A Short Course on Rearrangement Inequalities (original) (raw)

A symmetric density property: monotonicity and the approximate symmetric derivative

Proceedings of the American Mathematical Society, 1988

Let W and B be open sets of real numbers whose union has full measure. If for each x, the set {h > 0\x-h Ç.W, x + h € B} has density zero at zero, then these sets are all empty. This is then used to prove the following: If / is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then / is nondecreasing. Introduction. We will establish the following result. Let W and B be open subsets of an interval whose union has full measure in this interval. If for each x in the interval, the set {h > 0[x-h eW, x + h e B} has density zero at zero, then these sets are all empty. We then use this to prove the following. If / is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then / is nondecreasing. Early attempts to prove this theorem can be found in [1] and [5]. More recently, some partial results have been established [3, 4, 6]. A survey of this and related topics appears in Larson [2]. DEFINITIONS. We will use the following notation and definitions. All functions are real valued and A is Lebesgue measure on the real line. For a set A, xA (x) 1S tne characteristic function of A. For A measurable, the upper right density of A at x is d (A, x) = limsup A(A fl (x, x + h))/h as h-* 0+. The upper left, lower right, and lower left densities, d ,d+, and d~ are defined in a similar manner. If these four values are the same, their common value is d(A, x), the density of A at x. The upper right Dini derívate of / is D+f(x) = limsup(/(x + h)-f(x))/h as h-► 0+. The upper left, lower right, and lower left dérivâtes, D~f,D+f, and D-f are defined similarly. For sets A and B, we define AB(x) = {h > 0\x-h e A, x + h e B}. The lower approximate symmetric derivative of f,P*'(x), is the least upper bound of the set of a such that {h\(f(x + h)-f(x-h))/2h < a} has density zero at 0. The upper approximate symmetric derivative, fap (x), is defined similarly. If y'(x) fap (x), their common value is the approximate symmetric derivative of /, fàp(x).

Tamped functions: A rearrangement in dimension 1

2019

We define a new rearrangement, called rearrangement by tamping, for non-negative measurable functions defined on R+. This rearrangement has many properties in common with the well-known Schwarz non-increasing rearrangement such as the P{\'o}lya-Szeg{\"o} inequality. Contrary to the Schwarz rearrangement, the tamping also preserves the homogeneous Dirichlet boundary condition of a function.

Some Properties of Variational Measures

Real Analysis Exchange

Recently several authors have established a remarkable property of the variational measures associated with a function. Expressed in classical language, this property asserts that if a function is ACG * on all sets of Lebesgue measure zero then the function must be globally ACG *. This article is an exposition of some ideas related to this property with the intention of bringing it to the attention of a wider audience than these original papers might attract. If f : [a, b] → R then a necessary and sufficient condition for the identity f (x)−f (a) = x a f (t) dt in the sense of the Denjoy-Perron integral is that µ f is σ-finite and absolutely continuous with respect to Lebesgue measure on [a, b].

Rearrangement transformations on general measure spaces

Indagationes Mathematicae, 2008

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees. § Both authors have been partially supported by Grants MTM2004-02299 and 2005SGR00556.

A general rearrangement inequality à la Hardy–Littlewood

Journal of Inequalities and Applications, 2000

Let F F(vl ]m) smooth on (R)m with Fv, vj > 0 for #j. Furthermore, let Ul Um nonnegative and bounded functions on R with compact support. We prove the inequality fR" F (ul urn) dx < fR. F (u* U'm) dx, where * denotes symmetric decreasing rearrangement.