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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3
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1answer
18 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
1
vote
2answers
33 views

Show that the series converges ($l^2$)

I know that $\sum_{k=1}^\infty|y_n|^2=S<\infty$. I also have that $\lambda >1$. I need to show that $$ \sum_{k=1}^\infty \left| \frac{y_1}{\lambda^k} + ...
0
votes
0answers
26 views

Continuity of translation property [duplicate]

Let $u \in L^{p}(U)$ where $1 \leq p \lt \infty$ & $U \subseteq \mathbb R^{n}$ . Define : $F : \mathbb R^{n} \to L^{p}(U) $ by $ F(y) := u(x+y)$ . Prove that: as a function of $y$ ; $F(y) $ is ...
3
votes
1answer
35 views

Question on Lp spaces defining metric

First question here so really excited and hope you can help me, thanks! In my intro to functional analysis class we just now covered $L^p$ spaces and I was presented with this homework question: ...
0
votes
2answers
29 views

Absolutely continuous function whose derivative is in $L^2([0,1])$ etc., evaluate $\lim_{x\to0^+}\frac{f(x)}{\sqrt{x}}$

Suppose $f$ is absolutely continuous on $[0,1]$, $f(0)=0$, and $f'(x)\in L^2([0,1])$. Show that $$\lim_{x\to 0^+}\frac{f(x)}{\sqrt{x}}=0$$ So far I've got the following: Since $f$ is AC by the ...
2
votes
4answers
72 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
1
vote
0answers
33 views

$\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2$

I want to show that $\{x \mapsto e^{2\pi i k x} \mid k \in \mathbb{N}\}$ is orthonormal basis of $L^2((0,1); \mathbb{C}) =: X$. Of course the only problem is to show completeness. In our lecture we ...
1
vote
0answers
41 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
4
votes
2answers
80 views

Are the Hermite-Gauss functions linearly dense in $L^1(\mathbb{R})$?

The Hermite-Gauss functions ($t\mapsto H_m(t)e^{-t^2/2}$) are known to be an orthonormal basis for $L^2(\mathbb{R})$, a fortiori linearly dense in $L^2(\mathbb{R})$, and all are in the Schwartz space ...
4
votes
1answer
54 views

Is $A=\{x \in \ell^2 \mid \sum_{n=1}^{\infty} \frac{x_n}{n}=0 \}$ dense in $\ell^2$

I think that the answer is no I thought quite a bit about this problem. My idea was to build a sequence $(y_n)_{n \in \mathbb{N}} \subset A$ such that given a $x \in \ell^2$ we pick the first N ...
5
votes
2answers
64 views

Is an $L^p$ function in an annulus $L^p$ restricted to almost all planes?

Let $n\geq3$ and consider the annulus-like domain $A=B(0,1)\setminus B(0,r)\subset\mathbb R^n$. Take any number $p\in[1,\infty]$. If $f\in L^p(A)$, is it true that $f|_{P\cap A}\in L^p(P\cap A)$ for ...
0
votes
0answers
29 views

Behaviour of $L^2$ functions at infinity [duplicate]

Is it possible to prove that if $f\in L^2(\mathbb {R}) $ then $\exists\lim_{x\to\pm\infty}\lvert f\rvert^2$ and $\lim_{x\to\pm\infty}\lvert f\rvert^2=0$? If not, is it easy to find a counterexample?
0
votes
0answers
28 views

$L^2$ space (or just integration) on $\Omega \times \{0,1\}$

Let $\Omega$ be a smooth bounded domain. If $u \in L^2(\Omega \times \{0,1\})$, then $$\int_{\Omega \times \{0,1\}}|u(x,y)|^2 < \infty$$. How to interepret the integral $\int_{\Omega \times ...
0
votes
1answer
30 views

Existence of a Particular $L^1$ Function

I hope this question hasn't already been posted before in some other form (I couldn't find it, so if it has, please pardon me). I found this question on an old qualifier, but I am completely lost as ...
1
vote
1answer
23 views

Extending bounded linear functionals from L^q to L^p on finite measure spaces

I'm trying to give an example to show the following: Not every bounded linear functional on $L^3([0,1],\mu)$ is the restriction of a bounded linear functional on $L^2([0,1],\mu)$. (Here $\mu$ is ...
0
votes
0answers
21 views

question about $L^2([0,T]\times \Omega)$ and iterated integral [duplicate]

Let $X=[0,T]\times \Omega$ where $\Omega$ a bounded domain. Consider the space $L^2(X)$, so $u \in L^2(X)$ if $$\int_{[0,T]\times\Omega}|u|^2 < \infty.$$ Is it true that the integral $$\int_0^T ...
2
votes
1answer
57 views

A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
1
vote
2answers
56 views

Question about the Riemann-Lebesgue Lemma proof

Ok, so one of the formulations of the Riemann-Lebesgue Lemma says: $$ f\in L^1(\mathbb{R}) \implies \hat{f}(\omega)\to 0\;\mbox{ when } \;|\omega|\to\infty.$$ I get all the steps of the proof, except ...
2
votes
2answers
62 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...
1
vote
0answers
19 views

Prove that $\frac{\langle f^2,g\rangle_{L^2}}{\left\|f\right\|_{L^2}^2}\ge-\left\|g\right\|_{L^\infty}$ for $f\in L^2$ and $g\in L^\infty$

Let $\Omega\subseteq\mathbb{R}^n$ be bounded, $f\in L^2(\Omega)$ and $g\in L^\infty(\Omega)$. How can we show, that $$\frac{\langle ...
0
votes
0answers
31 views

Does $L^p(L^1([0,1]))$ make sense?

I'm examining several function spaces like $L^p(X,\mu)$ where $X$ is a Banach space. Is it possible to take $X = L^1([0,1])$ and then look at $L^p(X)$? The problem I have is that I don't know ...
0
votes
1answer
27 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
-1
votes
1answer
30 views

An ineqaulity involving critical Sobolev exponent

This is related to my previous question An inequality involving Sobolev embedding with epsilon. There I wished to get that, for given a nice bounded domain $\Omega$ in $\mathbb{R}^n$, $\forall ...
1
vote
1answer
23 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
28 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 ...
1
vote
1answer
23 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 ...
1
vote
1answer
27 views

Motivation for a specific basis for the space $L^2$ zero mean and piecewise constant functions?

Let $S$ be the space of $L^2$ zero mean and piecewise constant functions in $\Gamma$. In other words, $$S=\left\{\eta\in L_0^2 : \eta|_{I_i}\in \mathbb{R}\;\forall i=1,...,N\right\}$$where ...
3
votes
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 ...
0
votes
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
votes
1answer
94 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 ...
1
vote
1answer
41 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
27 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
votes
1answer
41 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
26 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
votes
1answer
67 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 ...
0
votes
0answers
15 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 ...
-1
votes
1answer
44 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
40 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
19 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 ...
0
votes
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
votes
2answers
63 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
votes
1answer
56 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
votes
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
59 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
38 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
36 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 $L^{p_0}$ ...