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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2
votes
0answers
33 views

$||f_n-f||_1 \to 0 $ iff $ \int| f_n|\ to \int|f| $ if $f_n \to f $ a.e. [duplicate]

Assume $f_n , f \in L^1 $ and almost every where we have $ f_n \to f$ then I want to show that $\int|f_n-f| \to 0$ iff $\int|f_n| \to \int|f|$ One side is abvious by trinagle inequality , for the ...
-1
votes
1answer
30 views

Showing a convergence in $ L^p$

If $ f_n \in L^p(X,m) $ and $f_n \to 0 $ a.e with respect to $m$ and $||f_n||_p \to ||f||_p $ as $n \to \infty$ then can we say that for all $r \in [1,\infty] $ we have $$ ||f_n-f||_r \to 0 $$ Can ...
3
votes
2answers
69 views

Are $L^\infty$ bounded functions compact in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a compact subset of $L^2$? (Compact in the topology induced by the ...
1
vote
1answer
24 views

Lower semicontinuity of ${\dot{H}}^1$ norm

I have a in $H^1(\mathbb{R^N})$ uniformly bounded sequence $u_n \in H^1$. I also know $u_n\to u$ in $L^p$ for every $2\leq p < 2^\ast$, where $\ast$ means the Sobolev exponent. Can I conclude that ...
1
vote
1answer
25 views

Proving a certain subset is closed in $L^1$

In an exam, I was asked to prove that, if $A=\{f\in L^1([0,1]):\int_0^1|f(x)|^2\mathrm{d}x\leq1\}$, then $A$ is closed in $L^1$. I tried this approach. $A$ is closed iff for all $f_n\to f$ in $L^1$ ...
2
votes
1answer
28 views

Dual space of weighted $L^p(\omega)$

Let $\omega \in A_p$, where $A_p$ is the family of Muckenhoupt weights. I'm wondering what is the topological dual space of $L^p(\omega)$. Is it isometrically isomorphic to $L^q(\omega)$? (1/p + 1/q = ...
1
vote
1answer
30 views

Has a $L^1$ bounded sequence a weak converging subsequence in $L^2$?

Let $f_n \in L^2(0,1)$ with the property that $\sup_n || f_n ||_{L^1}<A< \infty$, i.e. $f_n$ is a sequence in $L^2$ that is uniformly bounded in the $L^1$-Norm. Does $f_n$ then have a weak ...
3
votes
1answer
102 views

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)
1
vote
1answer
29 views

Does $\int 1_{|m|>A} m^2$ converge to zero for an $L^2$ function?

We assume that $m \in L^2(0,1)$, hence $\int_0^1 m(x)^2 dx< \infty$ but that $m \not \in L^\infty(0,1)$. Hence $\{ x: |m(x)|>A \}$ has always positive measure. Now the question: Does $\int ...
1
vote
0answers
29 views

$f\in L^1(\mathbb{R}^N)$ and Lipschitz continuous, then $f\in L^\infty(\mathbb{R}^N)$.

$f\in L^1(\mathbb{R}^N)$ and Lipschitz continuous, then $f\in L^\infty(\mathbb{R}^N)$. Denote the Lipschitz constant of $f$ as $C$, suppose $f$ is not bounded a.e., then $\mu(\{f> k+C)\}) ...
2
votes
1answer
38 views

Approximating the gradient of a function by $L^2$ functions (cut-offs)

Let $u:(0,\infty) \to \mathbb{R}$ be such that $u' \in L^2(0,\infty)$, but $u \notin L^2(0,\infty)$. However $u \in L^2(0,T)$ for all finite $T$. Is it possible to find a sequence of functions $u_n ...
0
votes
2answers
48 views

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

In this problem If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$ I don't understand this part: $$ \int_U \partial_i u^\epsilon\,v\,dx\to 0\tag w$$ for all $v\in L^2(U)$, but now we only ...
3
votes
0answers
27 views

lp and c0 little question

i want to explain that the inclusion $\bigcup_{p < \infty} l^p \subset c_0$ is true. that comes very quickly from the definition of $l^p$. The problem is, that Im sure that this inclusion is ...
1
vote
1answer
32 views

Analogue of Borel-Cantelli Lemma in $L^1(\mathbb{R})$

Let $\mu$ denote Lebesgue measure on the real line. In measure theory, the Borel-Cantelli Lemma states that if $\lbrace E_k \rbrace_{k\ge 1}$ is a collection of measurable sets such that $\sum_{k\ge ...
2
votes
1answer
64 views

If $f\in L^1(\mathbb{R})$, then $\sum_{n\ge 1}f(x+n)$ Converges for a.e. $x$.

I am given that $f\in L^1(\mathbb{R})$ (i.e., $\int_{-\infty}^\infty\vert f \vert<\infty$). I would like to show that $$\sum_{n\ge0}f(x+n)\tag{$*$}$$ converges for almost every (a.e.) $x$, and I am ...
2
votes
0answers
28 views

Discuss the duality relation between $L^p(X)$ and $L^q(X), 1/p + 1/q = 1, $ for $1 ≤ p < ∞.$

Let $(X,M, μ)$ be the measure space where $X = \{a, b\}, M= P(X),$ and $μ(\{a\}) = 1, μ(\{b\}) = ∞.$ Discuss the duality relation between $L^p(X)$ and $L^q(X), 1/p + 1/q = 1, $ for $1 ≤ p < ∞.$ ...
2
votes
1answer
37 views

Showing some Function $fg$ is Integrable when $g(x)\le e^{-\vert x\vert }$ and given some Conditions on $f$.

Suppose $\int_a^b \vert f \vert <\infty$ for all real $a,b$ and that $$\int_{-r}^r \vert f \vert \le (r+1)^a$$ for all real $r$ some real $a$, and that $$g(x)\le e^{-\vert x \vert}$$ I want to show ...
0
votes
2answers
29 views

$μ$ is $σ$-finite iff $L^p(X)$ contains a strictly positive function.

Let $(X,M, μ)$ be a measure space and $0 < p < ∞$. Prove that, $μ$ is $σ$-finite iff $L^p(X)$ contains a strictly positive function. My Work: If I suppose $L^p(X)$ contains a strictly ...
2
votes
3answers
73 views

Adjoint operator of $L^\infty$

Lets denote with $(\Omega,\Sigma,\mu)$ a $\sigma$-finite measurble space with a linear, continuous operator $$T : L^\infty \to L^\infty.$$ Does this always imply the existence of a linear, continuous ...
2
votes
1answer
33 views

Is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$?

If $\Omega$ is a bounded $C^1$ domain, is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$? Are they the same? I know this is true when instead of $(0,\infty)$ we have a bounded interval.
3
votes
1answer
63 views

If $(a_n)$ is such that $\sum_{n=1}^\infty a_nb_n$ converges for every $b\in\ell_2$, then $a\in\ell_2$

Please help me with this question. I've been thinking about it for almost two days. Let $a_n$ a real series that have the following property: for every series $b_n$ in $l_2$: $\sum_{n=1}^\infty ...
0
votes
2answers
28 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
53 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
44 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
3answers
59 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
48 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 ...
1
vote
0answers
23 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
73 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
83 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
44 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
51 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
42 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
25 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$=$ ...
1
vote
0answers
30 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...
2
votes
1answer
51 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
69 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
47 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
29 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
49 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
11 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
42 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
34 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
123 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
74 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
44 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
43 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
43 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. ...