Inequalities: A Journey into Linear Analysis

Abstract

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.

Correntropy: Implications of nonGaussianity for the moment expansion and deconvolution

“…= 9. Using in fact only 7 terms in (4), plot (a), there is agreement with (8). However, after about term 20 oscillations start to occur in the terms of (4), plot (b), and these grow and grow.…”

Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations

“…Then, the choice of 12) will make (2.10) hold. And we will use induction to prove that for all k ≥ s,…”

“…And let δU(ζ ), δW (ζ ), and δ (ζ ) be the difference of the two solutions, Using Hardy inequality [12], there exists C 1 independent of such that…”

On Khintchine type inequalities fork-wise independent Rademacher random variables

“…This short note concerns Khintchine's inequality, a classical theorem in probability, with many important applications in both probability and analysis (see [4,8,9,10,12] among others). It states that the L p norm of the weighted sum of independent Rademacher random variables is controlled by its L 2 norm; a precise statement follows.…”

“…Letε i , 1 ≤ i ≤ N , be independent copies of ε 0 and a ∈ R N . Khintchine's inequality (see, for example, Theorem 2.b.3 in [9] , Theorem 12.3.1 in [4] or the original work of Khintchine [7]) states that, for any p > 0…”