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| 书名： | Inequalities: A Journey into Linear Analysis | |
|---|---|---|
| 作者： | Garling, D. J. H. | |
| 出版时间： | 2007-07-05 | |
| ISBN： | 9780521876247(P-ISBN) ，9780511288685(O-ISBN) | |
| 摘要： | ||
| 摘要： | ||
书目详情：
Half-titleTitleCopyrightContentsIntroduction1 Measure and integral1.1 Measure1.2 Measurable functions1.3 Integration1.4 Notes and remarks2 The Cauchy--Schwarz inequality2.1 Cauchys inequality2.2 Inner-product spaces2.3 The Cauchy--Schwarz inequality2.4 Notes and remarksExercises3 The arithmetic mean–geometric mean inequality3.1 The arithmetic mean–geometric mean inequality3.2 Applications3.3 Notes and remarksExercises4 Convexity, and Jensens inequality4.1 Convex sets and convex functions4.2 Convex functions on an interval4.3 Directional derivatives and sublinear functionals4.4 The Hahn--Banach theorem4.5 Normed spaces, Banach spaces and Hilbert space4.6 The Hahn--Banach theorem for normed spaces4.7 Barycentres and weak integrals4.8 Notes and remarksExercises5 The Lp spaces5.1 Lp spaces, and Minkowskis inequality5.2 The Lebesgue decomposition theorem5.3 The reverse Minkowski inequality5.4 Hölders inequality5.5 The inequalities of Liapounov and Littlewood5.6 Duality5.7 The Loomis--Whitney inequality5.8 A Sobolev inequality5.9 Schurs theorem and Schurs test5.10 Hilberts absolute inequality5.11 Notes and remarksExercises6 Banach function spaces6.1 Banach function spaces6.2 Function space duality6.3 Orlicz spaces6.4 Notes and remarksExercises7 Rearrangements7.1 Decreasing rearrangements7.2 Rearrangement-invariant Banach function spaces7.3 Muirheads maximal function7.4 Majorization7.5 Calderóns interpolation theorem and its converse7.6 Symmetric Banach sequence spaces7.7 The method of transference7.8 Finite doubly stochastic matrices7.9 Schur convexity7.10 Notes and remarksExercises8 Maximal inequalities8.1 The Hardy--Riesz inequality…8.2 The Hardy--Riesz inequality p=1)8.3 Related inequalities8.4 Strong type and weak type8.5 Riesz weak type8.6 Hardy, Littlewood, and a batsmans averages8.7 Rieszs sunrise lemma8.8 Differentiation almost everywhere8.9 Maximal operators in higher dimensions8.10 The Lebesgue density theorem8.11 Convolution kernels8.12 Hedbergs inequality8.13 Martingales8.14 Doobs inequality8.15 The martingale convergence theorem8.16 Notes and remarksExercises9 Complex interpolation9.1 Hadamards three lines inequality9.2 Compatible couples and intermediate spaces9.3 The Riesz--Thorin interpolation theorem9.4 Youngs inequality9.5 The Hausdorff--Young inequality9.6 Fourier type9.7 The generalized Clarkson inequalities9.8 Uniform convexity9.9 Notes and remarksExercises10 Real interpolation10.1 The Marcinkiewicz interpolation theorem: I10.2 Lorentz spaces10.3 Hardys inequality10.4 The scale of Lorentz spaces10.5 The Marcinkiewicz interpolation theorem: II10.6 Notes and remarksExercises11 The Hilbert transform, and Hilberts inequalities11.1 The conjugate Poisson kernel11.2 The Hilbert transform on L2R)11.3 The Hilbert transform on LpR) for…11.4 Hilberts inequality for sequences11.5 The Hilbert transform on T11.6 Multipliers11.7 Singular integral operators11.8 Singular integral operators on LpRd) for…11.9 Notes and remarksExercises12 Khintchines inequality12.1 The contraction principle12.2 The reflection principle, and Lévys inequalities12.3 Khintchines inequality12.4 The law of the iterated logarithm12.5 Strongly embedded subspaces12.6 Stable random variables12.7 Sub-Gaussian random variables12.8 Kahanes theorem and Kahanes inequality12.9 Notes and remarksExercises13 Hypercontractive and logarithmic Sobolev inequalities13.1 Bonamis inequality13.2 Kahanes inequality revisited13.3 The theorem of Latala and Oleszkiewicz13.4 The logarithmic Sobolev inequality on…13.5 Gaussian measure and the Hermite polynomials13.6 The central limit theorem13.7 The Gaussian hypercontractive inequality13.8 Correlated Gaussian random variables13.9 The Gaussian logarithmic Sobolev inequality13.10 The logarithmic Sobolev inequality in higher dimensions13.11 Beckners inequality13.12 The Babenko--Beckner inequality13.13 Notes and remarksExercises14 Hadamards inequality14.1 Hadamards inequality14.2 Hadamard numbers14.3 Error-correcting codes14.4 Note and remark15 Hilbert space operator inequalities15.1 Jordan normal form15.2 Riesz operators15.3 Related operators15.4 Compact operators15.5 Positive compact operators15.6 Compact operators between Hilbert spaces15.7 Singular numbers, and the Rayleigh--Ritz minimax formula15.8 Weyls inequality and Horns inequality15.9 Ky Fans inequality15.10 Operator ideals15.11 The Hilbert--Schmidt class15.12 The trace class15.13 Lidskiis trace formula15.14 Operator ideal duality15.15 Notes and remarksExercises16 Summing operators16.1 Unconditional convergence16.2 Absolutely summing operators16.3 p,q)-summing operators16.4 Examples of p-summing operators16.5 p,2)-summing operators between Hilbert spaces16.6 Positive operators on L116.7 Mercers theorem16.8 p-summing operators between Hilbert spaces…16.9 Pietschs domination theorem16.10 Pietschs factorization theorem16.11 p-summing operators between Hilbert spaces…16.12 The Dvoretzky--Rogers theorem16.13 Operators that factor through a Hilbert space16.14 Notes and remarksExercises17 Approximation numbers and eigenvalues17.1 The approximation, Gelfand and Weyl numbers17.2 Subadditive and submultiplicative properties17.3 Pietschs inequality17.4 Eigenvalues of p-summing and p,2)-summing endomorphisms17.5 Notes and remarksExercises18 Grothendiecks inequality, type and cotype18.1 Littlewoods 4/3 inequality18.2 Grothendiecks inequality18.3 Grothendiecks theorem18.4 Another proof, using Paleys inequality18.5 The little Grothendieck theorem18.6 Type and cotype18.7 Gaussian type and cotype18.8 Type and cotype of Lp spaces18.9 The little Grothendieck theorem revisited18.10 More on cotype18.11 Notes and remarksExercisesReferencesIndex of inequalitiesIndex
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