Tagged Questions
1
vote
0answers
22 views
Gauss–Ostrogradsky formula for Distributions
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
1
vote
1answer
23 views
Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in ...
0
votes
1answer
50 views
A basic question about $\operatorname{supp}f$ (support of f).
Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0
$?
Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
2
votes
1answer
62 views
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
1
vote
2answers
79 views
Problem #23 pg-94, Stein and Shakarchi
As an application of the Fourier transform, show that
there does not exist a function $I\in L^1(R^d,m)$ such that
$f*I = f$ for all $f\in L^1(R^d,m)$.
1
vote
0answers
72 views
Problem # 25, page 95, from Stein and Rami [duplicate]
Let $(X,M,\mu)$ be a measure space with $\mu(X) < 1$. Show that for any $1\le p<q$, we have $$L^q (X,\mu)\subset L^p(X,\mu).$$ Let $\ell^p(Z)$ denote the $L^p$ space of the integers equipped ...
1
vote
1answer
39 views
Extension of Fourier Transform
We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
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.
1
vote
1answer
20 views
If a function is $L^p$ small, is its expectation with respect to a $\sigma$-algebra $L^p$ small?
This came up in my homework, but isn't strictly my homework. I've just gotten very curious, and I keep going in circles trying to prove it.
Consider a probability measure space $(X,\Sigma,\mu)$ and ...
1
vote
1answer
36 views
$L^p$ convergence proof check
I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it.
Theorem: Let $\{u_i\}$ ...
3
votes
1answer
57 views
Convergence of $L^p$ norm as $p \downarrow 0$ [duplicate]
Consider a measurable space $(\Omega, \mathscr{F}, P)$ with $P(\Omega) = 1$. Define for measurable functions $X$ the following $\| X \|_p := \left(\int |X|^p dP\right)^{1/p}$. We know that for $p \in ...
2
votes
2answers
63 views
Compact inclusion in $L^p$
Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$?
What is the counterexample if what I said is wrong?
Thank you.
8
votes
1answer
71 views
Various kinds of derivatives
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$.
Classical derivative. The unique function $f'_c$ defined pointwise by ...
1
vote
1answer
79 views
$\mathcal{L}^p$ spaces and convolution
Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that
(a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
1
vote
1answer
32 views
Measure and $L_\infty$ space
Consider a function $f \in L_\infty$. I am trying to see if the following statement is true and if so why.
$$ \mu\{\, \vert f \vert = \Vert f \Vert_\infty \} \stackrel{??}{>} 0 \text { and } \{\, ...
2
votes
2answers
73 views
$L_p$ space,convergence
Let $1<p<\infty$ and $h\in L_p(\mathbb{R})$,that is,$\left(\displaystyle\int_{\mathbb{R}}|h|^p\right)^{1/p}<\infty$. Define a sequence $(f_n)_{n\in\mathbb{N}}$ by $f_n(x):=h(x-n)$. How to ...
0
votes
1answer
70 views
When is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$? [duplicate]
Given a measure space $(X,\Sigma,\mu)$, when is it the case that $L^p(X,\mu)\subset L^r(X,\mu)$ for $p>r$, or for $p<r$ . Thanks.
2
votes
1answer
80 views
Question on $L^p$ spaces
Let $f:[0,1] \to \mathbb{R}$ be a measurable function. Prove that
$$
f \in L^\infty([0,1];\mathbb{R}) \iff f \in L^p([0,1];\mathbb{R}) \ \ \forall p \ge 1 \ \text{ and } \sup_{p\ge ...
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
98 views
Jensen's inequality and a estimate in $L^p$
In problem 3 we have:
If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and
$$\int_E e^{|f(x)|}dx =1,$$
then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and
...
4
votes
2answers
94 views
Finding $f\in L^{p}\left(\mathbb{R}\right)\setminus L^{q}\left(\mathbb{R}\right)$
Let $1\leqslant p<q\leqslant\infty.$ I know then that $L^{q}\left(\mathbb{R}\right)\subsetneq L^{p}\left(\mathbb{R}\right)$. How do I show that it is a proper subgroup? In other words, I am ...
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 $f(x)\in$ $L_1(\mathbb{R})$.
8
votes
1answer
152 views
Convergence in $L^1$ space
Suppose that $f_{n}$ is a sequence of measurable functions, in a finite measure space, $f_{n}\to f $ in $m$-measure and that there exists $g$ in $L^1$ such that $\vert f_n\vert \le g$.
Prove that ...
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 ...
0
votes
1answer
55 views
If f is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function g does that imply that f=0 a.e
If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$
function $g$ does that imply that $f=0$ a.e
4
votes
0answers
75 views
A question about functions in $L^p(E)$
I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
2
votes
1answer
71 views
How can I give a bound on the $L^2$ norm of this function?
I came across this question in an old qualifying exam, but I am stumped on how to approach it:
For $f\in L^p((1,\infty), m)$ ($m$ is the Lebesgue measure), $2<p<4$, let
$$(Vf)(x) = ...
4
votes
1answer
444 views
Limit of $L^p$ norm
Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
4
votes
1answer
69 views
Limit problem for $L^p$ function
I am having problems with proving the following:
Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that
$$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$
...
7
votes
1answer
188 views
How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function
We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$.
...
3
votes
2answers
139 views
Question from Folland, criteron for a function to belong to $L^p$
This question is from Folland 6.38,
Show that $f \in L^p $ iff $\sum_{k=-\infty}^ {\infty} 2^{pk} \mu \{{x: |f(x)|>2^{k}}\} \lt \infty$
If $f \in L^p $, I applied the Chebyshev's inequality
But ...
2
votes
1answer
77 views
Liapunov's Inequality for $L_p$ spaces
Let $1 \leq p,q < \infty$ and $0 \leq \lambda \leq 1$. If $r = \lambda p + (1 - \lambda)q$ and $f \in L_p \cap L_q $, then
$$||f||_r^r \leq ||f||_p^{\lambda p} ||f||_q^{(1 - \lambda)q} \tag{*}$$
...
3
votes
1answer
104 views
How is this book applying Fubini/Tonelli without assuming $\sigma$-finiteness?
I am reading about $L^p$ spaces on this google book and in proposition 1.1.4 (page 4) it writes
$$
p\int_0^\infty \alpha^{p-1} \int_X \chi_{\{x:|f(x)|>\alpha\}}d\mu(x)d\alpha = \int_X ...
2
votes
1answer
74 views
Does $\Vert f \Vert_p = \sup_{\Vert g \Vert_q=1}\int fg d\mu$ fail if $f \notin L^p$?
I know that for $p \in [1,\infty]$ if $X$ is $\sigma$-finite (for the $p=\infty$ case) we have
$$
\Vert f \Vert_p = \sup_{\substack{g \in L^q\\\Vert g \Vert = 1}} \int_X fg d\mu.
$$
I always see it ...
3
votes
1answer
62 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?
1
vote
1answer
89 views
Show a function is in $L_\infty$
Let's assume we're working on a measure space $(X,\Sigma,\mu)$, where $\mu$ is a $\sigma$-finite measure. Suppose that $g$ is a measurable function such that $\forall f\in L^2$, $||fg||_2\leq ...
3
votes
2answers
139 views
Lp Spaces and limits of translated functions
If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$
where $g_{(t)}(x):=g(t+x)$.
Any hints? Try to give me only ...
1
vote
2answers
81 views
$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$
If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$.
But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to ...
1
vote
1answer
52 views
How to bound $L^p$ norm of a product
I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate ...
1
vote
1answer
443 views
Using Lusin's Theorem to show that continuous functions are dense in $L^p$
Lusin's theorem says that in a finite measure space, given a measurable function $\varphi$, for every $\varepsilon \gt 0$ there exists a continuous function $g$ such that $$ \mu\left(\{x : ...
4
votes
3answers
328 views
Convergence in $L^{\infty}$ norm implies convergence in $L^1$ norm
Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. Assume the measure space $X$ has finite measure. If $f_n$ converges to $f$ in ...
2
votes
1answer
105 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 ...
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
4answers
686 views
Convergence of integrals in $L^p$
Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < ...
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 ...
