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 ...

learn more… | top users | synonyms

2
votes
1answer
50 views

Showing a function is in $L^1(\mathbb{R})$

Given that $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$, $1\le p,q\le\infty.$ Define $F(x)=\int_0^xf(t)dt$. How can one show $$(\vert x\vert+1)^{-a}F(x)g(x)\in L^1(\mathbb{R})$$ when ...
4
votes
2answers
63 views

$f\in L^1\cap L^2$ implies $\hat f \in L^1$?

Given $f\in L^1(\mathbb R^d)\cap L^2(\mathbb R^d)$. The Riemann-Lebesgue lemma and the unitarity of the Fourier transform on $L^2$ implies that $\hat f \in L^2\cap C_0$ where $C_0$ are continuous ...
2
votes
1answer
34 views

Limit of products in $L^p(\mathbb R^d)$

Fix $1 \leq p < \infty$. If $f_n \to f$ in $L^p(\mathbb R^d)$, $g_n \to g$ pointwise, and $\| g_n \|_{\infty} \leq M < \infty$ for all $n$, prove that $f_ng_n \to fg$ in $L^p(\mathbb{R}^d)$. ...
1
vote
1answer
45 views

Adjoint of Integral Operator in $L^p$

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Find the adjoint of $T$. I know how to this in the case $p=2$ as shown here. But in general $L^p$ is not an ...
0
votes
1answer
24 views

Prove convolution $f\ast g\in L^\infty(\mathbb{R})$

Let $f\in L^p(\mathbb{R}),g\in L^q(\mathbb{R})$ ($1\le p,q<\infty:\frac 1 p+\frac 1 q=1$). Prove that $L^\infty(\mathbb{R}) \ni f\ast g$ (the convolution of them) and also prove that $$\Vert ...
2
votes
0answers
48 views

Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
0
votes
1answer
10 views

Is $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$?

Define $f(u) := \int_\Omega u^2(x)h(x)$ weakly lower semicontinuous in $L^2(\Omega)$, where $h \in L^\infty(\Omega)$, but nothing is known about the sign of $h$? I do not believe it is weakly lower ...
0
votes
0answers
23 views

If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an integrable function?

Let $f_n,f$ be positive functions such that $f\in L^1(\Omega)$ and $f_n\in L^p(\Omega)\,\,\forall\,1\leq p<\infty.$ If $f_n\to f$ in $L^1$ can we derive that the functions $f_n$ are bounded by an ...
0
votes
1answer
40 views

$f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)\implies f_n\,g\in L^2?$

Let $f,g\in L^2(\Omega),\,$ $f_n\in L^p\,\,\forall 1\leq p<\infty$ such that $f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)$. I was trying to understand if we can derive that $f_n\,g\in L^2?$ My first ...
1
vote
0answers
33 views

Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...
0
votes
1answer
104 views

If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ Let $\phi$ be a smooth, bounded and nondecreasing ...
1
vote
1answer
43 views

If $u \in W^{1,p}(U)$, prove that $Du=0$ a.e. on the set $\{u=0\}$.

Assume $1 \le p \le \infty$ and $U$ is bounded. (a) Prove that if $u \in W^{1,p}(U)$, then $|u| \in W^{1,p}(U)$. (b) Prove $u \in W^{1,p}(U)$ implies $u^+,u^- \in W^{1,p}(U)$, and ...
4
votes
1answer
42 views

$f,g\in L^1(\mu)\implies fg\in L^1(\mu)$

Let $(X,\mu)$ be a measure space and suppose that $f,g\in L^1(\mu)$, i.e. $$\|f\|_1=\int_X|f|d\mu<\infty\quad\text{and}\quad\|g\|_1=\int_X|g|d\mu<\infty.$$ How to show that $fg\in L^1(\mu)$? ...
1
vote
1answer
42 views

is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$?

Consider the $L^p$ spaces. is $\|\cdot\|_p\leq \|\cdot\|_{p'}$ for $p<p'$? is it true if the domain of $L^p$ is finite measure? Thanks
2
votes
0answers
40 views

Verify that the unbounded function belongs to $W^{1,n}$ [duplicate]

Verify that if $n > 1$, the unbounded function $u = \log \log \left(1+\frac 1{|x|}\right)$ belongs to $W^{1,n}(U)$, for $U=B^0(0,1)$. This is PDE Evans, 2nd edition: Chapter 5, Exercise 14. ...
0
votes
0answers
4 views

Intersecting hyperballs in Lp space

I'm trying to manipulate hyperballs in Lp distance. What I would like to do is take the intersection/union of two hyperballs and then find the smallest hyperball which covers the intersection/union. ...
1
vote
1answer
77 views

Integrate by parts to prove this inequality

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 ...
1
vote
1answer
40 views

An exercise showing that $l^1$ is not the dual of $l^\infty$

there is a well known fact that $l^1$ is not the dual of $l^\infty$. An exercise Folland's Real analysis serves as an example for this.(Page 192 ex 19) Define $\phi_n \in (l^\infty)^*$ by ...
0
votes
0answers
49 views

When $1 \le p \le \infty, p\ne 2$, $L^p$ space is not a Hilbert space

It suffices to show that when $1 \le p \le \infty, p\ne 2$, $L^p$ norm does not arise from an inner product.(there is a hint saying that we can use the parallelogram law) I can proof a special case ...
0
votes
0answers
11 views

Approximating the weak gradient of the constant function in $L^p$

I want to find a sequence $u_n:(0,\infty) \to \mathbb{R}$ such that $u_n \to k$ pointwise and $\nabla u_n \to 0$ in $L^p$, where $k$ is the constant function equal to $k \in \mathbb{R}.$ Will the ...
1
vote
1answer
44 views

Deriving $|u(x)-u(y)|\le|x-y|^{1-\frac 1p}\left(\int_0^1 |u'|^p \, dt \right)^{1/p}$

Assume $n=1$ and $u \in W^{1,p}(0,1)$ for some $1 \le p < \infty$. (a) Show that $u$ is equal a.e. to an absolutely continuous function and $u'$ (which exists a.e.) belongs to $L^p(0,1)$. ...
3
votes
1answer
64 views

Sequence is not in any $\ell^p$ space

We know the sequence {$\frac{1}{ln(n)}$} such that $(n>=2)$ converges to $zero$ but is not in any $L_p$ space because of $$\sum_{n=2}^{\infty}\left|{\frac{1}{ln(n)}}\right|^p ={\infty}$$ for any ...
3
votes
1answer
56 views

Convergence in dual of Sobolev space

Hi please view the following question: Consider Sobolev space $W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^{n}$ is bounded. We also have a mapping $a: \Omega \times \mathbb{R} \times ...
2
votes
1answer
32 views

If $f \in L^1 \cap L^2$ is $L^2$-differentiable, then $Df \in L^1 \cap L^2$

Working with the definition that $f \in L^2(\mathbb{R})$ is $L^2$-differentiable with $L^2$-derivative $Df$ if $$ \frac{\|\tau_hf-f-hDf\|_2}{h} \to 0 \text{ as } h \to 0 $$ (where $\tau_h(x) = ...
1
vote
0answers
34 views

Almost everywhere convergence of convolution with mollifiers

I read that for $j\in L^1({\bf R}^n)$ with $\|j\|_1=1$ and $f\in L^1_{\rm loc}({\bf R}^n)$ the mollifiers $j_\epsilon(x):=\epsilon^{-n}j(x/\epsilon)$ exhibit $j_\epsilon\ast f\in L^1({\bf R}^n)$ and ...
8
votes
1answer
133 views

Problems with the proof that $\ell^p$ is complete

By struggling with the proof that $\ell^p$ is complete, I looked up different proofs by different authors, and I ended up focusing on the one given by Kreyszig in his classic book on functional ...
0
votes
1answer
47 views

L-p space: p-norm proof

Can somebody put me in the right direction to prove that: $\lim_{p \to 1} \lVert f \rVert_{p}^p=\lVert f \rVert_{1}$ ? Maybe this will be a beginning: If $f \in$ $\mathcal{L}^1(\mu)\cap ...
2
votes
0answers
37 views

Completeness of $ L^{p} $ spaces and “rapidly Cauchy” sequences

http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf In the book of Royden, the completeness of $ L^{p} $ spaces has been done using what he calls "rapidly ...
0
votes
1answer
44 views

Function in $\mathcal{L}^p(\mu)$ for all $1\leq p \leq 2$

Let (X, $\mathscr{A}$, $\mu$) be a measure space and f $\in$ $\mathcal{L}^1(\mu)\cap \mathcal{L}^2(\mu)$. Can I ask how to show that $f \in \mathcal{L}^p(\mu)$ for all $1\leq p\leq 2$?
3
votes
1answer
29 views

If $u \in L^1(0,\infty)$, then $|u(x)| \to 0$ as $x \to \infty$?

Let $u \in L^1(0,\infty)$. Does this mean necessarily that $|u(x)| \to 0$ as $x \to \infty$? I think it has to decay otherwise the integral will be infinite. Can I get a hint on how to prove this? ...
0
votes
0answers
13 views

Relation between $L^{p}$ norm of derivative $\|Dg\|_{L^{p}}$ and $\|g\|_{L^{1}}$

Let $\phi \in \mathcal{S}(\mathbb R)$(Schwartz space) with $\phi =1$ on $[-1, 1].$ Put $g:=\phi^{\vee}$(inverse Fourier transform of $\phi$). My Question: Can we show ...
2
votes
1answer
62 views

Characterizing when composition by a power function lies in an $L^p$-space

This is a past qual question that I have been struggling with: let $p,q,r \geq 1$. One would like to characterize the constants $q$ such that $f(x^r) \in L^q ((0,1))$ for all $f \in L^p((0,1))$, that ...
2
votes
1answer
54 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
3
votes
1answer
72 views

A function $f$ such that $f \in L_1$ but $f \notin L_p$ for $p>1$ [duplicate]

I want find a function $f: [0,1] \mapsto \mathbb{R}$ such that $f \in L_1[0,1]$ but $f \notin L_p[0,1]$ for all $p>1$. My attempts: First I thought in the family of functions $\frac{1}{x^\alpha}$ ...
1
vote
1answer
34 views

Is $\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}}$ a Cauchy Sequence in $L^p((0,1))$

Is $(\left(\frac{1}{x}\right)^{\frac{1}{p} - \frac{1}{n}})_{n\in N}$ a Cauchy Sequence in $L^p((0,1))$? and does it converge to $\frac{1}{x}^{\frac{1}{p}}$ (p is a real number bigger or equal to 1) I ...
1
vote
0answers
50 views

Dual of $L^p$ when $p = 0$?

I've spent some time searching for this online - both on this site and elsewhere - and even after consulting a considerable amount of literature, I can't seem to nail down an answer. Perhaps someone ...
1
vote
2answers
26 views

A basic question on the space of square integrable functions

I have seen in a book the following cliam: Let $f_m,f \in L^2[0,N]$ and $\frac{1}{m}\sum_{k=1}^{m}f_{n(k)} \to f$ in $L^2[0,N]$ for a subsequence $n(k)$ Then for any $g \in L^2[0,N] s.t. \|g\|=1$ ...
1
vote
1answer
30 views

Find $c_0$ for which a sequence is in $l_2$?

Let $y \in l_2$ and let $$x_k=\left[C_0+\sum\limits_{j=0}^{k-1}\frac{y_j}{(\lambda+1)^{j+1}}\right](\lambda+1)^k.$$ Does there exits unique constant $C_0$, such that $x \in l_2?$ I need to show the ...
1
vote
1answer
41 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
3
votes
0answers
32 views

Dual of $l^p$ Direct sum

I am asked to show that the $l^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $l^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ ...
5
votes
1answer
39 views

Integral with a compact supported function $0$ indicates the $L^2$ function $0$ almost everywhere

Suppose we have $f\in L^2([0,1])$,and for every $\varphi\in C_{0}^{\infty}((0,1))$, we have $$\int_0^1 f(x)\varphi(x)dx=0$$ Then how can I show $f=0$ a.e? I know when $f\in C^0([0,1])$ the results ...
5
votes
0answers
56 views

Alternate proof of a result on dual spaces: what is wrong with it?

I am familiar with Rudin's book's proof of the fact that, in $\sigma$-finite measure spaces and for $p\in[1,+\infty)$, the dual space of $L^p$ is $L^q$ where $p,q$ are conjugate, i.e. ...
0
votes
1answer
40 views

Show that $g=\sum_{n=1}^{\infty } |f _{n+1 }-f _n | $ has $||g ||_p\le 1 $ if $||f _{n+1 }-f _n ||_p <2 ^{-n } $

Minkowskis inequality implies that $g _k=\sum_{n=1}^{k} |f _{n+1 }-f _n | $ has norm less than $1 $, and there is a hint to use Fatou's lemma to $g _k ^p$. Then $\int \lim \inf g _k ^p \le \lim \inf ...
2
votes
1answer
33 views

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,.,1,0,.)$ and a bounded linear functional $\Phi$ find $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges?

For $(l^2,\|\cdot\|_2)$ and $e_n=(0,0,...,1,0,...)$ and a bounded linear functional $\Phi$ find a value of $p\geq 1$ where $\sum_{n=1}^\infty |b_n|^p$ converges for $b_n=\Phi(e_n)$? Ok so since ...
0
votes
0answers
26 views

The dual of the space $L^\infty$. [duplicate]

As we know the dual of $L^p$s are $L^q$s where $\frac{1}{p} + \frac{1}{q} =1$, and dual of $L^1$ is $L^\infty$. What is dual of the space $L^\infty (E)$ where E is a measurable subset of $\mathbb{R} ^ ...
1
vote
1answer
40 views

$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
2
votes
0answers
46 views

Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
2
votes
1answer
71 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
1
vote
0answers
30 views

Prove that a function lies in $L^1$ and in $W^{(1,1)}$ for some parameter

I want to do the following tasks Let $G:=B_1(0)\subset \mathbb R^2$ be the open ball around $0$ with radius 2 in the norm $||\cdot||$ and $u_{\rho}(x)=||x||^{\rho}_2$, $x\in G$. Show the following ...
2
votes
1answer
48 views

If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$

Problem: If $f_n \to f$ and $g_n \to g$ in $L^p$, and $g_n$ is uniformly bounded, then $f_ng_n \to fg$ in $L^p$. An official solution I saw for this problem looked very different. Here is my ...