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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9 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to ...
-1
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1answer
56 views

Why is $f(x) = \sin(x)$ an element of $L^2(-\pi, \pi)$ not $L^2(a,b)$ [on hold]

I am having some trouble understanding why some functions are members of $L^2(\mathbb{R})$ whereas other functions are members of some restricted subset of $\mathbb{R}$ such as $(-\pi, \pi)$ Can ...
1
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1answer
41 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 ...
2
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0answers
33 views

Factorization of $L^{p}$ spaces

Is the following theorem true and if so does it have a name? Theorem: Let $p$, $q$, $r\in\left(0, \infty\right]$ with $\frac{1}{q}=\frac{1}{p}+\frac{1}{r}$. Every $L^{q}$-function is the product of a ...
1
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1answer
41 views

Is σ-finiteness necessary for the “in measure” version of the dominated convergence theorem to hold?

Let $(X,\Sigma,\mu)$ be a measure space, $g\in L_1$, $|f_n|\le g$ and $f_n\to f$ in measure. I want to prove that $\int f_n\to f$, and $f_n\to f$ in $L_1.$ Now, this may be already solved in the ...
0
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1answer
29 views

Product of an $L^\infty$ function and an $L^1$ function is integrable

For every $f \in L^1(\mathbb{R^n})$ let $$ \hat{f} : \mathbb{R}^n \to \mathbb{C}, \quad \hat{f}(\xi) := \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) \exp{(-i \langle x, \xi \rangle)} ...
3
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2answers
87 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ? Edit: I am adding my ...
2
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1answer
36 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
0
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2answers
78 views

If $f_n\to f$ a.e. and are bounded in $L^p$ norm, then $\int f_n g\to \int fg$ for any $g\in L^q$

Suppose $p>1$ and $q$ is its conjugate exponent. Suppose $f_n \rightarrow f~a.e.$ and $\sup_n\|f_n\|_p < \infty $. prove that if $g \in L^q,$ then $\lim_{n \rightarrow \infty} \int f_ng=\int ...
3
votes
3answers
69 views

Multiplication operator on $L^1$

Let $\phi :X \rightarrow \mathbb{C}$ be measurable with respect to the measure space $(X,\mu)$. Suppose that $\phi f \in L^1(\mu)$ whenever $f \in L^1(\mu)$. Define $M_{\phi}(f)=\phi f$, for $f \in ...
4
votes
2answers
44 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 ...
30
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2answers
9k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, ...
3
votes
1answer
49 views

Question on $L^p$ 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: ...
2
votes
1answer
35 views

A form of Nash's inequality, $\|f\|_2\le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta$

For $f\in \mathcal{S}(\mathbb{R})$ can anyone help me prove the following Nash inequality, $$\|f\|_2 \le C \|f\|_{1}^{\alpha} \|f'\|_2^\beta.$$ I believe $\alpha$ and $\beta$ should be $2/3$ and ...
1
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2answers
37 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} + ...
1
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1answer
28 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 ...
0
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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 ...
1
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2answers
58 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 ...
0
votes
2answers
30 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
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4answers
75 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 ...
2
votes
2answers
41 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwarz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
2
votes
1answer
44 views

Uniform integrability and weak L1 convergence

I am working on exercise 4.14 in chapter 3 (on convergence) in the book "Probability and Stochastics" by Erhan Cinlar. The exercise can be found on page 109. First, let me give the necessary ...
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 ...
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 ...
2
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1answer
67 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
4
votes
2answers
100 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 ...
1
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0answers
43 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( ...
1
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1answer
56 views

show $\ell^p$ is dense in $\ell^q$

I am trying to show $\ell^p$ is dense in $\ell^q$, where p< q so i need to show that any $\epsilon > 0$ for any $(x_i)\in\ell^q$ there is $(y_i) \in \ell^p$ such that $d((x_i),(y_i)) < ...
4
votes
1answer
58 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 ...
0
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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
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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
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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 ...
1
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1answer
141 views

Estimate $L^{2p}$ norm of the gradient by the supremum of the function and $L^p$ norm of the Hessian

Prove $$\|Du\|_{L^{2p}(U)} \le C\|u\|_{{L^\infty}(U)}^{1/2} \|D^2 u\|_{L^p(U)}^{1/2}$$ for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise ...
4
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1answer
204 views

Intuition for a function that belongs to a $L^p$ space

Does a function $f(x):[0,\infty)\rightarrow R$ with $f\in L^p$ for $p<\infty$ have to die down to $0$ as $x\rightarrow \infty$? I some how feel that the $L^p$ norm exist only when the function dies ...
8
votes
2answers
309 views

Subspaces of $L^p$

So studying Qualifying Exam problems in Analysis I cam across this one: For $1\lt r \lt p \lt s \lt \infty$ where $\mu$ denotes Lebesgue measure, a) Construct a subspace of $L^p([0,1],\mu)$ such ...
2
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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 ...
2
votes
1answer
58 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 ...
0
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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 ...
4
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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
42 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
1
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0answers
20 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 ...
-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 ...
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
vote
1answer
24 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 ...
1
vote
1answer
24 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 ...
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 ...
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 ...
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 ...