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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.

2
votes
2answers
42 views

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$

Show that $g_n(x)=ne^{-nx}$ is bounded, where $x\in\Omega=]0,1[$ for $n\in\Bbb{N}.$ My trial I'm thought of it this way that if $g_n(x)=ne^{-nx}$ converges weakly in $L_1(\Omega)$ then it is bounded....
3
votes
1answer
45 views

How can I prove that the Hilbert transform on the 1-torus doesn't map $L^1(\mathbb{T})$ into itself?

Let $\mathbb{T}$ be the 1-torus. Then, it is well defined the Hilbert transform: $$\mathcal{H}:L^1(\mathbb{T})\to L^0(\mathbb{T}), \vartheta\mapsto\int_{-\pi}^\pi f(\vartheta-t)\cot\left(\frac{t}{2}\...
2
votes
1answer
27 views

If we have a sequence of measurable functions that is Cauchy with respect to the weak L^p norm, is it Cauchy with respect to the L^p norm?

If $(f_n)$ measurable on $(X,\mathcal{M},\mu)$, $f_n$ Cauchy with respect to weak quasi $L^p$-norm: $[f_n]_p=\sup_{\alpha>0}\alpha \lambda_{f_n}(\alpha)^{\frac{1}{p}} $ where $\lambda_{f}(\alpha)...
3
votes
1answer
34 views

$L^{q,\infty}(\mu) \subset L^p(\mu)$ with $1 \leq p<q <\infty$ and $\mu$ finite measure?

Let $\mu$ be a finite measure on a measurable space $(X,\Sigma)$. I want to prove that exists $C > 0$ so that $$ \| f\|_p \leq C \| f \|_{q,\infty}$$ when $1 \leq p < q < \infty$, where $$ \| ...
1
vote
0answers
16 views

Counting Balls in $L^2_m[0,1]$

Setup and Thoughts to Date Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a ...
0
votes
2answers
27 views

Weak Star Convergence of $u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)})$

Consider the sequence \begin{equation}u_n=\sin(\frac{nx}{n+1})(1+\exp{(-n|y|)}) \text{ where } (x,y)\in I=[-1,1]\times[-1,1], n\in\mathbb{N}.\end{equation} (a) Study the equicontinuity of $(u_n)$ ...
3
votes
2answers
79 views

A proposition about bounded and weak-$L^2$ integrable fucntion

I have this proposition that I have no idea how to start. any hints helps. If $f\in L^{2,\infty}\cap L^{\infty}$ then $f\in L^p$ for $p\geq2$ and the second part asks me to generalize this result. ...
2
votes
1answer
47 views

Embedding $L^p$ into weak $L^p$ and notion of boundary of $L^p$

I'm very much a novice to analysis, so this is a kind of conceptual question that is probably very much the wrong way to think about these spaces, but perhaps it fits my more topological brain. The ...
6
votes
2answers
100 views

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

Let $C^\infty_0$ denote the smooth functions on $\mathbb{R}^n$ which decay at infinity. Let $L^p_{\mathrm{weak}}(\mathbb{R}^n)$ denote the space of all functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$...
2
votes
0answers
260 views

Definition of weak $L^p$ spaces

Today I came across the notation $L_\text{weak}$ and I don't know what exactly it means. I can only find the weak Lp spaces , but here they don't use the same notation. Is this the same thing? I'm ...
0
votes
1answer
44 views

$L^2$ convergence given weak $L^2$ convergence

Let $K\in L^1(\mathbb{R}^d)$ with Lebesgue measure. Suppose that $\psi_n\in L^2(\mathbb{R}^d)$ is a sequence of functions such that $\psi_n\to \psi$ weakly in $L^2$ and $\psi_n\equiv 0$ when $|x|>...
2
votes
2answers
30 views

Deriving a particular Poincare equality by a lemma

What I want to prove is the following inequality $\Vert u \Vert_{L^2[0,1]} \leq \Vert u' \Vert_{L^2[0,1]}, \quad u \in W^{1,2}_0 [0,1], $ which I am trying to derive from $$ \int_\Omega \vert u(x+...
1
vote
0answers
49 views

A weakly open subset of the unit ball of the Read's space $R$ (an infinite-dimensional Banach space) is unbounded.

We know every weakly open subset of an infinite-dimensional Banach vector space X is unbounded. Now, Read's space $R$ (an infinite-dimensional Banach space) has the property: there is $ρ >0$ such ...
0
votes
1answer
87 views

$\lVert f_{1}+\cdots+f_{m}\rVert_{L^{p,\infty}}^{p}\le \frac{2-p}{1-p}\sum_{j=1}^{m}\lVert f_{j}\rVert_{L^{p,\infty}}^{p}$

Let $(X,\mu)$ be a measure space. Then for each $p$ between $0$ and $1$,we have $$\lVert f_{1}+\cdots+f_{m}\rVert_{L^{p,\infty}}^{p}\le \frac{2-p}{1-p}\sum_{j=1}^{m}\lVert f_{j}\rVert_{L^{p,\infty}}^{...
0
votes
1answer
31 views

For $0<p<\infty.$Show that $\left\lVert\max_{1\leq j\leq m}|f_j|\right\rVert_{L^{p,\infty}}^{p}\leq\sum_{j=1}^{m}\lVert f_j\rVert_{L^{p,\infty}}^p$

Let $(X,\mu)$ be a measure space and let $f_j$,$1\leq j\leq m$ be measurable functions on X. For $0<p<\infty.$Show that $$\bigg\lVert\max_{1\leq j\leq m}|f_j|\bigg\rVert_{L^{p,\infty}}^{p}\...
0
votes
1answer
31 views

Show that $f$ is belongs to weak $L^{1}(E)$ and $f$ is bounded on $E$. Then $f$ is $L^{p}(E)$ for each $1<p<\infty$

Show that $f$ is belongs to weak $L^{1}(E)$ and $f$ is bounded on $E$. Then $f$ is $L^{p}(E)$ for each $1<p<\infty$ ,where $E$ is a subset of $\mathbb{R^{n}}.$ Here is my proof : Now we take ...
2
votes
1answer
57 views

Show that$\,\,\int_{|f|\leq s}|f\,|^{q}\,d\mu\leq\frac{q}{q-p}s^{q-p}\lVert f\,\rVert_{L^{p,\infty}}^p\,\,,$ where $0<p<q<\infty$

Let $(X,\mu)$ be a measure space and let $s>0$ Let $f$ be a measurable function on $X$ .Show that $0<p<q<\infty $ we have $$\int_{|f|\leq s}|f\,|^{q}\,d\mu\leq\frac{q}{q-p}s^{q-p}\lVert f\...
1
vote
1answer
51 views

some inequality for weal-$L^p$

Let $(X,\mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E\,)<\infty$ Assume that $f\in L^{p,\infty}(X,\mu)$ for some $0<p<\infty$. Show that for $0<q<p$ we have $$\...
0
votes
1answer
49 views

the additivity for space of weak $L^p$

There is an exercise in my textbook with working by myself : Let $f_1,f_2,...,f_N$ be in $L^{p,\infty}(X,\mu)$ which is the well-known space of weak $L^p(X,\mu)$ Prove that for $1\leq p< \infty$ ...
1
vote
1answer
256 views

Normability of weak $L^p$-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 $\mu$-...
0
votes
1answer
220 views

Intuition of the Lorentz norm in an Wikipedia article

It is said in this Wikipedia article that the Lorentz space norm ${\|f\|_{L^{1,\infty }}}$ with ${f(x)=|x-1|^{-1}}$ is the area of the largest rectangle with sides parallel to the coordinate axes ...
0
votes
2answers
97 views

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 $\alpha=d/p$. ...
0
votes
1answer
38 views

Do we have $\lVert f\rVert_{L^p(X)}=\int_0^\infty p\alpha^{p-1}d_f(\alpha)\,\mathrm{d}\alpha$ for non $\sigma$-finite $X$?

Recall that for $\sigma$-finite measure spaces $(X,\mathscr{M},\mu)$, for any measurable function $f:X\to\mathbb{C}$, we can recover $\lVert f\rVert_{L^p(X)}$ by defining the distribution function $...
0
votes
1answer
60 views

Showing equivalence of definition of $L^p$ weak norm

Definition. Let $(X,\mu)$ be a measure space. For $f$ a measurable function on $X$, the distribution function of $f$ is the function $d_f$ defined on $[0,\infty)$ as follows: $$d_f(\alpha) := \mu\left(...
8
votes
3answers
693 views

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}+...
3
votes
1answer
223 views

Triangle inequality in weak $L^1$ space

I have some problems with this exercise: $\forall\varepsilon>0$ 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)...
1
vote
1answer
63 views

If $||g_j-g||_{1,\infty}\to0$, then $||g_j||_{1,\infty}\to||g||_{1,\infty}$

I have some problems with my notes: my teacher wrote that if a sequence $\{g_j\}_j\subseteq L^{1,\infty}(\Bbb R^n)$ (which is the weak $L^1$ space, endowed with the quasinorm $||\cdot||_{1,\infty}$) ...
1
vote
0answers
72 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
2
votes
1answer
199 views

hint on exercise about weak $L^p$ space

I'm working on a problem from Grafakos, Classical Fourier Analysis. Let $(X, \mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E) < \infty$. Assume that $f$ is in $L^{p,\infty}(...
2
votes
1answer
60 views

Problem with $L^p(\Bbb R^n)$ spaces (Marcinkiewicz Theorem)

Marcinkiewicz Theorem says that, if $1\le p<q$ given a bounded operator $T$ from $L^p(\Bbb R^n)$ to $L^{p,w}(\Bbb R^n)$ (the last one is the $L^p$ weak space) AND EVEN from $L^q(\Bbb R^n)$ to $L^{q,...
2
votes
0answers
125 views

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

Let $0<p_0<p_1<\infty$, $0<\theta<1$, and $1/p_\theta=(1-\theta)/p_0+\theta/p_1$. Show that $$L^{p_\theta,\infty}(X)\subset L^{p_0}(X)+L^{p_1}(X).$$ Suppose that $f\in L^{p_\theta,\...
1
vote
1answer
56 views

Why $L^p$ is strictly contained in $L^{p,\infty}$

I'm reading up on weak $L^p$ spaces (a.k.a. Marcinkiewicz spaces, or $L^{p,\infty}$ spaces), and I have a little trouble seeing why the function $|x|^{-n/p}$ lies in $L^{p,\infty}(\mathbb{R}^n)$ but ...
2
votes
1answer
592 views

Equivalent Definition of Weak $L^{p}$ (Quasi-) Norm

For a sigma-finite measure space $(X,\Sigma,\mu)$, the weak $L^p$ (hereafter denoted $L^{p,\infty}$) is defined by $$\|f\|_{L^{p,\infty}}:=\sup_{t>0}t\mu(|f|>t)^{1/p}, \qquad (1\leq p<\infty)...
1
vote
1answer
185 views

Proof of Hunt's Interpolation

I'm new to weak $L^p$ spaces and I'm doing a book exercise. Can someone enlighten me on the proof of the Hunt's interpolation theorem, which goes as follows: Theorem Let $\langle \,M, \mu \, \...
1
vote
1answer
43 views

Condition on $f$ in $L^{p, \infty} $ implies $f \in L^q$

If $f \in L^{p, \infty}$ and $\mu ( \{ x : f(x) \neq 0 \} ) < \infty$, then $f \in L^q$ where $q <p$. $\textbf{My Attempt:}$ Let $E = \{x:f(x) \neq 0 \}$ $$ || f ||_q^q = q \int_{0}^{\infty} \...
2
votes
0answers
48 views

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 ...
6
votes
1answer
1k views

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}}...
1
vote
1answer
391 views

Reference concerning weak-type (1,1) operator in $\mathbb{R}^d$

I wish someone give me some reference on weak-type (1,1) operator in the d-dimensional Euclidean space. Thanks for any kind help.
4
votes
1answer
133 views

Upper Bound for Operator Norm in Marcinkiewicz Interpolation Theorem

Exercise 1.3.3(c) Let $0<p_0<p<p_1<\infty$ and let $T$ be an operator as in Theorem 1.3.2($\|T(f)\|_{L^{p_0,\infty}(Y)}\leq A_0\|f\|_{L^{p_0}(X)}$ for all $f\in L^{p_0}(X)$ and $\|T(f)\|_{...
8
votes
1answer
680 views

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{...