Questions tagged [weak-lp-spaces]
This tag address to any question concerning weak-lp -spaces. which are larger spaces than classical lp-spaces. These spaces are particular cases of Lorentz-spaces.
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Convolution between $L^1$ function and a singular integral kernel
I meet a problem when reading Modern Fourier Analysis(3rd. Edition) written by L.Grafakos. On pg.82 he writes:
Fix $L\in\mathbb{Z}^+$. Suppose that $\{K_j(x)\}_{j=1}^L$ is a family of functions ...
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$L^\infty(\Omega)$ is dense in $L^{p,\infty}(\Omega)$ if $\Omega$ is compact
Given a compact set $\Omega\subset \mathbb{R}^N$, I am wondering if $L^\infty(\Omega)$ is dense in the weak $L^p$ space $L^{p,\infty}(\Omega)$ with $1< p<\infty$ (see here the definition).
I ...
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A proof of Pólya-Szegő inequality
Denote by $|A|$ the $N-$dimensional Lebesgue measure of a Borel set $A \subset \mathbb{R}^N$ and define
$$
A^\ast := B_{R}(0), \quad R = \left(\frac{N}{\omega_N}|A| \right)^{\frac{1}{N}},
$$
where $\...
- 1,219
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Proving an Integral Identity for the Laplacian in $W^{2,2}(\Omega)$ for Functions in $C_c^{\infty}(\Omega)$
Let $\Omega \subset \mathbb{R}^d$ be open and bounded. Consider the Laplace-Operator given by
$$
\Delta u=\sum_{j=1}^d \partial_j^2 u \quad \text { for all } u \in W^{2,2}(\Omega) .
$$
(i) Show that ...
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Can we find $f: [0,1] \to [0, \infty) \notin L^{\frac{q}{q-1}}[0,1]$ such that $\int_A f \le |A|^{1/q}$ for some $q \ge 2$ and all $A \subset [0,1]$?
Let $f: [0,1] \to [0, \infty)$ be a measurable function satisfying $$\int_A f \le |A|^{1/q}$$ for some $q \ge 2$ and all measurable subsets $A \subset [0,1]$. Show that $f\in L^p[0,1]$ for all $1 < ...
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Prove that simple functions are not dense in weak $L^{p,\infty}$
Let $p > 0$, and denote by $L^{p,\infty}(\mathbb{R})$ the space of all measurable functions
$f : \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
\|f\|_{p,\infty}:=\sup_{\alpha > 0} \alpha^p |\{x ...
- 1,575
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Confused by part of the proof of Stein's maximal principle
I'm reading through the proof of Theorem 1 in Stein's "On limits of sequences of operators", and I'm confused at a step. I'll try to replicate all the parts I think are relevant. At the end ...
- 8,769
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Weak L^p space and inequality concerning Fourier transform
The norm in $L^{p, \infty}$ is defined as
$$
\sup_{\alpha > 0} \alpha |\{ x \in \mathbb{R}^n: |f(x)| > \alpha \}|^\frac{1}{p}.
$$
Fourier transform and it's reversal is defined by
$$
\mathfrak{...
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a weak $L^p$ embedding inequality
For $1<p<\infty$ , if $f$ is a weak $L^p(\mathbb{R}^n)$ function , then prove that $f_\delta:=f.{\bf{1}}_{(|f|\geq\delta)}$ belongs to $L^q(\mathbb{R}^n)$ for $1\leq q<p$ and for all $\delta&...
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Grafakos Classical Fourier Analysis problem 1.1.14
I'll first type the problem.
Let $(X,\mu)$ be a measure space and let $s>0$.
(a) Let $f$ be a measurable function on $X$. Show that if $0 < p < q < \infty$, we have
$$ \int_{|f| \leq s} |f|...
- 520
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Boundedness of Fourier transform on weak $L^2$ spaces.
It is well-known that Fourier transform $\mathcal{F}$ is isometry on $L^2(\mathbb{R}^d)$.
I would like to know whether $\mathcal{F}$ is bounded or not on weak space $L^{2,\infty}(\mathbb{R}^d)$.
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The space of continuous function on $[0,1]$ in dense in $L^{\infty}$ respect to the weak$-*$ topology
I am reading Rudins' book on Functional Analysis for self study. I stumbled upon an exercise and I would like someone to revise my solution: exercise 7 chapter $3$ second edition.
The exercise asks to ...
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Why find weak solutions of PDE, and how to go from weak to strong?
In some math books, instead of finding a strongly differentiable solution to some PDEs, they find weak solutions (weakly differentiable). How can we say that the weak solution satisfies that PDE? This ...
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Brézis Exercise 3.18 [duplicate]
For every integer $n \geq 1$ let
$$e^n=(0,0,\dots,\underset{(n)}{1},0,\dots).$$
Prove that $e^n \underset{n \rightarrow \infty}{\rightharpoonup} 0$ in $\ell^p$ weakly $\sigma(\ell^p,\ell^{p'})$ with $...
- 2,379
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Is $f \mapsto (x\mapsto \int_0^x f\,\mathrm d\lambda)$ weak-strong continuous from $L^1([0,1])$ to $C([0,1])$?
A mapping $T : L^1([0,1]) \to C([0,1])$ is called weak-strong continuous at a point $f\in L^1([0,1])$, if for every sequence $(f_n)_n$ of functions in $L^1([0,1])$ that converges weakly to $f$ $\left(\...
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