Description
This book contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills out standard treatments, providing many diverse applications: for example, the Lebesgue decomposition theorem and the Lebesgue density theorem, the Hilbert transform and other singular integral operators, the martingale convergence theorem, eigenvalue distributions, Lidskii's trace formula, Mercer's theorem and Littlewood's 4/3 theorem. It will broaden the knowledge of postgraduate and research students, and should also appeal to their teachers, and all who work in linear analysis.
Chapter
4 Convexity, and Jensen's inequality
4.1 Convex sets and convex functions
4.2 Convex functions on an interval
4.3 Directional derivatives and sublinear functionals
4.4 The Hahn--Banach theorem
4.5 Normed spaces, Banach spaces and Hilbert space
4.6 The Hahn--Banach theorem for normed spaces
4.7 Barycentres and weak integrals
5.1 Lp spaces, and Minkowski's inequality
5.2 The Lebesgue decomposition theorem
5.3 The reverse Minkowski inequality
5.5 The inequalities of Liapounov and Littlewood
5.7 The Loomis--Whitney inequality
5.9 Schur's theorem and Schur's test
5.10 Hilbert's absolute inequality
6.1 Banach function spaces
6.2 Function space duality
7.1 Decreasing rearrangements
7.2 Rearrangement-invariant Banach function spaces
7.3 Muirhead's maximal function
7.5 Calderón's interpolation theorem and its converse
7.6 Symmetric Banach sequence spaces
7.7 The method of transference
7.8 Finite doubly stochastic matrices
8.1 The Hardy--Riesz inequality…
8.2 The Hardy--Riesz inequality (p=1)
8.4 Strong type and weak type
8.6 Hardy, Littlewood, and a batsman's averages
8.7 Riesz's sunrise lemma
8.8 Differentiation almost everywhere
8.9 Maximal operators in higher dimensions
8.10 The Lebesgue density theorem
8.12 Hedberg's inequality
8.15 The martingale convergence theorem
9.1 Hadamard's three lines inequality
9.2 Compatible couples and intermediate spaces
9.3 The Riesz--Thorin interpolation theorem
9.5 The Hausdorff--Young inequality
9.7 The generalized Clarkson inequalities
10.1 The Marcinkiewicz interpolation theorem: I
10.4 The scale of Lorentz spaces
10.5 The Marcinkiewicz interpolation theorem: II
11 The Hilbert transform, and Hilbert's inequalities
11.1 The conjugate Poisson kernel
11.2 The Hilbert transform on L2(R)
11.3 The Hilbert transform on Lp(R) for…
11.4 Hilbert's inequality for sequences
11.5 The Hilbert transform on T
11.7 Singular integral operators
11.8 Singular integral operators on Lp(Rd) for…
12 Khintchine's inequality
12.1 The contraction principle
12.2 The reflection principle, and Lévy's inequalities
12.3 Khintchine's inequality
12.4 The law of the iterated logarithm
12.5 Strongly embedded subspaces
12.6 Stable random variables
12.7 Sub-Gaussian random variables
12.8 Kahane's theorem and Kahane's inequality
13 Hypercontractive and logarithmic Sobolev inequalities
13.2 Kahane's inequality revisited
13.3 The theorem of Latala and Oleszkiewicz
13.4 The logarithmic Sobolev inequality on…
13.5 Gaussian measure and the Hermite polynomials
13.6 The central limit theorem
13.7 The Gaussian hypercontractive inequality
13.8 Correlated Gaussian random variables
13.9 The Gaussian logarithmic Sobolev inequality
13.10 The logarithmic Sobolev inequality in higher dimensions
13.11 Beckner's inequality
13.12 The Babenko--Beckner inequality
14.1 Hadamard's inequality
14.3 Error-correcting codes
15 Hilbert space operator inequalities
15.5 Positive compact operators
15.6 Compact operators between Hilbert spaces
15.7 Singular numbers, and the Rayleigh--Ritz minimax formula
15.8 Weyl's inequality and Horn's inequality
15.11 The Hilbert--Schmidt class
15.13 Lidskii's trace formula
15.14 Operator ideal duality
16.1 Unconditional convergence
16.2 Absolutely summing operators
16.3 (p,q)-summing operators
16.4 Examples of p-summing operators
16.5 (p,2)-summing operators between Hilbert spaces
16.6 Positive operators on L1
16.8 p-summing operators between Hilbert spaces…
16.9 Pietsch's domination theorem
16.10 Pietsch's factorization theorem
16.11 p-summing operators between Hilbert spaces…
16.12 The Dvoretzky--Rogers theorem
16.13 Operators that factor through a Hilbert space
17 Approximation numbers and eigenvalues
17.1 The approximation, Gelfand and Weyl numbers
17.2 Subadditive and submultiplicative properties
17.3 Pietsch's inequality
17.4 Eigenvalues of p-summing and (p,2)-summing endomorphisms
18 Grothendieck's inequality, type and cotype
18.1 Littlewood's 4/3 inequality
18.2 Grothendieck's inequality
18.3 Grothendieck's theorem
18.4 Another proof, using Paley's inequality
18.5 The little Grothendieck theorem
18.7 Gaussian type and cotype
18.8 Type and cotype of Lp spaces
18.9 The little Grothendieck theorem revisited
Inequalities: A Journey into Linear Analysis
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