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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 ...

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1answer
15 views

Showing that the almost uniform limit of functions with bounded $L_\infty$ norms is in $L_\infty$

Suppose that $(f_n)$ is a sequence of functions for which there exists a finite constant $C$ such that the $L_\infty$ norm of $f_n$ is less than or equal to $C$ for all $n$. Suppose further that $f_n$...
0
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1answer
9 views

Discrete convolution of two $\ell_p(\mathbb{Z})$ functions for $p > 2$

Given $f, g \in \ell_p(\mathbb{Z})$, we define the convolution of $f$ and $g$ as follows : $$(f\ast g)(x) :=~ \sum\limits_{y = -\infty}^\infty f(y)g(x-y).$$ It is easy to prove that : If $p = 1$ ...
0
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1answer
5 views

Coercive bilinear form for maximum norm

Let $f$ be a differentiable function. Denote a bilinear form by $$b(f,f) = \int_{0}^{1} \bigg( \frac{d f(x)}{dx} \bigg)^{2} dx.$$ Given $f(0) = 0,$ we want to show that $$a \cdot b(f,f) \geq ||f||_{\...
1
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2answers
35 views

Linear operator in Lp and its norm [on hold]

Given $$g \in L_\infty[0,1]$$ show that, for 1 ≤ p ≤ ∞, $$ f \rightarrow f\cdot g$$ is a linear continuous operator of $$L_p[0,1] \to L_p [0,1]$$ and compute its norm
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0answers
22 views

Can you prove Lusin's theorem without using approximation of $L_1$ functions?

Lusin's theorem states that if $f:[a,b]\rightarrow\mathbb{R}$ is a Lebesgue measurable function, then for any $\epsilon>0$ there exists a compact subset $E$ of $[a,b]$ whose complement has Lebesgue ...
2
votes
1answer
32 views

Compact operators on $\ell^1$

Let $T\in \ell^1$, $Tx = (\lambda_1x_1,\dots,\lambda_nx_n,\dots)$. Want to show that if $T$ is compact, then $\lambda_n\to0$. I know for $p\in(1,\infty]$, canonical basis $e_n \rightharpoonup 0$ (so ...
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1answer
31 views

Proving a property of $L^\infty$ spaces

Let $f$ be continuous and bounded on $\mathbb{R}^d$. Show that $\|f\|_\infty=$ sup$\{|f(\vec{x})|: \vec{x}\in\mathbb{R}^d\}$ relative to Lebesgue measure. Basically I need to prove that $\|f\|_\infty$...
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1answer
18 views

About Sobolev Embedding theorem in dimension one

If $a,b<\infty$ and $f \in H^{1}([a,b])$, is this inequality true ? $$ \Vert f \Vert_{L^{\infty}(a,b)} \leq C \Vert f\Vert_{L^2(a,b)}\Vert f'\Vert_{L^2(a,b)}.$$ I know it is true when $a=-\infty$ ...
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0answers
36 views

Norm of Fourier Sum operator $S_n$ on $\mathcal{L}^1$

Consider the operator $S_n: \mathcal{L}^1(\mathbb{T}) \to \mathcal{L}^1(\mathbb{T})$ defined by $S_nf(x)=\sum_{|k|\leq n} e^{inx}\hat f_n = f * D_n(x),$ where $\hat f_n$ are the Fourier coefficients ...
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1answer
31 views

Metric Space - A point outside unit ball in $l^1(\mathbb R^2)$ and inside unit ball in $l^\infty (\mathbb R^2)$.

A point outside unit ball in $l^1(\mathbb R^2)$ and inside unit ball in $l^\infty (\mathbb R^2)$. Is there a $p$ between 1 and infinity such that $||x||_p = 1$? How about in $\mathbb R^n?$ Since it ...
2
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1answer
32 views

Intuition (“resolving power”) of $L^p$ norm or space

I am now studying functional analysis, especially $L^p$ spaces and I'm wondering what kind of property of functions the $L^p$ norms measure. I know that when $\Omega$ has a finite measure, there ...
3
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1answer
33 views

Finding the norm of the operator $M_g : L_p \to L_p$ where $g \in L_{\infty}$

Suppose $(X,\,\cal{A},\,\mu)$ is a measure space where $\mu$ is a finite measure, and $p \in (1,\,\infty)$. Say that we take some $g \in L_{\infty}$, and we define $M_g : L_p \to L_p$ by $$M_g (f) = ...
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1answer
21 views

Showing a product of two Lebesgue integrals is $\geq 1$ if the product of the integrands is $\geq 1$

Let $\mu$ be a probability measure on a set $X$, i.e. $\mu(X)=1$, and let $f$ and $g$ be positive measurable functions on $X$. Show that if $fg\geq1$, then the integral of $f$ times the integral of $...
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1answer
22 views

Proof that the product of locally integrable function $g$ with any test function is zero implies $g=0$ a.e

How to prove that If for $g\in L_{p}$ $p\in(1,2]$ and for all $\phi \in C_{0}^{\infty}$ we have $\int_{\mathbb{R}} g \phi d\mu =0 $ then $g=0$ a.e where $\mu $ is the Lebesgue measure. I was ...
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1answer
20 views

$L^1$ norm submultiplicative?

Let $f,g:[0,1]\to[0,M]$ be measurable for some $M<\infty$. We know by the Hoelder inequality that for any $p\geq1$, $$\|fg\|_1\leq\|f\|_p\|g\|_{1/(1-1/p)}\,.$$ Do we also have (for this special ...
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1answer
23 views

Compactness of $A \subset l^2(\mathbb{N})$

I'm wondering about how to approach something like this: Let $c \in \ell^2(\mathbb{N})$, $A = \{x \in \ell^2(\mathbb{N}) : |x(n)| \le |c(n)|, \forall n \in \mathbb{N}\}$. Show that $A$ is compact in ...
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1answer
18 views

$X_n$ be a sequence of $L^1$ random variables converging a.s. to $X$ with $\inf_{n\ge 1} E(X_n) >-\infty$. To show $E(|X|)< \infty$

Let $\{X_n\}_{n\ge1}$ be a sequence of random variables such that $X_{n+1}\le X_n,\forall n \ge 1$ and $ E(|X_n|) <\infty,\forall n \ge 1 $, $\inf_{n\ge 1} E(X_n) >-\infty$ and $X_n \to X$ ...
4
votes
1answer
26 views

Is the Norm of the $L^{2}$ Function Equal to the Given Limit?

Let $g$ be an $L^{2}$ function on $[0,2]$, with respect to the Lebesgue measure $m$. Is it true that $$||g||_{1}=\lim_{p\to 1}\left(\int |g|^{p}~dm\right)^{1/p}?$$ I'm really not sure how to tackle ...
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0answers
11 views

Poisson Kernel $L^p$-functions

Suppose $f \in L^p(\partial \mathbb{D})$ where $\mathbb{D}$ is the unit disk in $\mathbb{R}^2$. Let $\Pi(x, y)$ be the Poisson kernel (of the unit disk). Is it possible to bound the $L^p(\partial\...
2
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1answer
23 views

Sum of Integrals and $L^{p}$ Spaces

Let $(X,\mathfrak{M},\mu)$ be a finite measure space and let $\{f_{n}\}$ be a sequence of real-valued measurable functions on $X$. Assume that $F(x)=\sum_{n=1}^{\infty}f_{n}(x)$ exists for ...
0
votes
1answer
24 views

Show $\lim\limits_{h\to0} \sup_{y\in B_h(0)} \int_{A_h} |f(x+y)-f(x)|^pdx=0$

Show $\lim\limits_{h\to0} \sup_{y\in B_h(0)} \int_{A_h} |f(x+y)-f(x)|^pdx=0$ where $A \subset\mathbb R^n$ is open. $f\in L^p(A)$. And for $h>0$ we set $A_h=\{x\in A: d(x, \partial A)>h\}$...
2
votes
1answer
22 views

Given Limit is the Essential Supremum

Let $(X,\mathfrak{M},\mu)$ be a measure space with $\mu(X)=1$ and let $f$ be an $L^{\infty}(\mu)$ function on $X$. Prove that $$\lim_{t\to\infty}\frac{1}{t}\log\left(\int_{X}e^{tf}~d\mu\right)=\...
3
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2answers
42 views

Limit in $L^{1}(\mathbb{R})$

I have the following problem concerning a limit in $L^{1}(\mathbb{R}$), the class of all Lebesgue integrable functions on $\mathbb{R}$ with the Lebesgue measure $m$. Let $f\in L^{1}(\mathbb{R})$. ...
1
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1answer
17 views

Approximation of compactly supported continuous functions that conserve the integral value

Say $f\in L^p((0,1))$ is a function satisfying $$ \int_0^1 f = 0 $$ Is it possible to find a sequence $(f_k)$ of compactly supported continuous function also satisfying $$ \int_0^1 f_k = 0 $$ such ...
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2answers
63 views

$\|\cdot \|_p$ is not equivalent to $\|\cdot \|_q$ in $l^p$

To clarify things, let $\textbf{x}=(x_n) \in l^p$, then: If $p\geq 1$, $$\| \textbf{x} \|_p=\Bigg(\sum_{n=1}^{\infty} |x_n|^{p}\Bigg)^{1/p}. $$ If $p=\infty$, $$\| \textbf{x} \|_\infty=\sup_{n\geq1}...
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1answer
20 views

$L_p$ Norms and Holder's Inequality question

Suppose that $-∞ < a < b < ∞$ and $1 < p < q < ∞$. Let $$L_p[a,b] = \{ f :\Bbb R \to \Bbb R : \left( \int_a^b\left|f\left(x\right)\right|^p~\mathrm dx\right)^{\frac{1}{p}} < ∞ \}....
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1answer
33 views

Examples of sequences in lp spaces

Give an example of a sequence in $l^{1}$ that: 1) converges a zero in $l^{\infty}$ but is not bounded in $l^{2}$; 2) converges a zero in $l^{2}$ but is not bounded in $l^{1}$. Can anyone help me? ...
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2answers
26 views

On almost sure convergence of sequence of random variables $\{X_n\}$ such that $\forall p>0, E(|X_n-X|^p)\to 0$ as $n\to \infty$

Let $(\Omega,\mathcal F, P) $ be a probability measure space. If $\{X_n\}_{n=1}^\infty$ is a sequence of random variables on that probability measure space such that for a random variable $X$ on it, $\...
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0answers
5 views

$f\in L_{x}^{t}(L_{y}^{u})$ for $(t,u)$ s.t. $(\frac{1}{t},\frac{1}{u})$ lies within line

If $f(x,y)\in L_{x}^{p}(L_{y}^{q})\cap L_{x}^{r}(L_{y}^{s})$, then $f\in L_{x}^{t}(L_{y}^{u})$ for every $(t,u)$ such that $\left(\frac{1}{t},\frac{1}{u}\right)$ lies within the line whose ...
0
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1answer
43 views

Is $\int_0^1 \int_0^1 \frac{|f(x,y)|^2} {|f(x,y)|^2 + |f(x-(1/2),y )|^2} dx dy$ independent of $f$?

Let $f\in L^2([0,1)^2).$ Define $|F(x,y)|^2:= \frac{|f(x,y)|^2} {|f(x,y)|^2 + |f(x-(1/2),y )|^2}$ for all $x, y \in [0,1].$ Assume that $f$ is nice so everything make sense. My question is: How to ...
1
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2answers
29 views

Ambiguous definition of a sequence in $l^2$

Show that the sequence $(x_n)_{n\in\mathbb{N}}$ in $l^2$ given by $$ (x_n)_{n\in\mathbb{N}} := \left(1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},0,0,0,...\right) $$ converges to $$x=\left(1, \frac{...
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2answers
34 views

If $f \in L^1(\mathbb{R})$, does $\lim_{s \rightarrow \infty}\int |f(x-s) - f(x)| dm$ exist?

So if it were to exist, clearly it is nonnegative. Then for $s\in \mathbb{R}$, $$\int |f(x-s) -f(x)|dm(x) \leq \int|f(x-s)|dm(x) + \int|f(x)|dm(x)$$ because $f \in L^1(\mathbb{R})$. My intuition is ...
0
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1answer
12 views

If $E$ is finite-dimensional and $\mu$ finite show that $\mathcal S(X,\mu,E)$ is dense in $\mathcal L_\infty(X,\mu,E)$

If $E$ is finite-dimensional and $\mu$ finite show that $\mathcal S(X,\mu,E)$ is dense in $\mathcal L_\infty(X,\mu,E)$ This is an exercise of the book Analysis III of Amann and Escher. Here $\mathcal ...
0
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1answer
18 views

Show that $\phi \circ f$ belongs to $L_p$ for each $f \in L_p$

Let $(X, \mathbb{X}, \mu)$ be a finite measure space and let $1 \leq p< \infty$. Let $\phi$ be continuous on $\mathbb{R}$ to $\mathbb{R}$ and satisfy the condition: $(*)$ there esists $K>0$ such ...
1
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0answers
18 views

Strengthening $L^p$ Interpolation Bound

Suppose that $f$ is a function such that $|f|\ge 1$ on $\text{supp}(f)$ so that $$\int |f|^p$$ is increasing with respect to $p$. By standard $L^p$ interpolation we know that $$\Big(\int |f|^{(p+q)/2}\...
2
votes
3answers
40 views

Inequality of real numbers with exponent

For $a,b>0$ are two real numbers and $p\geq 1$. Is the following inequality true $$|a^p-b^p|\leq|a-b|^p\;\;?$$
2
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1answer
36 views

Lower Bound on Product of $L^p$-norms

Is there an established way to get a single $L^p$-norm that bounds from below $$\int |f|^p \int |f|^q\;?$$
1
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1answer
42 views

Is $C_c^\infty(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be open and $1\le p < \infty$. I'd like to know whether $C_c^\infty(\Omega)$, the set of the test functions on $\Omega$, is dense in $L^p(\Omega)$. I guess that it ...
1
vote
1answer
29 views

Riesz representation Theorem by Brezis

I don't understand the following proof: Theorem: Let $1<p<\infty$ and let $\phi\in(L^p)^*$. Then there exists a unique function $u\in L^q$ such that $\langle\phi\,,f\rangle=\int uf$ for all $f\...
4
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1answer
82 views

Can $n+1$ distinct positive vectors in $\mathbb{R}^n_{>0}$ agree on $n$ distinct weighted $p$-norms?

Consider $n$ distinct positive numbers $\{p^{(1)},...,p^{(n)}\}\subset [1,\infty)$ along with weights $\{w^{(1)},...,w^{(n)}\}\subset\Delta^{n-1}$ and scalars $\{q^{(1)},...,q^{(n)}\}\subset(0,\infty)$...
1
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0answers
28 views

Proper way to check if a measurable function belong to $L^p$?

Studying I observed that a line is measurable and it's measure is $0$ (you can check it here: Show that a straight line has a lebesgue measure of zero). Let $\mu(x)$ be a measure. I know that a ...
1
vote
0answers
21 views

$L_\infty[0;1] \subset L_1[0;1]$?

Is it true that $L_\infty[0;1] \subset L_1[0;1]$ ? How can I prove it? If $f \in L_\infty[0;1]$, then it's essentially bounded, then we can change $f$ on a zero-measure set and get a bounded on $[0;...
1
vote
2answers
92 views

What does it mean to say $f\in L^{\infty}(\mathbb{R}^N)$?

I'm trying to understand $L^p$ spaces. I know that a function is $L^1(\Omega)$ if $\int_{\Omega}|f|<\infty$ and $L^2(\Omega)$ if $\int_{\Omega}|f|^2>\infty$. However, what does it mean to say $...
2
votes
0answers
16 views

Lp bound for affine mappings

Is it possible to show the following or something similar by adjusting the assumptions a little bit? Let $B \subset \Omega \subset \mathbb{R}^3$, $B$ open and nonempty, $\Omega$ bounded $f_n:[0,T]\...
3
votes
2answers
31 views

Question concerning specific part of proof that the dual of $L^p$ is $L^q$ where $p$ and $q$ are Hölder conjugates and $p<2$.

I want to show that if $p<2$ then the dual of $L^p([0,1])$ is $L^q([0,1])$ where $\frac{1}{p}+\frac{1}{q} = 1$ without referring to the fact that $L^p([0,1])$ is uniformly convex. My question ...
5
votes
1answer
42 views

What is the intersection of all $L^p(\mathbb{R}^n)$ spaces?

I wondered this, and tried to find an answer online, but the only thing I could find was a statement that the set of functions which are in all $L^p(\mathbb{R}^n)$ is well-studied. But what functions ...
1
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0answers
30 views

$u,v \in W^{1,2}(\Bbb{R})$, then $\int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx$

So I need to show that, given $u,v \in W^{1,2}(\Bbb{R})$, then \begin{equation} \int_\Bbb{R}{uv'}dx=-\int_\Bbb{R}{u'v}dx \end{equation} My attempt has been this: If $u,v \in W^{1,2}(\Bbb{R}) \...
3
votes
1answer
35 views

A inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator?

Does someone know any inequality between $||u||_{p}$ and $||\gamma (u)||_{p, \partial \Omega}$, where $\gamma$ is the Trace Operator? I need to find something like $||u||_{p}\leq C||\gamma (u)||_{p, \...
0
votes
0answers
23 views

Constrained optimisation by solving an intermediate optimisation problem

Let $g:\mathbb{R}^l \mapsto \mathbb{R}^m$ and $f: \mathbb{R}^m \mapsto \mathbb{R}^n$ be two continuous functions and let $x_0 \in \mathbb{R}^l$. Let $x'$ be the closest point (according to a certain $...
1
vote
1answer
35 views

Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by

Let $f_n \in L_p(X, \mathbb{X}, \mu)$, $1 \leq p < \infty$, and let $\beta _n$ defined for $E \in \mathbb{X}$ by $\beta _n = (\int_E |f_n|^pd \mu)^{\frac{1}{p}}$ and suppose that $f_n$ is a ...