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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.

Let $(X, \mathcal A,\mu)$ be a measure space, and $0<p<\infty$

Definition: The weak $L^p-$space on $(X, \mathcal A,\mu)$ denoted $L^{p,\infty}(X, \mu)$ is defined as the set of all $\mu$-measurable functions $f$ such that: $$\|f\|_{L^{p,\infty}} = \sup\{ t\mu\left(\{x\in X: |f(x)|>t\}\right)^{1/p}: t>0\}<\infty.$$

These spaces contain the classical lp spaces and are particular cases of Lorentz-spaces

78 questions
8
votes
3 answers

Triangle inequality fails in $L^{1,\infty}$

It can be proved that $\forall\varepsilon>0$ there exists $C(\epsilon)>0$ such that for all $f,g\in L^{1,\infty}(\Bbb R^n)$ we have that $$ ||f+g||_{1,\infty}\le(1+\varepsilon)||f||_{1,\infty}+C(\varepsilon)||g||_{1,\infty}$$ for example…
Joe
  • 12,091
8
votes
1 answer

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, \infty}} = \sup_{t\in\left(0, \infty\right)}…
user1321324
  • 606
7
votes
1 answer

Weak $L^{p}$ spaces are quasi-normed?

Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition $L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that \begin{eqnarray*} \|f\|_{p,\infty} &:=& \text{inf}\{c > 0:…
roo
  • 5,800
7
votes
2 answers

How to show that $C^\infty_0$ is not dense in $L^p_{weak} (\mathbb{R}^n)$?

Let $C^\infty_0$ denote the smooth, compactly supported functions on $\mathbb{R}^n$. Let $L^p_{\mathrm{weak}}(\mathbb{R}^n)$ denote the space of all functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ which satisfy $$ \Big| \big\{x \in \mathbb{R}^n…
JMill.
  • 403
5
votes
1 answer

Besov or Triebel-Lizorkin spaces versus Lorentz spaces

At the $0$ order of derivatives of Sobolev spaces, we find Besov spaces $\dot{B}^0_{p,q}$, Triebel Lizorkin spaces $\dot{F}^0_{p,q}$ and Lorentz spaces $L^{p,q}$, with in particular if $p≥ 2$ $$ \begin{align*} \dot{B}^0_{p,1} ⊂ \dot{B}^0_{p,2} ⊂ L^p…
LL 3.14
  • 13,938
4
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0 answers

Another definition for vector-valued distributions

Here, $\Omega\subseteq\mathbb{R}^n$ is an open set, $X$ is a Banach space, and $X^*$ its topological dual. I ended up on a problem for which I need to define a suitable notion for the gradient of a map $\mu:\Omega\to X$, and I would need this object…
rod
  • 905
4
votes
2 answers

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 $1 < p < \frac{q}{q-1}$. Is $f$ necessarily in…
stoic-santiago
  • 14,877
4
votes
1 answer

Best constant in Weak-$L^p$-triangle inequality

What is the best constant $C_p$ in the "triangle inequality" $$ \| f + g \|_{p,\infty} \le C_p ( \|f\|_{p,\infty} + \|g\|_{p,\infty})$$ for the weak $L^p$ spaces? Here, I am mostly interested in the case $p \in [1,\infty)$. Typical proofs show $C_p…
gerw
  • 33,373
4
votes
1 answer

Inclusion of $L^p$ and weak $L^p$ spaces

Let $0
Xiang Yu
  • 5,193
4
votes
1 answer

A proposition to prove the real interpolation of positive exponent

Let $p,A\in(0,\infty)$ $\|f\|_{L^{p,\infty}(X,\mu)} := \sup \left\{\lambda\mu(\{|f|\geq\lambda\})^{\frac{1}{p}}\right\}$ . Show that the following are equivalent (1) $\|f\|_{L^{p,\infty}(X,\mu)}\leq C_pA$ for some constant $C_p$ (2) For any set $E$…
user134927
  • 510
3
votes
1 answer

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 of pg. 148 we assume for contradiction that there exists a…
AJY
  • 9,086
3
votes
1 answer

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 $$ \int_{|f| \leq s} |f|^q \leq \frac{q}{q-p} s^{q-p} \|f\|_{L^{p,\infty}}^p. $$ (b)…
Franlezana
  • 552
3
votes
0 answers

Bounded gradient in weak $L^1$ implies compactness in $L^1$ (?) Relation to fractional Sobolev compactness.

Let $u_k$ be a sequence of functions in $C^{\infty}_c(B_1),$ where $B_1 \subset \mathbb{R}^n$ is the unit ball. Then, if $$\sup_{k} \|u_k\|_{L^1(B_1)} < \infty \quad \text{and} \quad \sup_{k} \|\nabla u_k\|_{L^{1,\infty}(B_1)}<\infty,$$ is it true…
Ignatius
  • 566
3
votes
0 answers

Show $[fg]_1\leq p^{\frac{1}{p}}(p')^{(\frac{1}{p'})}[f]_p[g]_{p'}$ in weak $L^P$ norm

Let $f: X \rightarrow \mathbb{R}$ be a measurable function where $(X,\mu)$ is a measure space and say that $f \in L^{p,\infty}$ $\iff$ $[f]_p < \infty$ where $[f]_p = \sup_{t>0} t \mu(\{x : |f(x)| > t \})^{\frac{1}{p}} < \infty$. In my measure…
user637978
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