For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a ...
3
votes
1answer
42 views
Monotonicity of $\ell_p$ norm
Consider a $n$ dimensional space, it is known (Wikipedia) that for $p>r>0$, we have
$$
\|x\|_p\leq\|x\|_r\leq n^{(1/r-1/p)}\|x\|_p.
$$
I have two questions about the above inequality.
$(\bf ...
1
vote
1answer
54 views
A bounded sequence in $L^\infty$ has a weak-$^*$ convergent subsequence
Suppose $u_n$ is bounded in $L^\infty(\Omega)$. $\|u_n\|_{L^\infty(\Omega)}<M$.
Then $u_{n_k}\to u$ weak star in $L^\infty(\Omega)$ for some $n_k\uparrow \infty$ and $u\in L^\infty$.
I want to ...
1
vote
1answer
33 views
Need help with notation: What is the “$L^2+L^{\frac{4}{3}}$-Norm” supposed to be?
I recently came across the following notation in a paper. The author first estimated \begin{equation} \vert u (1-\vert u \vert^2)\vert \leq 2 \left[ \left\vert 1-\vert u \vert^2 \right\vert 1_{\lbrace ...
2
votes
0answers
56 views
An almost orthogonality principle for $L^p$
If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
1
vote
1answer
65 views
Norm operator and compactness
For the operator $U\colon \ell_{p}\to\ell_{p},\;\left( 1\leqslant p<\infty \right) :$
\begin{equation*}
Ux=U\left( x_{1},x_{2},\dots \right) =\left( 0,x_{1},\frac{x_{2}}{2},\frac{%
...
0
votes
2answers
47 views
function in $L^1\setminus L^2$
I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded).
Does anyone know such a function?
16
votes
0answers
214 views
How many points of intersection between an ellipse and an $L_p$-circle?
Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.)
Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
4
votes
1answer
56 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 ...
2
votes
1answer
106 views
$L^p$ integrable but not $L^q$ integrable
Does there exist a continuous function on $[0, \infty)$ such that it is in $L^p(0,\infty)$ for some $p\in [1,2]$ but is not in $L^q(0,\infty)$ for any $q\in (2, 2/(2-p))$?
Thanks!
1
vote
2answers
28 views
Equality of expressions containing supremum of some double sums
I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions:
$\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} ...
2
votes
0answers
19 views
A question on Orlicz norms
I was reading Empirical Processes from "Weak Convergence and Empirical processes" by Van Der Waart and Jon Wellner. There I was studying Orlicz norms of random variables which are defined as follows:
...
1
vote
1answer
34 views
Inclusion of $L^p$ spaces, bounded or unbounded
Problem statement:
We have show that $L^q [a,b] \subseteq L^p [a,b]$, for $1 \leq p \leq q$. Show that the
inclusion is proper by showing that there exists a function $f \in L^p [a,b]$, but $f ...
8
votes
2answers
121 views
When exactly is the dual of $L^1$ isomorphic to $L^\infty$ via the natural map?
The dual space to the Banach space $L^1(\mu)$ for a sigma-finite measure $\mu$ is $L^\infty(\mu)$, given by the correspondence
$\phi \in L^\infty(\mu) \mapsto I_\phi$, where $I_\phi(f) = \int f ...
1
vote
2answers
67 views
Example of a continuous function which is bounded and not contained in any $L_p$-space ($p\gt 0$)
I'm struggling to find an example of a continuous function $f:(0,\infty)\to \mathbb R$ which is bounded, not contained in any $L_p$-space ($p\gt 0$) and goes to zero when x goes to infinity.
I need ...
0
votes
1answer
38 views
Embedded Lp spaces [duplicate]
Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that ...
0
votes
1answer
53 views
convergence in measure does not imply weak convergence
Suppose $\sup_n\|f_n\|_1<\infty$ and $f_n\rightarrow f$ a.e.. However it is not necessary that $f_n\rightarrow f$ weakly in $L^1$.
Can someone raise an example?
Thanks in advance.
0
votes
1answer
66 views
Show $\|f\|_p\leq \lim\inf\|f_n\|$
$\Omega$ is a bounded domain of $\mathbb R^n$. If $\{f_n\}\subset L^p(\Omega)$ and $f_n\rightarrow f\in L^p(\Omega)$ weakly, then
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$
2
votes
1answer
37 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
32 views
Regularizing a $L^p(B)$ function in $B$ and in $\partial B$.
Let $B_1=\{x\in\mathbb{R}^N:\ |x|<1\}$. Let $u\in L^p(B_1)$ with $p\in (1,\infty)$ and suppose that $u$ is also defined in the boundary of $B_1$ and satisfies $u_{|\partial B_1}\in L^p(\partial ...
0
votes
1answer
112 views
What will be the support of the convolution of two test functions.
If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$.
Regularization of $g$ is ...
2
votes
1answer
45 views
Regularizing a solenoidal vector field $u\in L^p(\Omega)^N$.
Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and suppose that $u\in L^p(\Omega)^N$, $p\in (1,\infty)$. Assume that in the sense of distributions, $\operatorname{div}u=0$ where ...
1
vote
1answer
33 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 ...
3
votes
0answers
48 views
Mapping $G$ into its group algebra as left multiplication. Continuous?
I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove:
Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
0
votes
1answer
58 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
65 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.
2
votes
1answer
30 views
If $X_n \to X$ in $L^1$ does $E[X_n|\mathcal{G}] \to E[X| \mathcal{G}]$?
We work on a probability space $(\Omega,\mathcal{F},P)$, and $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$. Suppose that $X_n\to X$ in $L^1$, i.e. that $E[|X_n-X|] \to 0$. When does this ...
1
vote
0answers
25 views
Estimation of a scalar product
I encountered the following, which shouldn't be that hard, but I can't get my head around it.
The problem is the following estimate (part of a bigger equation, but here's just the difficult part):
...
0
votes
1answer
36 views
Square integrable function with bounded derivative goes to zero pointwise?
Prove the following: $f \in L^2(\mathbb{R})$ and $f'$ bounded $\Rightarrow$ $\lim_{|x| \to \infty}f(x)=0$.
In general, is it also true that $f \in H^1(\mathbb {R} ^n)$ $\Rightarrow \lim_{|x| \to ...
2
votes
1answer
64 views
Jensen's inequality and $L^p$ norms
Let $(X,\Sigma,\mu)$ be a probability space; in particular, $\mu(X)=1$. The integral form of Jensen's inequality can be phrased in terms of permuting a convex function $\varphi$ (say, with the ...
0
votes
1answer
43 views
uniform integrability characterization
How to show the following:
When a family of random variables $ \{X_n\}_{n \geq 1}$ is $L^p$ bounded for some
$p > 1$ then $ \{X_n\}_{n \geq 1}$ is uniformly integrable.
Also why does the above ...
1
vote
2answers
87 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
75 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 ...
2
votes
1answer
50 views
When is a subset of $\ell^2$ compact?
I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
2
votes
1answer
53 views
Extension of a Bounded Operator on $L^p$ to $L^r$
Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
5
votes
1answer
51 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 ...
1
vote
1answer
74 views
Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.
$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
1answer
42 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
44 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
34 views
Continuity of conditional expectation in $L_p$
I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, ...
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 ...
2
votes
1answer
67 views
Continuous function bounded in $L^\infty$
Is a continuous (real-valued) function in $L^\infty$ a (everywhere-)bounded function?
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 $\{u_i\}$ ...
1
vote
1answer
56 views
Showing that one cannot continuously embed $\ell^\infty$ in $\ell^1$.
Is it possible to embed $\ell^\infty$ into $\ell^1$ continuously? I.e. can one find a continuous linear injection $I:\ell^\infty \to \ell^1$.
I have reduced a problem I have been working on to ...
3
votes
1answer
64 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 ...
1
vote
1answer
14 views
Remainder of a series converges uniformly?
Let $B \subset \Bbb R^{\Bbb N}$ and $p \geq 1$. Suppose
$$
\sup_{u\in B}\sum_{n=0}^\infty |u_n|^p \leq 1,\qquad \sup_{u\in B}\sum_{n=0}^\infty |u_{n+1}-u_n|^p\leq 1
$$
Is it true that
$$
\sup_{u\in ...
2
votes
2answers
70 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.
5
votes
1answer
84 views
Is $(l^1 ,\|.\|)$ a Banach space?
Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
2answers
59 views
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$.
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
1
vote
1answer
25 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
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 ...
