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 ...
2
votes
1answer
40 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 ...
0
votes
1answer
56 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
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$
0
votes
1answer
76 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 ...
4
votes
0answers
130 views
Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.
The following is from the book "Sobolev spaces" ...
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 ...
1
vote
0answers
19 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
2answers
183 views
Proof of Clarkson's Inequality
Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
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 ...
3
votes
1answer
178 views
$L^p$ function that has no compact support
I understand an $L^p$ function have in them as a dense subset the set of functions with compact support. But do there exist $L^p$ functions that do not have compact support. What are some examples ...
0
votes
1answer
49 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 ...
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 ...
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 ...
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):
...
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
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).$$ Let $\ell^p(Z)$ denote the $L^p$ space of the integers equipped ...
1
vote
2answers
78 views
Problem #23 pg-94, Stein and Shakarchi
As an application of the Fourier transform, show that
there does not exist a function $I\in L^1(R^d,m)$ such that
$f*I = f$ for all $f\in L^1(R^d,m)$.
2
votes
1answer
48 views
Extension of a Bounded Operator on $L^p$ to $L^r$
Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a ...
2
votes
1answer
49 views
When is a subset of $\ell^2$ compact?
I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà–Ascoli theorem that provides a ...
4
votes
1answer
36 views
$L^{p}$ functions from Rudin Exercises 3.5
I am attempting a question from Rudin's "Real and Complex Analysis" Chapter 3 question 5. I shall summarise the question as below: Suppose that $f$ is a complex measurable function on $X$, $\mu$ a ...
1
vote
1answer
73 views
Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.
$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$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)$, ...
4
votes
1answer
439 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}$, ...
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
62 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 ...
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 ...
2
votes
0answers
69 views
When does $|f*g|_{p}=|f|_{1}|g|_{p}$?
From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4
Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
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 ...
2
votes
1answer
149 views
Problem 4 chapter 2: functional analysis (Rudin)
$L^1$, $L^2$: usual Lebesgue spaces on the unit interval. Show that $L^2$ is of the first category (meager) in $L^1$, in three ways:
(a) Show that $F_n:=\{f:\int|f|^2 \leq n\}$ is closed in ...
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 ...
4
votes
1answer
231 views
Doubts about Dominated Convergence Theorem
I am following a course of real analysis and my teacher, while proving the continuity of translation of functions in $L^p$, used the dominated convergence theorem (DCT) in a strange way. I write the ...
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 $L^\infty(\mu)=L^1(\mu)^\ast$? ...
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
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 ...
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 ...
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
3answers
179 views
Convergence of functions in $L^p$
Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say ...
1
vote
1answer
88 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 ...
