2
votes
2answers
56 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
3
votes
2answers
40 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
0
votes
1answer
32 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
0
votes
2answers
24 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
1
vote
1answer
53 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
1
vote
1answer
62 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
0answers
68 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 ...
0
votes
1answer
32 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 = ...
4
votes
1answer
55 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
3answers
59 views

If $f\in L^1$ has a compact support and $0 \leq p \leq1$ then $|f|^p\in L^1$

My text proved that If $f\in L^1$ is bounded and $p \geq1$ then $|f|^p\in L^1$ I wanted to prove the seemingly very similar statement: If $f\in L^1$ has a compact support and $0 \leq p ...
1
vote
2answers
79 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
0
votes
1answer
37 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
2
votes
2answers
56 views

Subsequence of functions in $L^p$

On a problem sheet we were asked to find a sequence of functions $(f_n)_{n \geqslant 0} \in L^p [0,1]$ such that $\lim_{n \to \infty} ||f_n||_p = 0$ but $\lim_{n \to \infty} f_n (x)$ doesn't exist ...
1
vote
2answers
94 views

$L^p$ spaces and counting measure

currently I am working on the following two exercises as a revision for my exam. Let $\mu$ be the counting measure on $\mathbb N$. Show that if $1 \le p < s < \infty$ then $f \in L^p$ implies ...
1
vote
1answer
77 views

Is there a sequence $(f_n)\in\ L^2([0,1])$, s.t. $\lVert f_n\rVert_2=1$, $\forall n$, but it has no convergent subsequences in $L^2([0,1])$ ?

Is there a sequence $(f_n)\in\ L^2([0,1])$, s.t. $\lVert f_n\rVert_2=1$, $\forall n$, but it has no convergent subsequences in $L^2([0,1])$ ? We know at least $(f_n)$, is not convergent in the ...
3
votes
1answer
80 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
3
votes
1answer
140 views

A measurable function with $\int f^n$ bounded or converging as $n \to \infty$

(1) Show that, if $f^n$ is integrable for all integers $n\ge 1$ and $\limsup_{n\to \infty} \int f^n<\infty$ then $|f|\le1$ almost everywhere. (2) Show that, if $f^n$ is integrable for ...
5
votes
2answers
229 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
3
votes
1answer
62 views

convergence on $L^p$ space

Let $ \displaystyle{ f \in L^p (\mathbb R^n), 1\leq p <\infty }$ and let $ \upsilon \in \mathbb R^n$. For $h>0$ define $\displaystyle{ f_h(x) = \frac{1}{h} \int_0^h f(x+s \upsilon) ds }$. ...
2
votes
1answer
225 views

Properties of $||f||_{\infty}$ - the infinity norm

Prove that $||f||_{\infty}$ is the smallest of all numbers of the form $\sup\{|g(x)|: x\in X\}$, where $f=g$ ($\mu$ almost everywhere). In addition, if $f$ is a continuous function on the measure ...
2
votes
1answer
109 views

Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$

Let $1<p_0<\infty$. Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$, but $f_k$ does not converge in $L^{p_0}$. ...
2
votes
3answers
79 views

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$.

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose there exist $g\in L^{p_1}$ and $h\in ...
5
votes
1answer
170 views

Proving a few things about $ L^{p} $-spaces

I am new to $ L^{p} $-spaces and am trying to prove a few things about them. Therefore, I would like to ask you whether I have gotten the following right. Prove that $ {L^{\infty}}(I) \subseteq ...
2
votes
2answers
62 views

Integral construction on $L^1(0,a), a>0$.

I am working on the following problem: Let $a > 0$, $f \in L^1(0,a)$ and define $$ g(x) = \int_x^a f(t) t^{-1} dt, \quad 0 < x \leq a. $$ Show that $g \in L^1(0,a)$ and $\int_0^a ...
4
votes
1answer
307 views

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain ...
2
votes
1answer
67 views

Function bounded a. e.

I have a question: if $f$ is uniformly bounded in $L^2(0,T,X)$ , then $f$ is uniformly bounded a.e. in $X \times (0,T).$ If yes, how to prove it? Thank you.
2
votes
1answer
54 views

Lebesgue's convergence for $H(u_n)\nabla u_n$ where $H$ is not everywhere defined

Consider the Heaviside function that is undefined in zero, i.e. $$H(t)=\begin{cases} 1&t>0 \\ 0&t< 0\end{cases}$$ Now consider a sequence of $H^1(\Omega)$-functions $u_n\to u$ in the ...
1
vote
1answer
93 views

Continuous embedding of $L^p$ into $L^q + L^\infty$

Let $L^p$ denote the Lebesgue-space over a $\sigma$-finite measure space $(\Omega,\mu)$. It is known that $L^{p_0} \cap L^{p_1} \hookrightarrow L^p \hookrightarrow L^{p_0} + L^{p_1}$ continuously ...
3
votes
1answer
148 views

$L^\infty$ and the intersection of the spaces $L^p$

I'm trying to understand if it's true that: " if $f\in L^p\quad \forall p\in N\implies f\in L^\infty$"? My thoughts: Since $\int_R |f(x)|^p dx<\infty\quad\forall p\implies |f(x)|\to 0$? Can anyone ...
1
vote
1answer
423 views

Is a $L^p$ function almost surely bounded a.e.?

I just have a quick question related to $L^p$ spaces. Any help is greatly appreciated. Is it true that if a function $f$ belongs to $L^p$ space, absolute value of $f$ raise to the power of $p$ is ...
4
votes
0answers
96 views

$f(y-x)$ integrable implies $f=0$ a.e.

If $f(y-x)$ is in $L^p(\mathbb R^d\times\mathbb R^d)$, then I seem to conclude that $f=0$ a.e. (which seems wrong). My reasoning is that by Fubini and the integral's shift invariance (assume $p=1$ for ...
2
votes
0answers
85 views

An almost orthogonality principle for $L^p$

If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
0
votes
2answers
65 views

function in $L^1\setminus L^2$

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded). Does anyone know such a function?
0
votes
1answer
170 views

Embedded Lp spaces [duplicate]

Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that ...
8
votes
1answer
211 views

$L^{p}$ functions from Rudin Exercises 3.5

I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
0
votes
1answer
73 views

Absolute Convergence of a Function

I have got stuck with a question. Please help me. Prove that $\dfrac{\sin(x)}{x}$ belongs to $L^p$ for all $p>1$. Thank You.
2
votes
2answers
115 views

$L_p$ space,convergence

Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to ...
2
votes
1answer
136 views

Uniform convergence in $L^p$-spaces

Let $f\in L^p(0,\infty)$, $p>1$. Show that $\int_0^\infty f(x)\frac{\sin xy}{x} dx$ converges uniformly in $y$ in every finite interval. Show also that $|g(t+y)-g(y)|\leqslant M|t|^{\frac{1}{p}}$. ...
3
votes
1answer
59 views

Why $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$?

How show that $f(x + x^3, y + y^3) \in L^1(\mathbb{R}^2)$, when $f(x, y) \in L^2(\mathbb{R}^2)$? Can someone help me? Thank you!
3
votes
1answer
93 views

Function in $L^1([0,1])$ that is not locally in any $L^{\infty}$

Can we find a function such that $f\in L^1([0,1])$ and for any $0\leq a<b\leq 1$ we have that $||f||_{L^{\infty}([a,b])}=\infty$?
1
vote
1answer
62 views

How to show that a certain function is in $L^p$?

How can I show that the function $ F(x)= \dfrac{|x| ^{-n+1}} { \log \frac{1}{|x|} } $, for $ 0 < |x| \leqslant \large\frac{1}{2} $ and $ F(x)=0 $, if $ |x|>\large\frac{1}{2} $, is in ...
1
vote
1answer
182 views

Prove that $f(x)$ is integrable on $\mathbb{R}$.

Suppose $f(x)$,$xf(x)$ $\in$ $L_2(\mathbb{R})$. Prove that $f(x)\in$ $L_1(\mathbb{R})$.
3
votes
1answer
131 views

How to know a function is in $L^p$.

I am having trouble with this question. It is not an homework question, I am currently trying to practise different problems for an exam. Let $f$ be a nonnegative measurable function on $\mathbb ...
2
votes
1answer
474 views

Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$. I ...
15
votes
2answers
5k views

Limit of $L^p$ norm

Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
1
vote
1answer
192 views

$L^p$ spaces in integration measure

This question looks simple at the first glance but ... I have tried to combine the theorems and definitions on $L^p$ spaces to solve this question but I have not been able to do so. I need help to ...
23
votes
2answers
6k views

$L^p$ and $L^q$ space inclusion

Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
7
votes
3answers
224 views

Convergence of functions in $L^p$

Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
2
votes
1answer
193 views

Convergence of integrals in $L^p$ and $L^{p/(p-1)}$

Let $X$ be a measure space and let $f_{n}$ be a sequence of functions which converge pointwise to a function $f$ in $L^{p}(X)$ where $p>1$ and suppose $g_{n}$ is a sequence of functions which ...