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.

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Uniform convergence and continuous limit in $C^0$ and $L^{\infty}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be $2\pi$-periodic and $f_n \in C^0_{2\pi}(\mathbb{R})$ for all $n \in \mathbb{N}$ such that $\lim_{n \rightarrow \infty} \| f_n-f\|_{\infty} =0$. Then $f \...
Oscar210899's user avatar
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Calculus Identities

I am trying to write an expression to $\partial_t \|\nabla u\|_{L^p(\Omega)}^p.$ Here $\Omega$ is a smooth domain, the function $u$ has no regularity problems (you can take it smooth) and the normal ...
BGT_MATH's user avatar
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If $f \in L^{\infty}$, Is it true that if $\mu(E_n) \to 0 $ then $\int_{E_n} f d \mu \to 0$? [closed]

I am trying to solve the following problem but I'm stuck, Could someone help me? Let $(X, \mathcal{M}, \mu)$ be a measure space and $f \in L^1$ and let $\{E_n\}$ be a sequence of measurable sets such ...
MC2's user avatar
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If $f\in L^2(\mathbb{R^3})$, then $\frac{\hat{f}(\xi)\cdot \xi}{2\pi i |\xi|^2} \in L^1_{loc}(\mathbb{R^3})$.

I have that $f$ is an $L^2(\mathbb{R^3})$ function, so I know that $\hat{f}$ its Fourier transform is in $L^2(\mathbb{R^3})$ too. I want to prove that: \begin{equation} \frac{\hat{f}(\xi)\cdot \xi}{2\...
Hapa's user avatar
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1 answer
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$\log \log (\frac{2}{||x||})$ in $W^{1,2} \setminus L^\infty$ of a ball in $\mathbb{R}^2$

Let $\Omega = B(0,1/2) \subset \mathbb{R}^2$, and let $$ u(x) = \log \log \left(\frac{2}{||x||}\right) \quad for \quad x \in \Omega. $$ I have to show that $u$ is in $W^{1,2}(\Omega) \setminus L^\...
Contrad's user avatar
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2 votes
1 answer
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Showing a function is identically zero

Suppose I have a simple function (not compactly supported) $f$ on $\mathbb{R}^n$ that is orthogonal to all polynomials (with respect to the $L^2$ inner product) on $\mathbb{R}^n$. We further know that ...
Soumya Ganguly's user avatar
1 vote
1 answer
21 views

For every functional on $l_p$ it exists a sequence that determines it uniquely

I am trying to prove this excercise from my Functional analysis book: Let $p \in (1, + \infty ) \subset \Bbb{R}$, $q = (1- \frac{1}{p})^{-1}$ and denote $l_p$ the space of $p-$bounded sequences of a ...
Superdivinidad's user avatar
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Convergence in $L^\infty$($\Omega$) and almost everywhere

I have a question about the difference between the convergence in $L^\infty$ and convergence almost everywhere. Precisly, let $\mu(\Omega) < \infty$, $f_n \rightarrow f$ in $L^1(\Omega)$, then ...
Annabelle's user avatar
3 votes
1 answer
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Why does the Haar orthonormal system span the whole $L^2$?

I am reading "Real Analysis with an Introduction to Wavelets and Applications" because I want to understand wavelets better for my work. I got stuck on a detail about the Haar Basis in ...
Matteo Aldovardi's user avatar
1 vote
1 answer
31 views

Finding the conjugate operator of the following operator

Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k} $ , where $y^0 \in l^2$ — fixed element. Show, that conjugate operator $A^*$ exists and find it. Show, ...
Metal Sonic's user avatar
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When an operator with variable coefficient is bounded?

When the operator $Au(x):=\mathcal{F}^{-1}(m(x,\xi)\widehat{u}(\xi))(x)$ is bounded on $L^p(\mathbb{R}^n)$? I know that, by Mikhlin theorem, if $m(x,\xi)=m(\xi)$ (not dependence on $x$) is a fourier ...
eraldcoil's user avatar
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If $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u\in L^1_{\text{loc}}(\Omega)$, then $\nabla u \in L^1_{\text{loc}}(\Omega)$?

(And as well, a transcription for those unable to load images:) Remark 2. There is a local form of Corollary 1, namely if $u\in L^1_{\text{loc}}(\Omega)$ and $\Delta u \in L^1_{\text{loc}}(\Omega)$, ...
Kimura Leo's user avatar
2 votes
1 answer
49 views

$S \in \mathcal{L}(L^1(\Omega))$, find $T^* \in \mathcal{L}(L^\infty(\Omega))$ with $T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega)$

Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ...
Meta-chan's user avatar
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1 answer
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Move Infinite sum inside a limit $t \to \infty$.

This may be simple, but I want to know if my reasoning is ok. I came across a problem whose essential set up is: let $f_k$ be a sequence of functions in $L^1(\mathbb{R})$ (Lebesgue integrable ...
César VB's user avatar
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1 vote
2 answers
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Using Hahn Banach for switching between $L^p$ spaces

I want to understand the proof (or under which conditions a proof holds) of the following statement: Let $f$ be a function in $L^p$ and let $q$ be such that $\frac{1}{p}+\frac{1}{q}=1$. Then we can ...
proofromthebook's user avatar
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Prove operator $(Tf)(x)=sin(x)\cdot f(x)$ is not compact

Given the following operator in $L_2[0,1]$ $$(Tf)(x):=sin(x)\cdot f(x)$$ Prove or Disprove that the opertor is Compact. I thought it is compact and used arzelà–Ascoli theorem, but apparently I am ...
Its me's user avatar
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3 votes
1 answer
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The dual $(L^\infty)^{*}$ is not $L^1$ by constructing example

The problem statement is the same as this post: $L^{\infty *}$ is not isomorphic to $L^1$ . Let $L^\infty = L^\infty(m)$, where $m$ is Lebesgue measure on $I=[0,1]$ . Show that there is a bounded ...
Nazono Sumiko's user avatar
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1 answer
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Square integrable for universal approximation

Let's consider square-integrable functions $f \in L^2\left(I_n\right)$ with the definition of the $\textit{discriminatory}$: $\textbf{Definition:}$. The activation function $\sigma$ is called ...
BlizzardWalker's user avatar
4 votes
0 answers
74 views

Showing $\chi_A \ast \chi_B(x)$ is continuous if $m(A),m(B) \in (0,\infty)$. [duplicate]

Let $A,B \subset \Bbb{R}$ be Borel measurable with $m(A),m(B) \in (0,\infty)$, then prove $\chi_A \ast \chi_B$ is continuous. Attempt: Note $$f(x)=\chi_A \ast \chi_B (x)=\int \chi_A(x-y)\chi_B(y)dy.$$ ...
homosapien's user avatar
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1 answer
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Sequence of sequences $\{a^{(n)}\}_n \subseteq \ell^2$ with bounded members $|a^{(n)}_k| \leq 1$ has converging subseq.

I struggling to understand a partial step in the solution to an exercise: Given a seq. of seq. $\{a^{(n)}\}_{n \in \mathbb N} \subseteq \ell^2$ such that $|a^{(n)}_k| \leq 1 \forall n,k \in \mathbb N$...
plshelp's user avatar
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convergence of functions.

Let a sequence of functions $v_m$ that converges to $v$ in the space $L^{10/3}(\Omega)$, Where $\Omega$ is a bounded domain. Additionally, $\sup\limits_{m} |v_m|<C$, where $C$ is a constant. Can I ...
Andrés Ortiz 's user avatar
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1 answer
33 views

Some intersection of Schatten spaces gives a weak Schatten space?

Assume $1\leq p <\infty$. It is known that for all $q>p$ $$ S^{p} \subset S^{p,\infty} \subset S^{q}. $$ Out by curioisity, it is true that $\cap_{q>p} S^{q}=S^{p,\infty}$ ? Recall that $$ ...
Liam's user avatar
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2 votes
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$L^p$-estimates for one dimensional wave-equation with lower order pertubation

Suppose $b\in L^\infty(\mathbb{R})$ and $u\colon \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, where $(t,x)\mapsto u(t,x)$ is a solution to $$ \partial_t^2 u = \partial_x^2 u +b(x)\partial_x u,\quad (x,...
ym94's user avatar
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With finite $\mu$, suppose $f_n$ converges in measure to $f$, and for all $n$, $||f_n||_2\leq 1$. Prove $||f_n-f||_1\rightarrow 0$

This is actually a part b), where part a) was to show $f\in L^2$. This was simple: as $f_n$ converges in measure to $f$, there exists a sub-sequence which converges $\mu$-a.e., thus Fatou's gives: $\...
cable's user avatar
  • 128
1 vote
1 answer
39 views

Does the weak limit of a sequence in $L^2([0,1])$ vanish on the limit set of vanishing sets?

Suppose $h_n$ is a sequence of non-negative functions in $L^2([0,1])$ converging weakly to $h$ (i.e., for every $g\in L^2([0,1])$ it holds $\int g\cdot h_n \,d\lambda \to \int g\cdot h\, d\lambda$). ...
Michael's user avatar
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1 vote
0 answers
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Approximating a weakly convergent sequence "uniformly" by a dense subspace

For a fixed $p \in (1,\infty)$, consider $L^p(\Omega)$ for a bounded domain $\Omega$ in the Euclidean space and a sequence $\{ f_{n} \} \subset L^p(\Omega)$ converging weakly. That is, for each $g \in ...
Keith's user avatar
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1 vote
0 answers
24 views

For $1 \leq p<q<\infty$ if $X$ contains sets of arb small positive measure, then $L^p \not\subset L^q$.

Its an if and only if but I have one direction down. So suppose $(X,\mathbf{A},\mu)$ is a measure space and $X$ contains sets of arbitrarily small positive measure. Choose $A_1 \in \mathbf{A}$ such ...
homosapien's user avatar
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1 vote
0 answers
29 views

Amenable countable groups specifically for complex-valued functions in $l^\infty(G)$

Let $G$ be a countable group, and $l^\infty(G)$ the collection of functions $f:G \to \mathbb{R}$ such that $\sup |f(x)| < \infty$. I know that $G$ is amenable if and only if for all $\epsilon > ...
The Unique Operator's user avatar
1 vote
0 answers
139 views

$fg \in L^1$ for every $f \in L^2$ implies $g \in L^2$

Suppose that $f g$ is in $L^1([a,b])$ for every $f$ in $L^2([a,b])$, I understand that this implies that $g$ is in $L^2([a,b])$. Is this assertion true and, if so, how do I prove it? This question was ...
Alessandro's user avatar
2 votes
3 answers
315 views

How to compute the numerical radius of the right shift operator?

Let $T$ be the right shift operator on $\ell^2$ defined by $T(x_1,x_2,\ldots)=(0,x_1,x_2,\ldots)$. The numerical radius of $T$ is defined by $w(T)=\sup\{|\langle Th,h\rangle|:\, \|h\|=1\}$. It is well ...
DeltaEpsilon's user avatar
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1 vote
2 answers
47 views

What conditions are needed to have $F(f,g) \in L^\infty$ for $f, g \in L^\infty$?

Let $\Omega \subset \mathbb{R}$ and $f, g \in L^\infty(\Omega)$. Define $F: \Omega \times \Omega \rightarrow \mathbb{R}$ by $F(x) = \tilde{f}(f(x), g(x))$ for some $\tilde{f}$. I am interested in what ...
CBBAM's user avatar
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Density of $C_c^\infty(\Omega)$ in $L^p(\Omega)$ for a bounded open set $\Omega$ - any detailed proof?

Let $\Omega$ be an open, bounded domain in $\mathbb{R}^n$ (NOT necesasrily with smooth boundary). Now, let $C_c^\infty(\Omega)$ be the space of smooth functions on $\Omega$ with compact support. Then ...
Keith's user avatar
  • 7,673
2 votes
2 answers
86 views

Estimate norm of convolution operator

I'm trying to find the operator norm for $T: L^2([0,1])\to L^2([0,1])$, defined as $Tf(x)=\int_{[0,1]}|\sin(x-y)|^{-\alpha}f(y)dy$, where $0<\alpha<1$. Using an upper bound on $|\sin(x)|\geq |x|/...
mtcicero's user avatar
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2 votes
1 answer
138 views

Completion of some $C[0,1]$ functions with some inner product is a reproducing kernel Hilbert space

The problem Let $X$ be the space of $C^1[0,1]$ functions with the special property $f(0) = 0$. Consider the inner product defined in the following way: $$\langle f, g \rangle = \int_0^1 f'(x) \...
kodiak's user avatar
  • 520
1 vote
0 answers
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Convergence on $L^1$ space

I come across this problem while reading hints for Exercise 3.15 (Brezis): Let $\Omega=(0,1)$ and sequence $f_n$ defined by $f_n(x)=ne^{-nx}$. Prove that: $$\displaystyle \int_{\Omega} \varphi f_n\...
Hải Nguyễn Hoàng's user avatar
1 vote
0 answers
24 views

Integration by Parts for not so regular Sobolev functions

I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$ and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta ...
micha's user avatar
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2 votes
0 answers
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Existence and uniqueness of SDE solution in $L^p(\Omega\times [0,T])$, for $p\geq 2$

I have been studying stochastic differential equations (SDE) and came across the following questions. Let $T \in (0,\infty)$. Let also $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be any filtered ...
jffi's user avatar
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3 votes
1 answer
97 views

Summability of the Fourier Transform.

I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ ...
Matteo Aldovardi's user avatar
1 vote
0 answers
111 views

Define multiplication on $L^1(\mathbb{R})$

I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
InnocentFive's user avatar
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1 answer
57 views

Convergence in $L^p_{loc}$ implies convergence of a subsequence in $L^\infty$

Let $\Omega \subset \mathbb{R}^n$ be bounded or unbounded. Suppose we have a sequence $\{f_n\} \in L^p_{loc}(\Omega)$ such that $f_n \rightarrow f$ in $L^p_{loc}(\Omega)$ for $f \in L^p_{loc}(\Omega)$....
CBBAM's user avatar
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4 votes
1 answer
105 views

$L^p_{loc}(\Omega)$ is completely metrizable

Let $\Omega \subset \mathbb{R}^n$ be a (not necessarily bounded) domain and $1 \leq p \leq \infty$. Then define $L^p_{loc}(\Omega)$ to be the set of functions $f: \Omega \rightarrow \mathbb{R}$ such ...
CBBAM's user avatar
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0 votes
0 answers
38 views

Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space?

Most often (at least in probability), one defines the $L^p$ space as Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then $$ L^p(\Omega, \mathcal{F},...
Euler_Salter's user avatar
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Is the set of polynomials of degree $\le n$ closed in $L^2(a,b)$? [duplicate]

Let $a<b$ be real numbers. I'm asked to prove that for every $f\in L^2(a,b)$ there exists a unique polynomial $p_n$ of degree less than or equal to $n$ such that $$\|f-p\|_2\ge\|f-p_n\|_2,$$ for ...
Little Jonny's user avatar
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1 answer
69 views

$f \in C^1([a;b])$, prove that $ \forall x,y \in [a;b]$ we have $|f(x)-f(y)| < || f' ||_2 \sqrt{|x-y|} $

Question: Let $f \in C^1([a;b])$, prove that $ \forall x,y \in [a;b]$ we have $|f(x)-f(y)| < || f' ||_2 \sqrt{|x-y|} $ Answer: 1- $f \in C^1([a;b]) \Rightarrow f'(\chi) $ exists and is defined: $ |...
OffHakhol's user avatar
  • 708
1 vote
0 answers
34 views

Sequence in $\ell_p$ spaces

The sequence given by: $$x_n=(1^{-1/q},2^{-1/q}-1^{-1/q}, 3^{-1/q}-2^{-1/q}, \dots)$$ That is, $$\sum_{n=1}^{\infty}n^{-1/q}-(n-1)^{-1/q}$$ Is this sequence in the sequence space $\ell_p$ ? where for $...
User2427's user avatar
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1 vote
1 answer
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Equivalence of two forms of the Marcinkiewicz interpolation theorem

On this article and Stein & Weiss (statement 1) and the books by Linares & Ponce and Duoandikoetxea (statement 2), I found the following statements of the Marcinkiewicz interpolation theorem: ...
Derivative's user avatar
  • 1,568
1 vote
2 answers
141 views

Does $\sin (nx)$ converge in $L^2$?

I was just introduced the concept that if $(f_n)$ converges in $L^2$ topology to $g(x)\in L^2([0,2\pi])$ then $\lim_{n\to\infty}\int^{2\pi}_0|f_n(x)-g(x)|^2dx=0$. I would appreciate any hint to how to ...
Derewsnanu's user avatar
2 votes
0 answers
43 views

Why is it bad to translate from Lp norm to priors?

It seems like we should start with a probability distribution and infer a log likelihood based on that. Why is it a bad idea to start with regularizers and map this onto priors? For example, if I were ...
user1289239's user avatar
4 votes
0 answers
239 views

$L^1- L^\infty$ estimate for the semi group of wave equations

I am looking for a proof of the following lemma for the case where: $y= (y_1,\cdots, y_n)\mapsto P(y) = \|y\|_2= \sqrt{y_1^2+ \cdots + y_n^2}.$ In this case the rank of the mentioned matrix is $n-1$...
A. PI's user avatar
  • 639
4 votes
0 answers
38 views

Trying to understand the proof for the criterion of compactness in $l_p$ space

I have the following theorem about the criterion of compactness in $l_p$ space For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions are equivalent: i) $K$-totally bounded in $(l_p,||.||...
lee max's user avatar
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