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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0
votes
2answers
22 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
2
votes
1answer
45 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 $\| \Lambda \|_2$ ...
3
votes
1answer
28 views

Dense subset in sequence space

I'm trying to prove that $F=\{x=\{x_n\}_{n\in \mathbb{N}}\in l^2(\mathbb{N}):\sum_{n=1}^{\infty} x_n=0\}$ is dense in the sequence space $l^2(\mathbb{N})$. I think it should be an easy exercise, but ...
0
votes
2answers
43 views

Compute $\lim_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$

Suppose $f\in L^p(\mathbb{R}^n), 0<p<\infty$, and compute $\displaystyle \text{lim}_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$ I have no idea from where to start since ...
3
votes
1answer
39 views

Weak convergence - $f_n$ “goes up the spout”

Fix $1 < p < \infty$. Given $f \in L^p(\mathbb{R})$ define $f_n(x) = n^{1/p}f(nx)$ for $n = 1, 2, \dots$. Prove that $f_n$ converges weakly to $0$ in $L^p$. I'm really confised about this ...
0
votes
0answers
14 views

Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
1
vote
1answer
61 views

Show that $f=0$ a.e. if $|\int_I f|^p \leq c|I|^{p-1}\int_I |f|^p$ with $0<c<1$

Suppose an extented real valued function $f$ defined on $\mathbb{R}^n$ satisfies the following two properties: a) There is a $p$, $1\leq p < \infty$ such that $f\in L^p(I)$, for every ...
1
vote
3answers
54 views

Is it possible to have simultaneously $\int_I(f(x)-\text{sin} x)^2 dx\leq \frac{4}{9}$ and $\int_I(f(x)-\text{cos} x)^2 dx\leq \frac{1}{9}$?

Let $I=[0,\pi]$ and $f\in L^2(I)$. Is it possible to have simultaneously $\int_I(f(x)-\sin x)^2 dx\leq \frac{4}{9}$ and $\int_I(f(x)-\cos x)^2 dx\leq \frac{1}{9}$? I don't understand what this ...
0
votes
2answers
76 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|)^{\frac{1}{2}}$. I want to show that $A=\{ ...
0
votes
0answers
41 views

$\|f_n-f\|_p \rightarrow 0 \Rightarrow \|f_n\|_p\rightarrow \|f\|_p$?

Let $(X,M,\mu)$ be a measure space. Suppose $\|f_n-f\|_p \rightarrow 0$ where $1<p<\infty$. Does it imply that $\|f_n\|_p\rightarrow \|f\|_p$. Here $\|f||_p=\left(\int_X |f|^p \, ...
5
votes
1answer
45 views

Show that $ \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$.

Let $I=[0,\pi]$. Show that $\displaystyle \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$. My Work: I think this is an application of Holders inequality. But any positive power of $ ...
2
votes
1answer
35 views

$f\in L^p(X,\mu)$ , $f-1\in L^q(X,\mu)$ then $\mu(X) < \infty $

Can some one give a hint how to start to solve : Assume $ 1 \le p,q < \infty $ and $$f\in L^p(X,\mu)$$ now if we assume $$f-1\in L^q(X,\mu)$$ then we have $$\mu (X) < \infty $$ Thanks If ...
1
vote
0answers
20 views

A question about weak convergence in Lp space [duplicate]

Suppose $1 \leq p<\infty$, given $f \in L^p (\mathbb{R})$, define $f_n (x)=n^{1/p} f(nx)$ for n=1,2,3... Prove $f_n$ converges weakly to zero in $L^p$. Now I can just know the that $ \|f_n\|_p$=$ ...
2
votes
1answer
35 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
54 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
41 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
19 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
46 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
9 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
39 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
58 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
34 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
39 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
36 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
34 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
74 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
47 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
8 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
36 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
53 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
49 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
31 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
26 views

Almost everywhere convergence of convolution with mollifiers

I read that for $j\in L^1({\bf R}^n)$ with $\|f\|_1=1$ 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
130 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
38 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
35 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
28 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
9 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
55 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
52 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
62 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
32 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
46 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 $g \in L^2[0,N] s.t. \|g\|=1$ ...