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Questions tagged [decreasing-rearrangements]

For all question related to the decreasing rearrangements of measurable functions so for measurable set too. As well the Lorentz spaces are strongly include since there are exclusively defined through the decreasing rearrangement of function. The analogue of this notion higher dimension is: The **spherical (or symmetric, or radial)** decreasing rearrangement of a measurable function

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symmetrized rearrangement on sphere.

I am trying to undestand the Corollary 2.2 from Osgood, Phillips and Sarnak (see http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf), that is, if $u \in W^{1}(S^...
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Schwarz symmetrization is equimeasurable

Suppose $\Omega\subset\mathbb{R^2}$ is open and bounded, and let $f:\Omega\rightarrow [0,\infty)$ be measurable. Moreover, let $\Omega^{\ast}$ denote the closed disk with midpoint $0\in\mathbb{R}^2$ ...
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Nonnegativity assumption for the Schwarz rearrangement of a function

I recently tried to understand the proof of Faber-Krahn inequality and stumbled upon the Schwarz rearrangement. For example, (See, for example, Henrot's book "Extremum Problems for Eigenvalues of ...
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Decreasing rearrangement of trigonometric function(s)

a) I am trying to find the decreasing rearrangement (DR) $f^*$ of the following function: $$f(x)=\sin(2x)+\sin(x)+2$$ in the interval $[0,2\pi]$. Admittedly, the $+2$ serves to get rid of negatives. ...
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Intuition regrading the weak $L^p$ functions

In this set of notes in harmonic analysis by Tao, the following remark is made: On a Enclidean space ${\bf R}^d$, the power function $|x|^{-\alpha}$ lies in weak $L^p$ if and only if $\alpha=d/p$. ...
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Coulomb terms of Thomas Fermi model is a symmetric decreasing function

In the proof of theorem 2.12 in Thomas-Fermi and related theories (Lieb) it is stated that: if $$\int_{\mathbb{R}^{3}}\rho\leq Z $$ then $$f(x)=Z \vert x\vert^{-1}- \int_{\mathbb{R}^{3}}\frac{\rho^{*}(...
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what is the intuition behind equimeasurable decreasing rearrangement of a function

I am very new to the these topics. I am in a section about Lorentz spaces in Adam's book on Sobolev spaces. What is the intuition in equimeasurable decreasing rearrangement of a function. Given a ...
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Inequality of Schmidt: $\int_S |x-y|^{-s} dy \leq \omega_n (n-s)^{-1} R^{n-s},$ where $|S| = e_n R^n = | B_R(x) |$ and $e_n = \frac{\omega_n}{n}$

If $0<s<n$ and $S$ is a measurable set with $|S| < \infty$. Then $$\int_S |x-y|^{-s} dy \leq \omega_n (n-s)^{-1} R^{n-s},$$ where $|S| = e_n R^n = | B_R(x) |$ and $e_n = \frac{\omega_n}{n}$ ...
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Calculation of symmetric-decreasing rearrangement

I can't image the symmetric-decreasing rearrangement, so , I want to calculate some example, but fail. For example,how to calculate the symmetric-decreasing rearrangement of $f(x)=x$ on $[0,10]$, zero ...
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How to show $\lambda_1(\Omega)\ge \lambda_1(B)$ when $|\Omega|=|B|$

$B$ is the unit ball in $\mathbb R^n$ , and $\Omega\subset \mathbb R^n$ is a domain which has same volume with $B$. $\lambda_1$ is the first eigenvalue of Laplace with Dirchlet bound condition. Then, ...
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Rearrangement of Laplacian function

It is known that if $\nabla u$, in the sense of distributions, is a function that satisfies $\|\nabla u\|_{L^2(\mathbb{R}^n)}<\infty$, then its Symmetric decreasing rearrangement $\nabla u^\ast$ ...
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On rearrangement of level set: $\{f>t\}^* = \{f^*>t\}\,\,\text{?}$

Let $A$ be a subset of $\mathbb{R}^n$ then the rearrangement of $A$ denoted by $A^*$ is the ball $B(0,r)$ having the same volume as $A$ i.e if $|A| =|B(0,r)|$ with respect to Lebesgue measure then $...
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A stronger form of the weak $(1,1)$ inequality for the Hardy-Littlewood maximal function

I am trying to show that for $f \in L^1(\mathbb R^d)$, if $f^*(x)$ is the Hardy Littlewood Maximal function, then the following inequality is satisfied:$$|\{x : f^*(x)> \alpha\}|\leq \dfrac{c}{\...
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Derive Hausdorff-Young inequality from Paley's inequality

Given a sequence $(c_j)_{j\in\mathbb{Z}}$ of complex numbers with $\lim_{|j|\to\infty}c_j=0$. Define the rearrangement $c_j^*$ as follows: for $j\geq 0$, $c_j^*$ is the $j+1-$th largest element of the ...
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Measure-preserving map between a function and its symmetric rearrangement

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can ...
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Which 2D domain with fixed area has the lowest laplacian eigenvalue?

Which 2D domain with fixed area has the lowest laplacian eigenvalue? I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?
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Does the symmetric decreasing rearrangement of a smooth function preserve smoothness?

Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of a ...
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Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
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The graph of symmetric-decreasing rearrangment of some function eg. $e^{x}$

$A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing rearrangement of a measurable function $f:\mathbb{R}^n \to \mathbb{R}$ is then defined by $$f^*(x):=\...
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Spherical rearrangement

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: ...
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Distribution function and decreasing rearrangement

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement $$\alpha_k=\inf\{\...
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Decreasing rearrangement of trigonometric function

If $f(x)=|\arctan(x)|$, how does one constructively prove that its decreasing rearrangement is given by the constant function $f^\star(y)=\pi/2$ defined on $[0,\infty)$? The decreasing rearrangement ...
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Increasing rearrangement and Hardy-Littlewood inequality

Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the ...
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The decreasing rearrangement is non-expansive

I found this statement about rearrangement from analysis Lieb and Loss in chapter 3. Suppose f, g are nonnegative functions in $L^2(\Bbb{R^n})$, then $\|f^*-g^*\|_2 \le\|f-g\|_2$ Where $f^*$ is the ...
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Symmetric-decreasing rearrangement of a function

I'm studying section 3.3 of Analysis by Lieb and Loss, about symmetric-decreasing rearrangement of functions. Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ ...
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nondecreasing rearrangement is equimeasurable

Two functions $f(x)$ and $g(x)$ are called equi-measurable if $m(\{x:f(x)>t\})=m(\{x:g(x)>t\})$. Nondecreasing rearrangement of a function $f(x)$ is defined as $$f^*(\tau)=\inf\{t>0:m(\{x:f(x)...