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 ...
1
vote
2answers
43 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
32 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
44 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
59 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
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$
1
vote
0answers
25 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
31 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
86 views
+50
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
42 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
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.
2
votes
1answer
26 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
24 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
34 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
60 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
42 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
81 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 ...
2
votes
1answer
49 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
48 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 ...
4
votes
1answer
36 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
73 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
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
29 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
61 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
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 ...
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
$$
\sup_{u\in ...
2
votes
2answers
63 views
Compact inclusion in $L^p$
Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$?
What is the counterexample if what I said is wrong?
Thank you.
5
votes
1answer
82 views
Is $(l^1 ,\|.\|)$ a Banach space?
Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
2answers
57 views
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
Let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $\ell^1$, suppose $x=\{x_n\}\in\ell^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$.
Are the norms $\|\cdot\|_1$ and $\|\cdot\|$ equivalent?
1
vote
1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
8
votes
1answer
71 views
Various kinds of derivatives
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$.
Classical derivative. The unique function $f'_c$ defined pointwise by ...
11
votes
1answer
79 views
Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$
Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$.
I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
3
votes
1answer
45 views
Monotonicity of $\mathcal{l^p}$ spaces using only Hoelder inequality
For $p > 0$, let $\ell^p$ be the space of sequences for which
$$\sum_{i=1}^{\infty} |a_i|^p$$
is finite ($a_i \in \mathbb{R}$). It is well-known that, for $q > p$, $$\ell^p \subset \ell^q.$$
...
2
votes
1answer
74 views
Lebesgue space - $L^p$ spaces
Let $X=\{a,b\}$,and $\mu(\{a\} )=1$, and $\mu(\{b\} )=\mu(X)=+\infty$ and $\mu(\emptyset)=0$. Is it truth that $L^\infty(\mu)$ is the dual space of $L^1(\mu)$. Whether $L^\infty(\mu)=L^1(\mu)^\ast$? ...
1
vote
1answer
89 views
Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$
Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$.
My friends and I have literally been pouring over this problem for days now without success. We've been using Hölder's ...
1
vote
1answer
95 views
Examples of $f \in L^p$ iff $p_0 < p < p_1$, $p_0 \le p \le p_1$ or $p = p_0$
Hi how to show the following:
Suppose $0 < p_0 < p_1 \leq \infty$. Find examples of functions $f$ on $(0,\infty)$
with Lebesgue measure such that $f \in L^p$
if and only if
(a) $p_0 < p ...
1
vote
0answers
61 views
Inclusion maps on $L^p$
How to show for $1 \leq p < q < r$
the inclusion maps
$$L^p \cap L^r \rightarrow L^q$$
$$L^q \rightarrow L^p + L^r$$
are continuous.
where the norms are defined in the following:
$L^p$ ...
1
vote
1answer
79 views
$\mathcal{L}^p$ spaces and convolution
Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that
(a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
1
vote
1answer
32 views
Measure and $L_\infty$ space
Consider a function $f \in L_\infty$. I am trying to see if the following statement is true and if so why.
$$ \mu\{\, \vert f \vert = \Vert f \Vert_\infty \} \stackrel{??}{>} 0 \text { and } \{\, ...
1
vote
1answer
88 views
$L^p$ space, paralellogram law
Let $(X,\mathcal{M},\mu)$ be a measure space and $1\leq p\leq\infty$. Suppose $f,g\in L^p(X,\mathcal{M},\mu)$ with $f(x)g(x)=0$ for almost all $x\in X$ and $\Vert f\Vert_p=\Vert g \Vert_p=1$. How to ...
1
vote
1answer
65 views
$l^{p}$ is not finite dimensional
Well, the exercise was to prove that $l^{p}$ is not finite dimensional space for $p$=2.
I did it proving that the unit ball is not compact. Easy.
However, i was trying to build an element $x \in ...
3
votes
1answer
77 views
$W^{1,p}$ compact in $L^\infty$?
Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
2
votes
3answers
83 views
Completeness proof of $\ell^p$
Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct:
Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
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 ...
3
votes
0answers
74 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
