Tagged Questions

4
votes
1answer
35 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
39 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
73 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 ...
1
vote
0answers
71 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
39 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
77 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
51 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
155 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
74 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
107 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 ...
4
votes
1answer
429 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
174 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 ...
11
votes
2answers
2k 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
179 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
146 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 ...