Symmetric decreasing rearrangement is sometimes continuous

  title={Symmetric decreasing rearrangement is sometimes continuous},
  author={Frederick J. Almgren and Elliott H. Lieb},
  journal={Journal of the American Mathematical Society},
  • F. Almgren, E. Lieb
  • Published 13 January 1989
  • Mathematics
  • Journal of the American Mathematical Society
This paper deals with the operation .9R 'of symmetric decreasing rearrangement which maps Wl,p (Rn) to Wi ,p (Rn) . We show that even though it is norm decreasing, .9R is not continuous for n ~ 2. The functions at which .9R is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in Wi ,P(Rn ). Curiously, .9R is always continuous in fractional… 


The operation R, of symmetric decreasing rearrangement maps W1,P( Rn) to W1,p( Rn). Even though it is norm decreasing we show that R is not continuous for n>2. The functions at which R is continuous

Symmetric Decreasing Rearrangement Can Be Discontinuous

Suppose f(xl,x2) ≥ 0 is a continuously differentiable function supported in the unit disk in the plane. Its symmetric decreasing rearrangement is the rotationally invariant function f*(xl,x2) whose

A ] 17 J un 2 00 5 Rearrangement inequalities for functionals with monotone integrands

The rearrangement inequalities of Hardy-Littlewood and Ri esz say that certain integrals involving products of two or three functions increase under sy mmetric decreasing rearrangement. The

On estimates of constants for maximal functions

In this work we will study Hardy-Littlewood maximal function and maximal operator, basing on both classical and most up to date works. In the first chapter we will give definitions for different

Local $T$-sets and renormalized solutions of degenerate quasilinear elliptic equations with an $L^1$-datum

In this paper we study essentially the questions of uniqueness and stability of solutions of boundary value problems associated with equations of the type: div(â(x, u,ru)) + b(x)|u| u = μ 2 L(⌦) on

Sobolev's imbedding theorem in the limiting case with Lorentz space and BMO

We consider the Gagliardo-Nirenberg type inequality in R. Let Ω be an arbitrary domain inR. It is well known that the Sobolev space H 0 (Ω), 1 < p < ∞, is continuously embedded into L(Ω) for all q


Kinnunen and Lindqvist proved in (9) that the local Hardy{Littlewood maximal function, dened in an open set in the Euclidean space, is a bounded operator from W 1;p () to W 1;p () , provided p > 1.

Some matrix rearrangement inequalities

We investigate a rearrangement inequality for pairs of n×n matrices: Let $\|A\|_p$ denote (Tr(A*A)p/2)1/p, the Cp trace norm of an n×n matrix A. Consider the quantity $\|A+B\|_p^p+\|A-B\|_p^p$. Under

Continuous rearrangement and symmetry of solutions of elliptic problems

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also



Symmetric Decreasing Rearrangement Can Be Discontinuous

Suppose f(xl,x2) ≥ 0 is a continuously differentiable function supported in the unit disk in the plane. Its symmetric decreasing rearrangement is the rotationally invariant function f*(xl,x2) whose

Minimal rearrangements of Sobolev functions.

where α (n) is the volume of the unit η-ball in H$ and μ(ί)< oo is the Lebesgue measure of the set Et = {x: u(x)>t}. Note that μ(ι) = \Ε*\ where E* = {x: ii*(x)>f} and |E*| denotes the Lebesgue

On the definition and properties of certain variational integrals

Here R denotes an open set in the number space En and u = u(x) is a realvalued function defined on R. The integrand f(x, it, p) is assumed to be nonnegative and continuous. Moreover, we suppose

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on Rd can be symmetrized to a function f* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are

Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

A maximizing function, f, is shown to exist for the HLS inequality on R': 11 IXI - * fIq < Np f A , Iif IIwith Nbeing the sharp constant and i/p + X/n = 1 + 1/q, 1 <p, q, n/X < x. When p =q' or p = 2

A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to

A General Integral Inequality for the Derivative of an Equimeasurable Rearrangement

  • G. Duff
  • Mathematics
    Canadian Journal of Mathematics
  • 1976
The theory of non-increasing (decreasing) equimeasurable rearrangements of functions was introduced by Hardy and Littlewood [6] in connection with their studies of fractional integrals and integral

Some relations between nonexpansive and order preserving mappings

Abstract : It is shown that nonlinear operators which preserve the integral are order preserving if and only if they are nonexpansive in L(1) and that those wich commute with translation by a

Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

The equation dealt with in this paper is in three dimensions. It comes from minimizing the functional which, in turn, comes from an approximation to the Hartree-Fock theory of a plasma. It describes