Tagged Questions

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...

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3
votes
1answer
42 views

Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...
0
votes
2answers
15 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
1
vote
1answer
39 views

a question about $L^p$ functions on domains in Euclidean spaces

Let $\Omega$ be an open set in $\mathbb{R}^n$ and $f\in L^p(\Omega)$, $1\leq p<\infty$. Define $||f||_{p,\Omega}=\inf\{||f-a||_p: a\in\mathbb{R}\}$. Prove that there exists $a\in\mathbb{R}$ such ...
0
votes
1answer
55 views

Am I wrong ? (2)

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and ...
2
votes
0answers
31 views

Proof verification-density of smooth compactly supported functions

I am trying to show that $C_{c}^{\infty}(\mathbb{R})$ (smooth compactly supported functions) is dense in $C_{c}(\mathbb{R})$ (in the $L^{p}$ sense). Can anyone check if my proof is correct? Let $f ...
2
votes
1answer
54 views

Is a non-compact Riemannian manifold a “measure space”?

One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times ...
1
vote
1answer
23 views

Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
2
votes
1answer
48 views

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$

How to show that the sequence $(x^{(n)})$ weakly convergent in $l_p$, $1\le p\lt \infty$. where $(x^{(n)})=(\underbrace{0,0,..0}_{n-1},1/n,1/(n+1),...,1/(2n),0,0,...)$ for $n\in\mathbb{N}$
1
vote
1answer
31 views

How to give a criterion for strong convergence of in $L_p[0,1]$ for this example

let $x_n=\alpha_n e^{-nt}$ for $n\in \mathbb{N}$ and $1<p<\infty$. How to give a criterion for strong convergence of in $L_p[0,1]$ for this example: $x_n\rightarrow 0$ (strong convergence). ...
0
votes
0answers
51 views

Generalisation of vector-valued Marcinkiewicz interpolation theorem

Given a compatible couple $(X,Y)$ of Banach spaces and some measure space $\Omega$ Lions and Peetre identified the real interpolation space between the vector-valued Lebesgue spaces $L_1(\Omega;X)$ ...
1
vote
1answer
67 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
votes
1answer
41 views

function $L_p$ iff $1\leq p<2$

Let $X=<0,1>$, take the borel sigma algebra, and the lebesgue measure. Consider $g(x)=\dfrac{1}{x^{\frac{1}{2}}}$. Show that $g\in L_p$ iff $1\leq p<2$. I have done this: ...
0
votes
0answers
13 views

$L^p(0,T;L^p(M)) = L^p((0,T)\times M)$?

I thought it was true that $L^p(0,T;L^p(M)) = L^p((0,T)\times M)$ for $p \neq \infty$, but can't find a proof. Can someone assist me with this? Thank you
0
votes
1answer
42 views

Bound on uniform norm of convolution of $L^p$ functions

This is Proposition 8.8 in Folland's Real Analysis: If $p$ and $q$ are conjugate exponents, $f \in L^P$, and $g \in L^q$, then $f*g(x)$ exists for every $x$, $f*g$ is bounded and uniformly ...
4
votes
2answers
75 views

If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?

Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$? What I actually want is that $X$ and ...
1
vote
1answer
46 views

Nontrivial functionals on $l^\infty$ vanishing on $c_0$

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?
2
votes
0answers
69 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
2
votes
2answers
78 views

Subsequence Properties and Lp Spaces

Exercise: Let $E$ be a measurable set in $\mathbb{R}$ under Lebesgue measure, and let $1<p<\infty$. Suppose $\{f_n\}$ is a bounded sequence in $L^p(E)$ and $f \text{ belongs to } L^p(E) $. ...
2
votes
1answer
37 views

Question on Lp Spaces

Exercise: Assume $E$ has finite measure and $1\le p_{1}<p_{2}\le\infty$. Show that if {$f_{n}$}$\space\rightarrow f$ in $L^{p_{2}}(E)$ then {$f_{n}$}$\space\rightarrow f$ in $L^{p_{1}}(E)$ ...
1
vote
1answer
29 views

$l^p$ norms and the relationship between $l^{p_1}$ and $l^{p_2}$ where $p_1<p_2$

Let $a=\lbrace a_n \rbrace_{n=1}^\infty$ be a sequence of real numbers. We define $\| a\|_p = \left( \sum_{n=1}^\infty a_k^p \right)^{1/p}$ for $0<p<\infty$ and $\| a\|_\infty = \sup_k |a_k|$. ...
0
votes
1answer
33 views

Functions in $L^p$ and $L^q$ spaces

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$. Solution: let $$f(x)=x^{-1/p}(1+|\log x|)^{-2/p}$$ Then $$\int|f|^p = ...
0
votes
1answer
46 views

Measure theory $L^p$ and $L^q$ spaces

For any two different numbers $p,q\in[1,\infty)$ find functions $f\in L^p \setminus L^q$ and $g\in L^q \setminus L^p$. Idea: This and the function $f=x^{-1/p}(1+|log x|)^{-2/p}$ and then I need to ...
1
vote
0answers
33 views

Convergence in $L^1$ and $L^p$ [duplicate]

Assume: (a) $\{f_n\} \subset L^p$, (b) $f_n \to f \text{ }\mu\text{-a.e.}$ and (c) $\|f_n\|_{L^p} \to \|f\|_{L^p}\to0$. Then show that $$\|f_n - f \|_{L^p} \to 0$$ using Fatou's Lemma, first for ...
1
vote
0answers
49 views

Condition for a product of two function sequences in $L^1$ to be in $L^1$

We have: 1. $\{f_n\} \subset L^1(E, \Sigma, \mu)$ 2. $g_n \subset L^\infty(E, \Sigma, \mu)$ 3. $\|f_n-f\|_{L^1}\to0$ 4. $g_n \to g \text{ }\mu\text{-a.e.}$ 5. $\{g_n\}$ is uniformly bounded. ...
2
votes
0answers
29 views

References on Weak Convervenge

I am looking for a good reference on weak convergence in L^p spaces, can anyone recommend anything? Thanks a lot in advance!
4
votes
1answer
74 views

Function compositions that are in $L^p$

We have $f \in L^p$. The goal is to show that $\exists \psi \in C(\mathbb{R^+}, \mathbb{R^+})$ such that $$ \lim_{s \to +\infty} \frac{\phi(s)}{s}=+ \infty \text{ and } \phi(|f|) \in L^p$$ I neeed ...
2
votes
1answer
64 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
0
votes
1answer
32 views

If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
0
votes
1answer
49 views

Sequence of piecewise constant functions converging to any $L^2$ function

Let $\{P_i\}_{i=1}^\infty$ be a sequence of partitions of the interval $[0,1]$ with a vanishing mesh. Additionally $H_i$ be the space of piecewise constant functions (step functions) with pieces ...
1
vote
1answer
60 views

1<p<q, norm inequalities

Prove or disprove the following. (Recall that $c_{00}$ is the space of sequences with only finitely many nonzero entries.) Conjecture. Let $1<p<q<\infty$. Then there exists a function ...
2
votes
1answer
19 views

a function that is in $L^2$ (the right version)

Sorry I made a mistake when posting the last question. Actually my question is: can you give a $f(x) $ such that $ f \in L^2 ( \mathbb R)$ but $ x^{-\frac{1}{2}} f \notin L^1 ( \mathbb R ) $. ...
2
votes
1answer
47 views

a function that is in $L^2$

Can anyone give me an example of $f(x) $ such that $ f \in L^2 ( \mathbb R)$ but $ x^{\frac{1}{2}} f \notin L^1 ( \mathbb R ) $. Thanks! It seems that $f(x) = x^\alpha$ doesn't work...
0
votes
1answer
42 views

No Continuous Mapping?

Exercise: For [a,b] a nondegenerate closed, bounded interval, show that there is no continuous mapping from $ L^{1}[a,b]$ onto $ L^{\infty}[a,b] $. My initial thought is to prove by contradiction. ...
0
votes
1answer
35 views

Converse of existing question on L^p convergence

My question is about this: Convergence in $L^p$ of $f_n$ implies convergence in $L^1$ of $|f_n|^p$ and $f_n^p$ It was shown that the author's question was indeed true by the use of MVT. Is the ...
3
votes
1answer
99 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
0
votes
0answers
9 views

Existence of uniformly continuous function on $L^p$

Suppose $f\in L^\infty(\Bbb R)$, and $f_h(x)=f(x+h)$ and $\lim_{h \to 0}||f_h-f||_\infty=0$. Prove that there exists a uniformly continuous function $g$ on $\Bbb R$ such that $f=g$ a.e. Problem is ...
5
votes
1answer
147 views

Showing that $\|f\|_p\to\|f\|_{\infty}$

I know this question has been asked a lot in this site, I've been checking those questions myself, however according to the theory we use in class there are things that I can't use and/or I don't know ...
0
votes
0answers
60 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
1
vote
1answer
34 views

Existence of a function in $L^{p}$ with a certain property

Is there a function $f \in L^{p}(\mathbb{R}^{n})$ such that $\|If\|_{L^{p}(\mathbb{R}^{n})} = \infty$ where $If(x) = \int_{\mathbb{R}^{n}}\frac{f(y)}{|x - y|^{n}}\, dy$? Such a case isn't covered by ...
2
votes
1answer
46 views

Convergence of function in $L^1$ space

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
0
votes
1answer
43 views

Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
1
vote
1answer
23 views

Convergence of series in L^p

Let $\langle f_n\rangle$ be a sequence of functions from the space $L^p(X,\mathcal M,\mu)$, $1\le p<\infty$, such that the series $$ \sum_{n=1}^\infty \|f_n\|_p \ \ \ (*) $$ converges. Prove that ...
0
votes
1answer
42 views

Why is $\overline{L^\infty(\Omega)} \subset L^\infty(\Omega) $ where the closure is in norm of $L^1(\Omega)$?

Let $\Omega$ be a domain which may or may not be unbounded (eg. $\Omega = B_1(0)\times (0,\infty)$). Why is $$ \overline{L^\infty(\Omega)} \subset L^\infty(\Omega)$$ where the closure is in norm of ...
2
votes
1answer
25 views

$f_n$ converges to $f$ in $L^3(X,\mathcal{M},\mu)$ , prove that $f_n^3$ converges to $f^3$ in $L^1(X,\mathcal{M},\mu) $

Suppose $(X,\mathcal{M},\mu)$ is a complete measure space. If $f_n$ converges to $f$ in $L^3(X,\mathcal{M},\mu)$ , prove that $f_n^3$ converges to $f^3$ in $ L^1 (X,\mathcal{M},\mu) $. Actually my ...
2
votes
1answer
36 views

$\|f'(x)\|_{L^p} \le C \|f(x)\|_{L^p}^{1/2} \|f''(x)\|_{L^p}^{1/2}$ for smooth $f$ with compact support

I'm trying to prove the following Let $f: \mathbb{R} \to \mathbb{R}$ be a smooth function supported on $[a, b]$ where $-\infty < a < b < \infty$. $2 \le p < \infty$. Then $$ ...
4
votes
1answer
69 views

$L^p$-space inclusions

Let $1\leq p<q<\infty$. Which of the following inclusions are true? $L^p(0,1)\subset L^q(0,1)$ $L^q(0,1)\subset L^p(0,1)$ $L^p(0,\infty)\subset L^q(0,\infty)$ $L^q(0,\infty)\subset ...
2
votes
1answer
81 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
1
vote
1answer
59 views

Weak $L_1$ norm is different from $L_1$ norm on a probability space

Can any one give an example of a probability space $(X , \mu )$ and functions $f_1, ...,f_n : X \to \mathbb R$ such that $ \| f_i \|_{ L^{ 1, \infty }} : = \sup_{ t > 0} t \lambda_{f_i} (t) \le 1$ ...
1
vote
1answer
38 views

weak $L_p$ implies bounded integral on finite measure set

Let $(X, \mu)$ be a measure space which is $\sigma$-finite. $ 1 < p < \infty $. $f : X \to \mathbb C$ is a measurable function. If we know $f$ is in the weak $L_p$ space, i.e. $ ||f||_{L^{(p, ...
1
vote
0answers
49 views

the dual space of $L^p$ [duplicate]

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...