0026lt; 1    Two open problems posed by Hardy and Littlewood    12.3 Subharmonic behavior of smooth functions     12.3a Quasi-nearly subharmonic functions    12.3b Regularly oscillating functions   12.4 A generalization of the Littlewood\/Paley theorem     12.4a Invariant Besov spaces and the derivatives of the integral means   12.4b Addendum: The case of vector valued functions    12.5 Mixed norm spaces of harmonic functions    13 Littlewood\/Paley theory     13.1 Some more vector maximal functions   13.2 The Littlewood\/Paley g-function     Calderon’s generalization of the area theorem (p \u0026gt; 0)    A proof of a the Littlewood\/Paley g-theorem (p \u0026gt; 0)   13.3 Applications of Cesàro means   13.4 The Littlewood\/Paley g-theorem in a generalized form     An improvement    13.5 Proof of Calderon’s theorem   14 Tauberian theorems and lacunary series on the interval (0,1)     14.1 Karamata’s theorem and Littlewood’s theorem     14.1a Tauberian nature of Λp1\/p   14.2 Lacunary series in C[0,1]    14.2a Lacunary series on weighted L∞-spaces   14.3 Lp-integrability of lacunary series on (0,1)     14.3a Some consequences   Bibliography","brand":"De Gruyter","offers":[{"title":"Default Title","offer_id":53516453151063,"sku":"9783110281231","price":134.42,"currency_code":"GBP","in_stock":true}],"url":"https:\/\/bookcurl.com\/products\/function-classes-on-the-unit-disc-an-introduction-9783110281231","provider":"Book Curl","version":"1.0","type":"link"}