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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 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 problem is a $L^p$ space.

6
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0answers
31 views

Can we approximate a.e. invertible matrices with everywhere invertible matrices in $L^2$ sense?

Let $\mathbb{D}^n=\{ x \in \mathbb{R}^n \, | \, |x| \le 1\}$ be the closed unit ball, and let $A:\mathbb{D}^n \to \mathbb{R}^{n^2}$ be smooth (on the entire closed ball). Suppose that $n \ge 2$, and ...
1
vote
0answers
16 views

Finite-dimensional copies of an unconditional basis uniformly complemented in $(\oplus\ell_\infty^n)_p$

Fix $1\leqslant p<\infty$, and let $(e_n)_{n=1}^\infty$ be an unconditional Schauder basis with some special properties which I'll discuss below. I'm trying to prove the following conjecture: ...
5
votes
1answer
36 views

Pointwise multiplication by unbounded function throws us out of $L^2(\mu)$

Let $(X,\mathcal{A},\mu)$ be a $\sigma$-finite measure space and let $\phi$ be a measurable function that is not an element of $L^\infty(\mu)$, i.e. $\phi\not\in L^\infty(\mu)$. I am trying to ...
1
vote
0answers
24 views

Is my proof about the behavior of Convolutions as $||x|| \rightarrow \infty$ correct?

I've written a proof to the following statement and would appreciate if somebody could look over it and give some feedback. My apologies if this is the wrong place for this. Statement Let $f \in L^...
1
vote
2answers
46 views

If $u \in L^\infty((0,T)\times \Omega)$, does it follow that $u(t) \in L^\infty(\Omega)$ for a.e $t$?

Let $u \in L^\infty((0,T)\times \Omega)$, where $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. We know that it is not necessarily true that $u$ belongs to $L^\infty(0,T;L^\infty(\Omega))$. ...
3
votes
1answer
37 views

Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
2
votes
1answer
60 views

Why can we consider this subsequence?

Why can the sequence be considered as it is? $\{f_n\}$ is Cauchy and $\{f_{n_k}\}$ is a subsequence with the property $\Vert f_{n_k}-f_{n_{k+1}}\Vert_p\le2^{-k}$ On the youtube video https://www....
2
votes
1answer
46 views

Grothendieck's lemma in $L^p$ spaces

So I am currently working on the proof of Grothendieck's Lemma : Let S $ \subset L^{\infty}(X) $, of finite measure, be a closed vector subspace of $L^p $ for a certain p such that $ S \subset L^{\...
1
vote
3answers
66 views

If $f\in L^1(\mathbb R)$ and $f'\in L^1(\mathbb R)$, then $\lim_{x\to \infty }f(x)=0$.

Let $f,f'\in L^1(\mathbb R)$. Prove that $$\lim_{x\to \infty }f(x)=0.$$ First of all, is $f'$ defined a.e. ? Because there are no assumption on the fact that $f$ is derivable. So, is $$f'(x)=\lim_{h\...
4
votes
0answers
50 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
1
vote
1answer
36 views

If $\|x_n\|_2 \to \infty$ in $L_2$-norm, does $\|x_n\|_{1+\varepsilon} \to \infty$ in $L_{1+\varepsilon}$-norm, for all $\varepsilon > 0$?

Question: If $\|x_n\|_2 \to \infty$ in $L_2$-norm, does $\|x_n\|_{1+\varepsilon} \to \infty$ in $L_{1+\varepsilon}$-norm, for all $\varepsilon > 0$? Details/Progress: This should follow trivially ...
1
vote
1answer
37 views

How to understand the convergence of Fourier Series in $L^p$

My professor told me that Suppose that $f \in L^p(-\pi, \pi)$ (i.e. $f$ is 2$\pi$-periodic and $\|f\|_{L^p} < \infty$). If $1<p<\infty$, then the Fourier series of $f$ converges to $f$ in $...
6
votes
3answers
205 views

If $T:L^p[0,1] \to L^p[0,1]$ bounded for $1 < p < \infty$ with continuous image, then it's compact

Is the following statement true? Let $T:L^p[0,1] \to L^p[0,1]$ be a bounded operator for $1 < p < \infty$ and suppose that $\operatorname{Im}(T) \subset C[0,1]$ consists of continuous functions....
0
votes
1answer
14 views

Weak Convergence of sequence in a Sobolov Space.

Consider the question asked in here. I understood most of the answer in the question but the part about the weak convergence I did not get. To show that $u'_{n_k}$ converges weakly to $u$ in $L^p$, ...
-1
votes
1answer
60 views

Why is the integral finite?

In this proof why integral is finite? My though was that as the partial sum $𝑆_𝐾(𝑔)$ converges. Hence the complete series $𝑔=|𝑔|$ converges which can be seen as $|𝑔|<∞$ thus $|𝑔|^𝑝<∞$, ...
2
votes
1answer
37 views

Showing that for $\{u_n,u\}_{n \geq 1} \subseteq L^p(\Omega)$ it is $u_n \to u$ in $L^p(\Omega)$.

Exercise : Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $\{u_n, u\}_{n \geq 1} \subseteq L^p(\Omega)$ with $1<p<\infty$ and we assume that $\|u_n\|_p \to \|u\|_p, \; u_n \...
0
votes
1answer
50 views

Why finite integral implies convergence almost everywhere?

In this proof why finite integral implies convergence almost everywhere? Note there sum symbols $\sum$ are missing in the complete proof; not sure why Why having $\int g^p<\infty$ implies the ...
-1
votes
1answer
28 views

Help proving that these functions don't converge in $L_1$ norm

so consider this $f_n(x)= \sin(nx)$ $f=0$ $f_n$ and $f$ $\in$ $L^{1}(X)$ $X=[0,2\pi]$ Now it's true that $ \int_E f_n$ $\rightarrow$ $\int_E f$, when $E$ contained in $X$ makes the integral ...
0
votes
1answer
27 views

Complete proof, $L_p$ complete

Does someone know about a proof for $L_p$ completeness that is not difficult to follow? I started to watch a YouTube video about it https://www.youtube.com/watch?v=BWnV8fBlDQQ start [10:20] minute. ...
0
votes
1answer
50 views

Is $\int_{\Omega} \bigg( \sum_{n=1}^{\infty} |f_n| \bigg)^p d \mu$ really in $L^p$?

Is $\int_{\Omega} \bigg( \sum_{n=1}^{\infty} |f_n| \bigg)^p d \mu$ really in $L^p$? What confuses me that I think that $|f_n|$ should have some power of $p$. $f_n$ are elements of $L^p$. $\sum_n ...
1
vote
1answer
47 views

Convolution of functions $f,g\in L^1([0,1])$

Problem: convolution ($f*g) $ of functions $f,g\in L^1([0,1])$, where: $$f(x) = \frac{3}{5-4\cos{4\pi x}},$$ $$g(x) = \frac{2\cos{2\pi x}}{5-4\cos{4\pi x}},$$ and $$(f * g)(x) = \int_{0}^{1}f(x-y)g(...
1
vote
1answer
29 views

Question about sequences in $l^p$ spaces and does this hold?

Let $X\in{l^p}$, with $1\le{p}\le{q}\le{\infty}$ and $X=(x_j)_{j=1}^\infty$ s.t. $\sum_{j=1}^{\infty}{|x_j|^p}<\infty$ (by definition). We choose $X$ s.t $\|X\|_{l^p}=\big(\sum_{j=1}^{\infty}{|x_j|^...
0
votes
1answer
29 views

$L^2$ functions with compactly supported Fourier transforms form a Hilbert space

Given a fixed compact subset of $\mathbb{R}$, I want to show that square integrable functions on the real line whose fourier transforms are supported in the given compact set form a Hilbert space in ...
2
votes
1answer
29 views

Norm defined by a conditional expectation

Let $\Omega$ be a probability space and $\mathbb{E} \colon L^\infty(\Omega) \to L^\infty(\Omega)$ be a conditional expectation such that $\mathbb{E}(|f|^2)$ implies $f=0$. Suppose $1<p<\infty$. ...
0
votes
0answers
14 views

Automorphism induced automorphism of Lp spaces

Let $(\mathbb{R}^d,\mathbb{B}(\mathbb{R}^d),\mu)$ where $\mu$ is a $\sigma$-finite Radon measure. If $\Phi:\mathbb{R}^d\rightarrow \mathbb{R}^d$ is a homeomorphism, then does $\Phi$ induce a ...
1
vote
0answers
35 views

Weakly compact set in $L^1$ not bounded in $L^p$

The Dunford-Pettis theorem states that a family $\mathcal{F}\subset L^1$ is relatively weakly compact if and only if $\mathcal{F}$ is bounded in norm and uniformly integrable, i.e. $\sup_{f\in\mathcal{...
2
votes
0answers
31 views

$L^1$ and $L^2$ spaces, how to determine which to work with?

This is from a standing point of a new student of measure-theoretic probability. For example, we have the following two definitions of conditional expectation: Definition 1 ($L^2$): Let $(\...
4
votes
0answers
39 views

Determining if $f\in L^{p}(\mathbb R)$ from a bound on the measure of the level sets $\{|f|>\lambda\}$ for all $\lambda>0.$

$\textbf{The Problem:}$ Let $f$ be a measurable function on $\mathbb R$ with respect to the Lebesgue measure $m$. $\textbf{a)}$ Suppose that $$m(\{\vert f\vert>\lambda\})\leq(1+\lambda)^{-1}$...
0
votes
1answer
30 views

Piecewise function in $L^p$ spaces

Consider the space $C[0,3]$ for piecewise function such that $$f_a(x)= \begin{cases}a^3(2-a^3x),& 0\le x \le \frac2{a^3}\quad\text{and} \\[1ex] 0 , & \...
0
votes
1answer
35 views

About $L^p$ space

In Rudin's book Real an complex analysis, we have Let $(X,\mu)$ be a measure space and $1\leq p<q<\infty$, then $\mu(X)<\infty$ if and only if $L^q(X)\subset L^p(X)$. Now, I want to know ...
0
votes
1answer
25 views

Probability distribution in $L_p$

I am stuck on this exercise. Let $F(x) = 1-1/x^a$ for $x\geq 1$ be a distribution. For which values of $a$, $F(x) \in L_p$? I tried to study vários integral functions, but I do not really know how ...
0
votes
1answer
26 views

Is this functional $f \mapsto \int |f|^2 dx $ Frechet differentiable?

Suppose $(X, \mathcal M, \mu)$ is a fixed measure space. For $f \in L^2(X)$, i.e., $f : X \to \mathbb C$ is measurable and $\int |f(x)|^2 d\mu(x) < \infty$, we define a functional $\phi : L^2(X) \...
1
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0answers
34 views

Clarkson inequality for complex numbers

Let $1<p<2$. I'm trying to prove the inequality $$ |a+b|^q+|a-b|^q\leq 2\big( |a|^p + |b|^p \big)^{q-1} $$ where $\frac{1}{p}+\frac{1}{q}=1$. Following this paper, I am able to prove the ...
0
votes
1answer
59 views

Obtaining orthogonality from a variational inequality in $L^2$

I'm working on the following problem: Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and let $H$ be a closed linear subspace of $L^2(\Omega)$. Let $\gamma : \mathbb R \to \mathbb R$ be a ...
1
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0answers
20 views

Are the derivatives of the orthogonal polar factor locally integrable?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a real-analytic map, satisfying $\det df>0$ everywhere except on a set of Hausdorff dimension not greater ...
8
votes
0answers
130 views

Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
1
vote
1answer
41 views

Does $L^p(E)$ with $m(E)<\infty$ with smaller norm preserve the Banach

If $1\leq p < q <\infty$ and $E$ a subset of $\mathbb{R}$ with finite measure if we consider the space $L^q(E)$ is it a Banach space with the norm $||.||_p$. I know that $L^p$ space is a Banach ...
2
votes
1answer
40 views

Understanding a proof that a compact multiplication operator is zero

This answer gives a proof of the fact that if $g\in L^\infty(0,1)$ and the multiplication operator $T_g:L^2(0,1)\rightarrow L^2(0,1)$ is compact, then $g=0$ almost everywhere: We show that if $g$ ...
2
votes
0answers
34 views

Lp space is a Hom functor

Is there way to see $L^1$ a functor from the category with objects as metric measure spaces $(X,d,\mu)$ and morphisms Lipschitz maps to the category of Banach spaces (or something which has a ...
0
votes
1answer
29 views

Show that $|T_{\epsilon}(f)-T_0(f)|\leq \frac{\epsilon}{2}||f||_E,\; \epsilon\in]0,1]$

I have the next exercice: For $\epsilon \in ]0,1]$, we have the function $$T_{\epsilon}(f)=\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}f(t)dt$$ and $T_0(f)=f(0).$ Show that for $\epsilon \in ]0,...
2
votes
0answers
29 views

Are Ito integrals equivalence classes or conrete random variables?

Technically speaking, are Ito integrals of stochastic processes $S$ with respect to Brownian motion $B$ $$ \int_{0}^{T} S_s dB_s $$ random variables or equivalence classes of almost surely equal ...
0
votes
1answer
46 views

What is the definition of this $L^p$-space?

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. What is the definition of the space $$ L^p(S^1,\mathbb{R}^{2n}) $$ Thank you for your kind help.
0
votes
2answers
25 views

How can I prove that $f\in L^p\cap L^q\implies f\in L^r$ for all $r\in [p,q]$?

I have a result that says that since $f\in L^p(\mathbb R)\cap L^q(\mathbb R),$ we have that $f\in L^r(\mathbb R)$ for all $r\in [p,q]$. I don't really know how to prove this. I know that if $r\in [p,q]...
0
votes
0answers
18 views

Singular integral and differentiability properties of functions. Stein pp. 4.

Let $g:\mathbb{R}^n\to\mathbb{R}$ and let $\lambda(\alpha)=m \left\{x:|g(x)|>\alpha\right\}$ If $g\in L^p$. Why $\int_{\mathbb{R}^n}|g(y)|^pdy=-\int_{0}^{\infty} \alpha^p d\lambda(\alpha)$?
1
vote
1answer
18 views

Limit of norm of indicator function

It is well known that for any function $f \in L^p(\mathbb{R}^n)$, we have $$\lim_{|h|\to 0}\, \lVert f(x-h)-f(x)\rVert_{p} = 0 $$ for $1 \leq p < \infty$. I am trying to find a counterexample to ...
0
votes
1answer
42 views

Proof of Plancherel Theorem

In Real analysis by Folland, the Plancherel Theorem is as follows: If $f \in L^1 \cap L^2$, then $\hat{f}\in L^2$, and $\mathcal{F}|(L^1 \cap L^2)$ extends uniquely to a unitary isomorphism on $L^2$. ...
0
votes
1answer
29 views

Linear operator image is not closed

Studying functional analysis, I have to prove that the image of the operator $S:\ell^1(\mathbb{N})\rightarrow \ell^1(\mathbb{N})$ given by $(S\xi)_n=\xi_n/n$ is not closed, even though $S$ is limited....
0
votes
0answers
23 views

$L^p$-space on the circle, question about the definition

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. Here's what is written in the book: Start by considering integrable functions $f \in L^1(S^1)$ which are ...
0
votes
1answer
46 views

Functional Analysis, spaces [closed]

If p $\neq$ q, Show that it implies $\ell_p$ $\neq$ $\ell_q$ $\\$ I am new to functional Analysis, I don't know how to go about this.
0
votes
0answers
25 views

Bounding the maximum of a sequence of continuous functions using integrals

I have absolutely non clue on how to solve this one. First, recall that $$\lim_{p \rightarrow +\infty }\left ({\int\limits_a^b |f(x)|^{p}dx)} \right) ^{\frac{1}{p}} = \max_{x \in [a,b]} |f(x)|$$ ...