# Tagged Questions

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### 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 $u \to 0$. ...
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### 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 ...
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### $f\in L^2(0,1)$ 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 $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
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### 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 ...
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### 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.
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### 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 $\Omega_\epsilon$ ...
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### 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 ...
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### 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 ...
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### 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.
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?
### 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 ...