2
votes
2answers
54 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 $u \to 0$. ...
0
votes
0answers
41 views

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? [duplicate]

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise ...
2
votes
2answers
50 views

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and some condition.

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and ere exists an increasing function $g:[0,1]\rightarrow \mathbb{R}$ such that $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
1
vote
0answers
20 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
4
votes
2answers
44 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
2
votes
1answer
67 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
1
vote
0answers
51 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
1
vote
1answer
37 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
46 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
3
votes
1answer
58 views

The asymptotics of $\int_x^1 |f|$ as $x\to 0$ for a nonintegrable function with $\int_0^1|f|^p<\infty$, $0<p<1$

Suppose that $0<p<1$. Let $h:(0,1]\to\mathbb C$ be a Lebesgue-measurable function such that $$\int_{x}^1|h(t)|\,\mathrm dt<\infty\quad\forall x\in(0,1].$$ Suppose also that ...
2
votes
2answers
39 views

Rate of divergence of the integral of an $L^q$ function

Let $p,q\in(1,\infty)$ be conjugate exponents (i.e., $1/p+1/q=1$) and let $f:(0,\infty)\to\mathbb R_+$ be a Lebesgue-measurable function such that $\int_0^{\infty} f(x)^q\,\mathrm dx<\infty$. It ...
0
votes
2answers
23 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 ...
0
votes
0answers
24 views

does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
1
vote
0answers
39 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
1
vote
1answer
28 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
2
votes
1answer
32 views

Exercise on L^p spaces

Let $f$ be a function of $L^p([0,2]) \>\> \forall p \in [1, \infty )$ and suppose $||f||_p \leq 1$. Show that $f$ belongs to $L^{\infty}([0,2])$ and $||f||_{\infty} \leq 1$.
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 ...
3
votes
1answer
71 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 ...
0
votes
0answers
42 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 ...
0
votes
1answer
101 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
0
votes
0answers
28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
2
votes
1answer
25 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
1
vote
1answer
37 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
1
vote
1answer
23 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...
2
votes
1answer
60 views

$\int_E f_n\to\int_E f$ implies $f_n\to f$ pointwise

The problem is to show that for a bounded sequence $\{f_n\}$ in $L^p[a,b]$ that $$ f_n\rightharpoonup f\,\Longleftrightarrow\, \int_E f_n\to\int_E f,\,\, \forall E_\text{(measurable)}\subset [a,b]. ...
2
votes
1answer
39 views

Are the $L^p$ norms ordered by $p$?

A question left over from this post is: Are the $L^p$ norms ordered by $p$ like the power means are?
1
vote
0answers
21 views

Criteria for a parameter-dependent integral to be squared integrable

I'm currently stumbling over a criteria of integrability. Let's suppose we have a real-valued function $h$ defined on the positive real axis which is integrable. We then define the function $K ...
7
votes
2answers
169 views

What does the $L^p$ norm tend to as $p\to 0$?

This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge ...
3
votes
1answer
93 views

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ ...
0
votes
1answer
164 views

Continuous functions vanishing at infinity is always integrable?

Let $$C_{0}(\mathbb R)= \big\{\,f:\mathbb R \to \mathbb C\,\,\, \text{continuous and}\,\, \lim_{x\to \pm \infty}f(x)=0 \big\}.$$ Assume that $f\in C_{0}(\mathbb R)$. My question is: Is it always ...
1
vote
1answer
136 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
3
votes
1answer
51 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
5
votes
1answer
160 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 ...
1
vote
2answers
124 views

convergence in $L^1$ for product of functions

If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
1
vote
1answer
50 views

Convergence in $L^p$ of an approximation to a function

Suppose you have a function $f\in L^p(\mathbb{R}^n)$ and some bounded set $\Omega$ of measure 1. Define $$ f_\epsilon = \frac1{\epsilon^n}\int_{\Omega_\epsilon} f(x + y)dy $$ Where $\Omega_\epsilon$ ...
2
votes
1answer
66 views

Integral on $\ell^{\infty}$

I begin with measure and integral theory. I want to give answer on the following statement: Suppose $l^{\infty}$ is the Rieszspace of all bounded functions on $\mathbb{N}$. Define ...
4
votes
1answer
67 views

Proving an inequality about an $L^2$ function

Let $u \in L^2(\mathbb{R}^2)$ be a function of two variables $x$ and $y$. I want to know if there is a relation between the Fourier tranform (with respect to $x$) of the $L^2$ norm (with respect to ...
0
votes
1answer
72 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.
3
votes
1answer
113 views

Negative integral on intervals implies negative function?

Let $f \in L^1([0,1])$ be such that for all $t \geq s$, $\displaystyle \int_s^t f(u)du \leq 0$. Is it true that $f\leq 0$ almost everywhere?
2
votes
1answer
183 views

Continuity and $L^p$ spaces

I have been wondering how to solve this question I saw in a textbook. Given $ g \in \bigcup _{1\leq p\leq \infty} L^{p}$ define, for $ r \in [ 0,1]$ , $$ G(r) = \int_{0}^{r} g(t) dt \;.$$ Show that ...