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 ...

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1answer
13 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Is it true for $\ell_q\big(\mathbb{N}\big)$ or only on ...
0
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1answer
17 views

application of positive linear functionl

The space $L^{1}(\mathbb R)$ is a commutative Banach algebra under convolution. A linear functional $F$ on $L^{1}(\mathbb R)$ is positive if $F(f^*f)\geq 0$ for $f\in L^{1}(\mathbb R).$ (where ...
2
votes
1answer
24 views

If ${f_n}$ converges to $f$ in $L_p$ sense and to $f'$ point-wisely, does it mean $f=f' a.e.$?

The question came into my mind when I read a theorem from Kubrusly's "Measure Theory: a First Course", saying that if $f_n\rightarrow f'$ uniformly and $f_n\rightarrow f''$ in $L_p$ sense, then ...
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1answer
19 views

Series Test for Integrability via the Distribition Function

I imagine that the following question has a well known (and perhaps, easily obtainable) answer, but I can't find it by myself nor along the references that I have in mind so far. So, if $f$ is a ...
3
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1answer
21 views

Question regarding convergence in $L^p$ spaces.

When solving an exercise regarding $\ell^p$ spaces, I came up with the following question. The exercise said, Let $1<p\leq \infty$ and let $p'=\frac{p}{p-1}$. Let $b\in \ell^{p'}$ and define ...
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0answers
26 views

Is a function $f \in \mathbb{C}^{ \infty}[0,l]$ always in $L^2(0,l)$?

I was trying to find a function that is not in $L^2(0,l)$ but that it is in $\mathbb{C}^{\infty}[0,l]$ for l>0. But if the function is continuous at both sides of the interval then it is integrable, ...
4
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1answer
82 views

Prove that a series converges in $L^2(\Omega)$

Let $\Omega$ be a bounded smooth domain and let $\varphi_k$ be eigenfunctions of the Neumann Laplacian with eigenvalues $\lambda_k$. Let $u \in H^1(\Omega)$. I want to show that for all $y \in ...
1
vote
1answer
28 views

Convergence in probability implies boundedness in $L^1$?

Suppose we have a sequence of positive random variables wgich converges to 0 in probability, i.e. $X_n=o_P(1)$. I want to prove that $E[X_n]$ is bounded. My idea: In particular $X_n$ is bounded in ...
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1answer
38 views

A question on Hölder inequality [duplicate]

Let $p, q > 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Then $$|\sum\limits_{i = 1}^n x_i y_i| \leq ||x||_p ||x||_q, \;\; \forall x, y \in \mathbb{R}^n.$$ I have to prove it considering $$u = ...
1
vote
1answer
25 views

About equi-integrability

Suppose $\Omega\subset \mathbb R^N$ is bounded and lipschitz boundary. Suppose $u_n, u\in H^1(\Omega)$ such that $u_n\to u$ weakly in $H^1$. Then can I conclude that $$ ...
2
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1answer
38 views

Exercise 8.1 in Brezis' Functional Analysis

Consider the function $$u(x) = \frac{1}{(1+x^2)^{\frac{\alpha}{2}}} \frac{1}{\ln(2+ x^2)} \qquad\; x\in \mathbb{R}$$ with $0<\alpha<1$. Check that $u\in W^{1,p}(\mathbb{R})$ for all $p\in ...
1
vote
1answer
25 views

Relations between the topologies of $L^0$ and $L^1$ on finite measure space

Let $(\Omega, \mathcal F, P)$ be a probability space and denote by $L^0(\Omega, \mathcal F, P)$ the space of all random variables $X : \Omega \to \mathbb R$ (i.e. measurable functions between ...
2
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0answers
61 views

a Bound for functions in $L^p$ after convolution with a $G_\lambda$ almost a heat Kernel

The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 ) We consider the operator $G_\lambda$ $$G_\lambda ...
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0answers
13 views

Composition of Lp convergent function and continuous function

Let $f$ be a continuous function on $\mathbb R$ such that for $U\subset \mathbb R ^n$ bounded it holds that $\forall w\in L^p(U) ~~ f(w)\in L^q(U)$. Let $~u_k \rightarrow u$ in $L^p(U)$ . Does ...
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1answer
40 views

Exercise 4.1 in Brezis' Functional analysis [closed]

I'm trying to solve exercise 4.1 (page 118) from H. Brezis' "Functional Analysis ...". Let $\alpha > 0 , \beta > 0$. Set $$f(x) = \{ 1 + |x|^\alpha \}^{-1} \{ 1 + |log|x||^\beta \}^{-1}, ...
2
votes
1answer
31 views

Counterexample to Rellich-Kondrachov Compactness Theorem, case $q=p^*$

I was searching some counterexample for I was searching some counterexample for Rellich-Kondrachov Compactness Theorem (You can see: PDE, Evans, chapter 5), for the case $q=p^*$ and I found this ...
4
votes
1answer
30 views

Dimension of a certain $L^p$ quotient space.

Define $L^p_0 := \{ f \in L^p : \int f = 0 \}$. I am trying to calculate the dimension of the cokernel of the inclusion operator $i:L^p_0 \to L^p$. That is, I am trying to calculate $$\dim ( L^p / ...
0
votes
1answer
18 views

Giving a bound of the norm of a convolution

Let $f:(-1,1)\to \mathbb{R}$ be a smooth function with compact support. Suppose that $f(x)\geq 0$ for all $x$ and $\int f =1$. Extend $f$ to $\mathbb{R}$ by $f(x) = 0$ if $x\not \in (-1,1)$. Show that ...
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0answers
24 views

Is $\sin(kx) $ a complete system in $L^{2}(0,b)$ for every k in $\mathbb{R}$?

Is the family $\{ \sin(kx ) \}$ a complete system in $L_{2}(0,1)$ for every $ k \in \mathbb{R}$, $k\geq1$ ? And less specifically, Is it a complete system in $L_{2}(0,b)$ for every b$\in \mathbb{R}$ ...
2
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2answers
61 views

A question in ${l^\infty }$

Let ${l^\infty }$be the space of all real bounded functions $x$ on the positive integers. let $\tau $ be the translation operator defined on ${l^\infty }$ by the equation Let 'r be $$(\tau x)(n) = ...
2
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1answer
54 views

Riesz-Fischer theorem

The aim of this exercise is to prove the Riesz-Fischer theorem for Hilbert spaces that aren't separable. Let $I$ an index set and $1\leq p \leq \infty$. Let $\mathcal{F}=\{F\subset I: F$ is ...
2
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1answer
25 views

Equivalences of weak convergence in $\mathcal{L}_p$ spaces with the Lebesgue measure

Let $\Omega =(0,1)$, and $f,f_n\in \mathcal{L}_p(\lambda)$. Prove that if $\sup_n{\| f_n \|}<\infty$ and $$\int_{(0,t]}f_n \, \,\mathrm{d}\lambda\rightarrow \int_{(0,t]}f \, ...
2
votes
2answers
56 views

Infinity norm of continuous function.

Let $f$ be a continuous function on the measure space $\mathbb{R}^n,\mathcal{L},\lambda$(Lebesgue measure). Prove that $\|f\|_\infty = \sup\{|f(x)|$ $|$ $x \in \mathbb{R}^n\}$ I saw same ...
0
votes
0answers
15 views

Function in Lp Space [duplicate]

Assume that $1<p<∞ $, f is absolutely continuous on $[a,b]$, $f′$ ∈ $L^{p}$ and $a=\frac{1}{q}$, where $q$ is the exponent conjugate to $p$. Prove that $f ∈ Lip$ $a$. I was thinking of using ...
1
vote
1answer
35 views

Convergence of sequence of function in norm.

Let $1\leq p<\infty$. Suppose that $\{f_k\}$ is a sequence in $L^p(X,\mathcal{M},\mu)$ such that the limit $f(x)=\lim_{k \to \infty}f_k(x)$ exists for $\mu$-a.e. $x\in X$. Asumme that ...
4
votes
3answers
78 views

If $f$ is in $L^{p}$, prove that $\lim \int_{x}^{x+1} f(t) dt = 0$

If $f$ is in $L^p$, prove that $\lim_{x \to \infty} \int_{x}^{x+1} f(t) dt = 0$ It is easy to think that integration must be vanish as $x \to \infty $ but I cannot write them with math. Suppose ...
1
vote
1answer
35 views

Sequence does not converge in $L^{p_0}$ but converges in $L^p\ \forall 1\le p<p_0$

Let $1<p_0<\infty$ Find a sequence $\{f_k\}$ such that $f_k \in L^p$ for $1 \leq p < \infty,$ $f_k \rightarrow 0$ in $L^p$ for $1 \leq p <p_0$, but $f_k$ does not converge in $L^{p_0}$ ...
2
votes
2answers
71 views

$\mathcal{L}_2$ continuous functions with $f(0)=\alpha$ are dense in $\mathcal{L}_2 [-1,1]$

Let $X=\mathcal{L}_2 [-1,1]$ and for any scalar $\alpha$ we define $E_\alpha=\{f\in \mathcal{L}: f \text{ continuous in } [-1,1] \text{ and } f(0)=\alpha \}$. Prove $E_\alpha$ is convex for any ...
3
votes
3answers
54 views

Proving that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$.

I want to prove that $x,y \in \ell^2(\Bbb N) \implies x+y \in \ell^2(\Bbb N)$. I'm damn sure that there is a quick way to do this, but I'm not seeing it. I am capable of proving Young, Hölder and ...
2
votes
2answers
55 views

Convergence in $L^p$ spaces.

Prove that for all integrable functions $g_n, g$, we have the implication $\|g_n-g\|_1\to 0\Rightarrow \|g_n\|_1\to \|g\|_1$ as $n\to \infty$. Is the converse true? It seems like $|g_n-g|_1 \to 0$ ...
1
vote
1answer
33 views

Is the Cesaro Operator normal?

The Cesàro operator $T:ℓ_p→ℓ_p$ is defined by $$(Tx)_k=(1/k)\sum_{j=1}^k x_j$$ where $x=(x_j)$. Is this operator normal?
4
votes
1answer
52 views

Dual space of $L^{\infty}$ - Where is the mistake?

Today I thought about this for the first time and I really cannot see what is going on. I think it is a very stupid question but I really cannot see it. Consider the space $L^{\infty}(\mathbb{R})$ ...
0
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0answers
15 views

$L_+^p(X,\mu)$ is a closed and convexe subset of $L^p(X,\mu)$.

I have a problem with an exercise: Let $(X,A,\mu)$ a measure place, $p\in[1,\infty)$ and $\mu(X)<\infty$.Prove that the set $$L_+^p(X,\mu):=\{f\in L^p(X,\mu):f(x)\geq 0\ \mu-\text{a.e.}\}$$ is a ...
1
vote
1answer
70 views

Prove $\sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$ [duplicate]

I want to show that $u_k(x)= \sin(kx) \rightharpoonup 0$ as $k \to \infty$ in $L^2(0,1)$. We know trivially that $0 \in L^2(0,1)$. I need to show that $\langle u^*,\sin(kx) \rangle \to \langle ...
3
votes
0answers
34 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
1
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1answer
44 views

Set derived from definition of $\Vert f \Vert_\infty$

Someone told me that the set $B_n := \{x \in X : \vert f(x) \vert > \Vert f \Vert_\infty - \frac{1}{n}\}$ for $n \in \mathbb{N}$ (where $B_n$ has finite positive measure), is derived from the ...
0
votes
0answers
14 views

Step of proof Hardy's Inequality [duplicate]

I trying to prove the Hardy's Inequality, by Evans book. I need of a little help in the step: If $u\in L^1(B(0,r))$ satisfying $$(2-n)\int_{B(0,r)}\frac{u^2}{|x|^2}dx=2\int_{B(0,r)}u ...
3
votes
1answer
27 views

Convolution product, Lp spaces

I wonder how to prove the following statement, Let p,q be real numbers s.t $1\leq p \leq\infty$, $1\leq q \leq\infty$ and $ \frac{1}{p}+ \frac{1}{q}=1$ Let $f \in L^p(\mathbb R^n)$ and $g \in ...
1
vote
1answer
32 views

About the definition of Sobolev Spaces

I'm studying Sobolev Space and I have a question about the definition: Def.: The Sobolev Space $W^{k,p}(U)$ consists of all locally summable functions $u:U\to \mathbb{R}$ such that for each ...
0
votes
1answer
24 views

Are continuous functions dense in $L^1$?

It is a well known fact that the continuous compactly supported functions are dense in $L^1(\mathbb R)$. An immediate counterexample to this fact for a non locally compact space is $\mathbb R ...
1
vote
2answers
50 views

$L^p$ space corollary

I'm confused about the proof for this theorem: let $E$ be a measurable set s.t. $mE<\infty$ and $1 \leq p_1 < p_2 \leq \infty$. Then, $L^{p_2}(E) \subset L^{p_1}(E)$. Also, $\|f\|_{p_1} \leq ...
3
votes
1answer
45 views

Definition of functions in $L^p$ space

I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$. But I feel as though I'm missing some important ...
1
vote
2answers
34 views

$L^1$ functions at $\infty$ to $\infty$

I am having trouble with a practice prelim question: If $f \in L^1(\mathbb{R})$ then $\lim_{n \rightarrow \infty} \int_n^{\infty} f(x)dx = 0$ I know that $f$ is bounded, but I am not if I should add ...
0
votes
1answer
22 views

$f_n$ converges in duality with $C_b$ and is uniformly bounded then it converges in duality with $L^1$

Let $(X,m)$ be a metric measure space, $(f_n)_n$ a sequence in $L^\infty, f \in L^\infty$ s.t. $$ \int gf_n \ dm \rightarrow \int g f $$ for every $g : X \rightarrow \mathbb R$ continuous and bounded. ...
4
votes
1answer
93 views

Converge of a sequence in $L^p(\mathbb{R}^3)$

Let $f(x)\in L^p(\mathbb{R}^3)$ for every $p\in [1, \infty]$. Let $B(n)\subset \mathbb{R}^3$ be the ball of radius $n$ centered at the origin. I want to show that the sequence ...
2
votes
2answers
42 views

Compactness and (global) convergence in measure

Let $B$ denote the unit ball of $L^\infty$. Question: is $B$ sequentially compact for the topology of convergence in measure ? I am not necessarily assuming that the measure is finite (but $\sigma$ ...
4
votes
1answer
34 views

Defining a bounded operator on $l^p$

Let $(c_{jk})_{j,k \in \mathbb{N}} \subset \mathbb{C}$ be such that $a:=\sup_{k \in \mathbb{N}} \sum_{j \in \mathbb{N}}|c_{jk}|<\infty$ and $b:=\sup_{j \in \mathbb{N}} \sum_{k \in ...
2
votes
1answer
110 views

Showing an integral is in $L^1$

Let $0<a<1$ and $f\in L^1([0,1])$. Show $g(x)=\int_0 ^x\frac{1}{(x-t)^a}f(t)dt$ exists a.e. in $[0,1]$ and $g\in L^1([0,1])$. Using Fubini, $$\int_0 ^1 \vert g(x) \vert dx=\int_0 ^1 \int_0 ...
1
vote
1answer
26 views

Is the Fourier transform a conformal map on $L^{2}$?

I read that a conformal map is one that preserves the angles. I know nothing more about conformal maps. I don't know where to find a generalized definition of a conformal map, but I guess that if ...
3
votes
1answer
47 views

Showing a sum of $\vert f(x+k)\vert $ belongs to $L^{\infty}$ if $f,f'\in L^1$

I am working on this Suppose that $f,f'\in L^1(\mathbb{R})$. Then $\sum_{k= 0} ^{\infty}\vert f(x+k)\vert\in L^{\infty}([a,b])$ for any $a,b\in \mathbb{R}.$ Idea: Let $i$ be any integer. $\int_i ...