Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
5,255
questions
1
vote
0
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18
views
Weak star convegence in $L^{\infty}(0,T;L^2(\Omega))$ implies almost everywhere convergence? [duplicate]
We know that $u_n \rightarrow^* u$ in $L^{\infty}\left(0, T ; L^2(\Omega)\right)$ to some function $u \in L^{\infty}\left(0, T ; L^2(\Omega)\right)$, i.e.
$$
\int_0^T \int_{\Omega} u_n(t, x) v(t, x) d ...
3
votes
0
answers
23
views
Brezis's Exercise 4.6: is the assumption $\mu(\Omega) < \infty$ necessary?
I'm trying to solve below exercise in Brezis's Functional Analysis, i.e.,
Let $(\Omega, \mathcal F, \mu)$ be a finite measure space. Assume $f \in \bigcap_{p \in [1, \infty)} L^p (\Omega)$ and there ...
- 13.8k
0
votes
0
answers
19
views
Sub-level sets of $BV$ functions - references and results
I am interested in understanding the set
$$
A := \lbrace x \in \mathbb{R}: f(x) \leq 0\rbrace
$$
where $f \in BV(\mathbb{R})$. I know that there is a decomposition of the variational measure in jump ...
- 5,773
0
votes
0
answers
29
views
If $\Omega$ has finite measure and $f \in L^\infty (\Omega)$, then $\lim_{p \to \infty} \|f\|_p = \|f\|_\infty$.
I'm trying to prove a result mentioned in this thread, i.e.,
Let $(\Omega, \mathcal F, \mu)$ be a finite measure space. If $f \in L^\infty (\Omega)$ then $\lim_{p \to \infty} \|f\|_p = \|f\|_\infty$....
- 13.8k
0
votes
0
answers
39
views
$\lim_{p \to \infty} \bigg ( \sum_{i=1}^m \lambda_i |x_i|^p \bigg )^{1/p} = \max _{1 \leq i\leq m} |x_{i}|$ [duplicate]
Let $\mathbb K$ denote $\mathbb R$ or $\mathbb C$. I'm trying to prove the discrete version of this result, i.e.,
Let $x= (x_1, \ldots, x_m) \in \mathbb K^m$ and $\lambda=(\lambda_1, \ldots, \...
- 13.8k
1
vote
1
answer
47
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Brezis's Exercise 4.5
I'm trying to solve below exercise in Brezis's Functional Analysis, i.e.,
Exercise 4.5 Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $p\in [1, \infty)$ and $q \in [1, \infty]...
- 13.8k
0
votes
0
answers
12
views
Show equivalence of 2 specific norms in sobolev space
I have this problem:
Let $\Omega\subset\mathbb{R}^2$ be a bounded open set. Let $u\in\mathbb{W}_2^{(5)}(\Omega)$
$$
\| u \|_1 = \sum_{|\alpha| \leq 5} \| \mathcal{D}^\alpha u \|_{L_2(\Omega)}
$$
and
$$...
- 3
2
votes
1
answer
39
views
The map $f*g$ is uniformly continuous
Let $p, q \in [1, \infty]$ such that $\frac{1}{p}+\frac{1}{q} = 1$. We define the convolution operator
$$
* : L^{p} (\mathbb R^d) \times L^{q} (\mathbb R^d) \to L^\infty (\mathbb R^d)
$$
by
$$
(f*g) (...
- 13.8k
1
vote
0
answers
30
views
If $f \in L^1 (\mathbb R^d)$ then $\lim_{h \to 0} \int |f(x+h)-f(x)| d x = 0$ [duplicate]
A sequence of mollifiers $(\rho_n)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^d$ such that
$$
\rho_n \in C_c^{\infty} (\mathbb{R}^d), \quad \operatorname{supp} \rho_n \subset \overline{B(...
- 13.8k
0
votes
1
answer
33
views
Brezis's Theorem 4.26: how to obtain $\|\rho_n \star f\|_{L^\infty (\mathbb{R}^N)} \le C_n\|f\|_{L^p(\mathbb{R}^N)}$?
A sequence of mollifiers $\left(\rho_n\right)_{n \geq 1}$ is any sequence of functions on $\mathbb{R}^N$ such that
$$
\rho_n \in C_c^{\infty}\left(\mathbb{R}^N\right), \quad \operatorname{supp} \rho_n ...
- 13.8k
1
vote
0
answers
14
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Question on finding a Hilbert space isomorphism between $L^2([0,R]^n)\otimes L^2([0,R]^m)$ and $L^2([0,R]^{n+m})$
Let $R > 0, n,m\in\mathbb{N}$ and suppose that we have already showed that the map $f:L^2([0,R]^n)\otimes L^2([0,R]^m)\to L^2([0,R]^{n+m})$ is a linear bijective isometry between the basis vectors ...
- 810
0
votes
0
answers
20
views
$\mathcal C_c^\infty (\Omega)$ is dense in $L^\infty (\Omega)$ w.r.t. the weak topology $\sigma(L^\infty, L^1)$
I'm trying to prove a result mentioned in this thread, i.e.,
Let $\Omega$ be an open subset of $\mathbb R^d$. Then $\mathcal C_c^\infty (\Omega)$ is dense in $L^\infty (\Omega)$ w.r.t. the weak ...
- 13.8k
0
votes
0
answers
26
views
Corollary 4.24 in Brezis's Functional Analysis
I'm trying to prove below result in Brezis's Functional Analysis, i.e.,
Corollary 4.24. Let $\Omega$ be an open subset of $\mathbb R^d$. Let $u \in L_{\text{loc}}^1 (\Omega)$ such that
$$
\int_{\...
- 13.8k
0
votes
1
answer
32
views
Is the canonical map $T : L^1(\Omega, \mu, \mathbb R) \to (L^\infty(\Omega, \mu, \mathbb R))^*$ injective?
Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. We define a map
$$
T : L^1(\Omega, \mu, \mathbb R) \to (L^\infty(\Omega, \mu, \mathbb R))^*
$$
by
$$
(T u) (f) := \int_{\Omega} uf \ ...
- 13.8k
1
vote
1
answer
39
views
$\sigma$-finiteness of measures and separability of $L^p$ spaces
The fact that a measure $\mu$ is $\sigma$-finite determines or not the separability of $L^2(\mu)$? I proved that $L^2([0,1])$ endowed with the counting measure $m$ is not separable since it admits an ...
- 13
1
vote
1
answer
51
views
Proposition 4.19 in Brezis's Functional Analysis
I'm trying to prove Proposition 4.19. in Brezis's Functional Analysis, i.e.,
Theorem Let $f \in \mathcal C_c (\mathbb R^n)$ and $g \in L_{\text{loc}}^1 (\mathbb R^n)$. Then $(f*g) (x)$ is well-...
- 13.8k
0
votes
0
answers
42
views
Generalization of the L^p space?
I've recently had a look at https://helios2.mi.parisdescartes.fr/~jdedecke/p1.pdf . In chapter 3, Definition 3.1, they defined: For any $p \geq 1$, let $\mathbb{L}^p$ be the class of real-valued ...
0
votes
2
answers
29
views
Rudin's RCA Fourier Coefficients of $L^1$ - functions
we associate to every $f$ $\in$ $L^1(T)$ a function $\hat f$ on $Z$ defined by
$\hat f $ $=$ $\frac {1}{2\pi}$ $\int_{-\pi}^{\pi}$ $f(t)$$e^{-int} dt$ $(n \in Z)$.
It is easy to prove that $\hat f $ $...
- 845
2
votes
0
answers
19
views
If $X$ is locally compact, then $\mathrm{Lib}_c (X)$ is dense in $L^p (X, \mu)$ for $p \in [1, \infty)$
Let $(X, d)$ be a metric space. Let $\mu$ be a Radon measure on $X$, i.e.,
$\mu$ is locally finite,
$\mu$ is tight on every Borel set, and
$\mu$ is outer regular on every Borel set.
As a result, $\...
- 13.8k
-1
votes
1
answer
21
views
Mean square convergence of the squares
Let $X_n$ be a sequence of random variables converging in mean square to $X$, that is,
\begin{equation}\tag{1}
\lim_{n\to\infty}\mathbb{E}\Big(\big|X_n-X\big|^2\Big)=0
\end{equation}
or, in short-hand ...
- 51
3
votes
0
answers
65
views
What are all $L^pL^q$ estimates for the heat equation (with gain of derivatives)?
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\quad\forall x\in\mathbb R. \end{aligned}\right. $$
It is ...
- 2,523
1
vote
0
answers
50
views
Functions that can be approximated by derivatives of test functions
Let $I \subseteq \mathbb{R}$ be a compact interval. We know that functions in $L^p(I)$, $(p \geq 1)$ can be $L^1$-approximated by a sequence $(\varphi_n)_{n \in \mathbb{N}}\subseteq C_0^\infty(I)$ (...
- 5,773
4
votes
1
answer
48
views
Is this Hardy-like operator $f \mapsto x^{-\frac{1}{2}} \int_{x}^{2x} f(y) dy$ bounded from $L^2$ to $L^2$?
I'm interested in knowing whether
$$Tf(x) = x^{-\frac{1}{2}}\int_{x}^{2x}f(y)dy = \int_{\mathbb R} K(x,y)f(y)dy, \quad x > 0$$
where $K(x,y) = x^{-\frac{1}{2}} 1_{0 < y \leq x \leq 2y}(x,y)$, ...
- 1,875
1
vote
1
answer
55
views
Is the inclusion $L^{\infty}L^{p} \cap L^{q}L^{\infty} \subset L^{\infty}L^{\infty}$ true?
Suppose we have the time interval $[0,T]$, a domain $\Omega = [0,1]$. I am particularly interested in the case $p=1, q=2$. In other words if we have a function
$$u \in L^{\infty}(0,T; L^{1}(\Omega)) \...
- 672
2
votes
0
answers
29
views
Interpolation between $L^p(\mathbb R^n)$ and $\operatorname{BMO}(\mathbb R^n)$
Consider a measurable, real function $f$ defined on $\mathbb R^n$ which belongs to $L^p(\mathbb R^n)\cap \operatorname{BMO}(\mathbb R^n)$, for some $1\leq p<\infty$. An interpolation inequality ...
- 2,523
3
votes
1
answer
74
views
Non equivalent norms on $L^2$
This might seem silly as the main issue is the question is vague, but it did lead to an interesting question.
So an old exam paper asks the following
Let $|| . ||$ be a norm on $L^2(\mathbb{R})$ s.t. ...
- 245
3
votes
1
answer
76
views
Total sets for $L^p$ for every $1\leq p < \infty$.
Consider $L^p[ 0,1]$ for $1\leq p < \infty$ or, if you prefer, $L^p(\mu)$ where $\mu$ is a finite Borel measure with compact support. Let $(\phi)_{i\in I}$ be a subset of measurable functions that ...
- 945
2
votes
0
answers
25
views
Dual of $L^p_{loc}$ and weak star convergence.
I was reading a paper and stumbled across the terms
$$f^\varepsilon\to f \text{ in } L^{p}_{loc} \Bigl( [0,+\infty) , \left[ L^p_{loc}(\mathbb{R}^2) \right]^* \Bigr)$$ and $$f^\varepsilon\...
- 21
3
votes
0
answers
33
views
Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.
Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
- 31
3
votes
0
answers
58
views
Under which assumptions on $f$ we can deduce that $g\in L^\infty$?
Let $f:\mathbb{R}^*_+\to\mathbb R$ be of class $C^\infty$ satisfying
$$|f(x)|\le c_0 |\log(x)| \quad\text{ for } 0<x\ll 1,$$
for a positive constant $c_0>0$.
Let $c_1>0$ be a constant. As an ...
- 3,128
1
vote
1
answer
31
views
Interpolation in Lebesgue-Bochner $L^p-L^q$ spaces.
Consider $\Omega \subset \mathbb{R}^n$ open and bounded, $I$ some bounded interval of $\mathbb{R}$. Let $2^\ast := \frac{2n}{n-2}$ (the critical Sobolev exponent).
Let $u \in L^{\infty}(I; L^2(\Omega))...
- 115
0
votes
0
answers
17
views
Examples of dense $G_\delta$ sets in $L^p$ spaces
The Baire Category Theorem states that the countable intersections of open dense subsets of a complete metric space (called dense $G_\delta$ sets) are dense. Any open set is $G_\delta$, so any dense ...
- 499
3
votes
0
answers
32
views
Relatively Compactness in $L^{\infty}(-T,T;L^1(B_r))$
I am reading 'Ordinary differential equations, transport theory and Sobolev spaces' by DiPerna and Lions. I am stuck at the following step in page $533$ in the paper:
$\frac{\partial}{\partial t}(\...
- 31
1
vote
0
answers
52
views
Convergence in $L^2(\mathbb{R})$ to interchange integral and limit.
I am trying to understand the proof of the second Paley-Wiener theorem, which states sufficient and necessary conditions for a function in $L^2(\mathbb{R})$ to have Fourier Transform with compact ...
- 765
0
votes
1
answer
59
views
Why isn't this proof sufficient for showing that $L^\infty$ is a Banach space?
Suppose we have already shown that $\|\cdot\|_\infty$ is a norm. In order for $L^\infty$ to be a Banach space, if $\{f_n\} \subset L^\infty$ is a Cauchy sequence such that $\| f_n - f\|_\infty \...
- 2,797
1
vote
2
answers
56
views
If $f \in L^{1}(0,\infty)$ and $x \in (0, \infty)$, then $xf \in L^{1}(0,\infty)$? [closed]
Suppose $f \in L^{1}(0,\infty)$. Is it true that
$$
\int_{0}^{\infty}xf(x)dx < \infty \ \
$$
- 75
1
vote
0
answers
26
views
Bounded difference property of $\arg\min$
I'm trying to perform some calculations regarding expectation value of a minimization problem. Specifically, I'm tackling the following inequality using McDiarmid inequality:
$$ \mathbb{P}\left(\left|...
- 133
1
vote
1
answer
79
views
Linear subspace of $\ell^\infty([0,1]):=L^\infty([0,1],\mathcal{P}([0,1]),\#)$
This question could sound naive, but I honestly did not found anything around the Net (just some related topics on https://mathoverflow.net/q/177806/95288).
Consider the Banach space $\ell^\infty([0,1]...
- 89
2
votes
1
answer
57
views
Prove that $\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z) ≤ ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$
Hey I have this problem where I am stuck on solving it.
I Think it is very easy but I dont know how to proceed.
The Exercise is
Let $f, g, h ∈ L_2(\mathbb{R}^2 )$.
To show is:
$\int_{\mathbb{R}^3} f(x,...
- 454
2
votes
2
answers
106
views
Under what conditions is an infinite matrix a bounded linear transformation in $\ell^2$?
Let $(e_n)_{n\in\mathbb N}$ be the canonical basis of $\ell^2$, and $\mathcal L(\ell^2)$ be the set of bounded linear transformations from $\ell^2$ to itself. If $A\in\mathcal L(\ell ^2)$ we can set
$$...
- 1,160
0
votes
1
answer
46
views
$f=g+ih$ is $L^p$ if and only if $g$ and $h$ are $L^p$?
Let $X$ be a measure space, $f:X\to\mathbb{C}$ a measurable function, and $1<p<\infty$.
$f$ is said to be $L^p$ if $|f|^p$ is integrable.
Now, let $f=g+ih$, where $g$ and $h$ are real-valued. Is ...
- 6,339
0
votes
1
answer
47
views
Find function $f(x)$ so $x\hat{f}(x)\in L^1(\mathbb{R})$
Suppose we have $f\in L^1(\mathbb{R})$ so $\xi\mapsto \xi\hat{f}(\xi)$ is in $L^1(\mathbb{R})$, where $\hat{f}$ is the Fourier Transform for the function $f$. I'm trying to show that there exists some ...
0
votes
1
answer
58
views
Showing $\phi(x) = (1 − |x|)_+$ is $L^p$ for all $p$
Consider the function $\phi$ defined by $\phi(x) = (1 − |x|)_+$, I have been tasked by an exercise in my textbook to verify both $\phi,\phi'\in L^p([-2,2])$ and deduce $\phi\in W^{1,p}((−2, 2))$ for ...
0
votes
0
answers
50
views
Littlewood–Paley theory in case $p=\infty$
What happen with Littlewood–Paley theorem in the case $p=\infty$. Exists some result or paper for this case?, by https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley_theory we can estimate bound the ...
- 1,135
0
votes
0
answers
20
views
Connecting piecewise smooth functions using mollification.
Let's say I am given a function $f: \mathbb{R}\setminus (0,1) \to \mathbb{R}$ defined piecewise as
$$f(x)= \begin{cases}
g(x) \quad x \geq 1 \\
h(x) \quad x \leq 0
\end{cases},$$
where $g$ and $h$ ...
- 961
3
votes
1
answer
88
views
$L^2(\mathbb{R})$ is separable
I would like to prove that $L^2(\mathbb{R})$ is separable. Using the hint provided in Reed & Simon's book, I was able to show that:
the set of simple functions on $[a,b]$ is dense in $L^2([a,b])$
...
- 2,797
1
vote
0
answers
25
views
Is there a standard notational convention for $L_p$ ($\ell_p$) spaces, norms, and metrics?
In reading about $L_p$ spaces and $\ell_p$ norms and metrics, I have seen a wide variety of different notations for the same thing.
Is there a standard convention for when to use lowercase $\ell$ vs ...
- 1,773
1
vote
0
answers
26
views
Norming subspace of $L_\infty(\Omega, \mu)$
Let $(\Omega, \mu)$ be a semifinite measure space. I know that for any $1 < p \leq \infty$, it holds
$$
\|f\|_p = \sup\left\{\left|\int fg \ {\rm d}\mu\right| \ : \ \|g\|_{p^\prime} \leq 1, \ g \ \...
- 51
0
votes
0
answers
21
views
Problem with convergence (application of Banach-Steinhaus theorem) [duplicate]
If $\sum_{n=0}^\infty x_ny_n $ converges for all $x_n\in l^p$ then $y_n\in l^q$.
I tried using uniform boundedness principle, but I didn't get given result. I defined operators $A_n(x)=\sum_{k=0}^n ...
- 65
1
vote
0
answers
24
views
Liapunov's Inequality for $L_p$ vector spaces
Suppose $1 < p \leq q \leq r$ and $x \in \mathcal {R}^n $, by Liapunov's Inequality, if for $\lambda \in (0,1)$ and $q=\lambda p+(1-\lambda)q$, then
$$
\lVert x\rVert_q^q \leq \lVert x\rVert_p^{\...
- 127