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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3
votes
1answer
24 views

Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with bounded support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
0
votes
1answer
48 views

Weak convergence of scaled elements implies norm convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
2
votes
1answer
24 views

Find the values of $p$ for which the given sequence converges in $l^p$ norm or weakly

We consider $E_p=(c_{00}, ||\cdot||_{p})$ (where $c_{00}$ are the sequences that are zero except in a finite number of values and $1\leq p \leq \infty$) and the sequence:$$(x_n)_{n\in ...
2
votes
1answer
27 views

Show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$.

This should be a relatively easy question, but I can't seem to figure it out. I want to show that $S = \{f \in L^1(\mathbb{R}) \mid \int_{\mathbb{R}}f dm = 0\}$ is closed in $L^1(\mathbb{R})$. As ...
0
votes
1answer
31 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
0
votes
2answers
42 views

Integrable function with given condition is in $L^p$

Suppose $f:\Bbb R \to \Bbb R$ is integrable and there exist constant $c\gt 0$ and $\alpha \in (0,1)$ such $$\int_A |f(x)|dx\le cm(A)^\alpha$$ for every Borel measurable set $A\subset \Bbb R,$ where ...
3
votes
1answer
37 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
1
vote
2answers
46 views

Showing a sequence is in $\ell^2$ [duplicate]

I am working on the following problem. Suppose that $\{a_j\}_{j=1}^{\infty}$ is a sequence with the property that, whenever $\{b_j\}_{j=1}^{\infty} \in \ell^2$, one has ...
2
votes
2answers
54 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
0answers
45 views

When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten -- von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, ...
4
votes
1answer
80 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
2
votes
1answer
44 views

Maximum value of a mapping on a compact subset of $\ell^{2}$

Let $A = \{x \in \ell^{2}: \sum_{n = 1}^{\infty}n|x_{n}|^{2} \leq 1\}$. What is the largest value $\frac{1}{2\pi}\int_{0}^{2\pi}\left|\sum_{n = 1}^{\infty}x_{n}e^{in\theta}\right|\, d\theta$ can take ...
2
votes
1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
1
vote
1answer
32 views

Prove this inclusion: $\bigcup_{k<p}\ell^k\subsetneq\ell^p$

Let $1<p<\infty$. I have to prove that $$ \bigcup_{k<p}\ell^k\subsetneq\ell^p. $$ I am not able to find a counterexample to prove the inequality.
3
votes
1answer
29 views

The existence of conditional expectation with respect to a sub-$\sigma$-algebra

I was trying to solve the exercise 3.17 from the book of real analysis by Folland and I've found a problem. The first part of the exercise is the following: Let $(X, M, \mu) $ be a $\sigma$-finite ...
3
votes
0answers
31 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
1
vote
0answers
28 views

Is Sobolev space $H^{s}(\mathbb R),$ for $s>\frac{1}{2},$ closed under point wise multiplication? [duplicate]

We note that, $L^{2}(\mathbb R)$ is not closed under point wise multiplication. Let $s>\frac{1}{2};$ and we define Sobolev space, as follows: $H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb ...
2
votes
2answers
67 views

Properties of a set in $\ell^2$ space

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$ and $e_n \in \ell^2 $ be the sequence whose $n$-th element is 1 and all other elements ...
2
votes
1answer
51 views

Domain of multiplication operator

Edit: This question arose due to a misunderstanding, which has now been resolved. Let $\psi \in L^{2}(\mathbb{R})$ be a continuous function. Let $M_{\psi}$ be the multiplication operator on ...
1
vote
0answers
22 views

Why does duality imply, that it is enough to consider p>2.

Let $f\in L^p(\Omega, \nu).$ Let $L$ be a self-adjoint operator on $L^p.$ Suppose we want, for every $p>1,$ to prove an inequality $$||Lf||_p\leq C(p)||f||_p,$$ where $C(p)$ is some function ...
3
votes
1answer
58 views

Weak convergence $f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$ implies $f^2\leq g$ a.e.

$f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$, then $f^2\leq g$ a.e. Could you guys help me check the proof please, thanks! Proof: to show $f^2 ...
1
vote
0answers
38 views

Use the uncertainty principle to prove that $\int_{\mathbb R^n} (f(x))^2|\hat{f}(x)|^2 \,dx=0$ if and only if $f\equiv 0$.

Let $f \in L^2(\mathbb R^n)$. Use the uncertainty principle to prove that $\int_{R^n} (f(x))^2|\hat{f}(x)|^2 \,dx=0$ if and only if $f\equiv 0$. $\leftarrow$: If $f \equiv 0$ then $\int_{\mathbb R^n} ...
0
votes
1answer
33 views

Canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact?

Does there exist $q>p$ such that the canonical inclusion $L^q(0,1) \to L^p(0,1)$ is compact? My answer is no. Since we know that $L^\infty (0,1) \to L^p(0,1)$ is not compact, take $\{\sin(nx)\}$ ...
2
votes
1answer
36 views

Questions about $L^p$ spaces and convergences

I would like to sort out the relations for strong/weak convergences for $L^p(X)$ mainly between $[p=1; p>1]$ and $[\mu(X) <\infty ; \mu(X) = \infty]$ For the purpose of strong/weak ...
0
votes
1answer
36 views

$\ell^p$ spaces' inclusion

$$ \ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q $$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
0
votes
0answers
41 views

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? [duplicate]

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise ...
0
votes
2answers
68 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
2
votes
1answer
57 views

Is a linear operator on $\ell^2$ defined by the inner product necessarily bounded? [duplicate]

If $a=\{a_n\}\in \ell^\infty(\mathbb{R})$ and $\langle a,x \rangle<\infty$ for all $x\in \ell^2(\mathbb{R})$, (where $\langle a, x\rangle=\displaystyle \sum_{k=1}^\infty a_kx_k$), then is $a\in ...
3
votes
2answers
44 views

Convergence in $L^p$ by using Holder's inequality

Let $1\lt p \lt \infty$ and $f\in L_p[0,\infty )$. Show that a) $$\left\vert\int_0^x f(t)\,dt\right\vert\le\|f\|_px^{1-\frac{1}{p}},$$ for $x\gt 0$. b) $$\lim_{x\to \infty} ...
1
vote
0answers
37 views

Proof of separability of Lp spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof. Questions: It says 'it is easy to construct a function $f_{2} ...
1
vote
1answer
25 views

For what values of $a > 0$ does $f(x,y)=(x^{2}+y^{a})^{-1} $ belong to $ L^{1}([0,1]^{2})?$

I am trying to understand for what values of $a>0$ the function $$f(x,y) = \frac{1}{x^2+y^a}$$ belongs to $L^1([0,1]^2)$. I think $a \geq 2$ should work. But how to show that it is not the case ...
1
vote
1answer
40 views

Shortest p-distance

We are given two points, $(0,0)$ and $(1,1)$. How many shortest paths exist between these points? For $p=2$ (Euclidean) distance, the answer is $1$, as the shortest path is a straight line. However, ...
1
vote
1answer
20 views

Showing that a function is bounded in $L^1$ given a bound on its distribution function

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold. Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is ...
2
votes
2answers
50 views

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and some condition.

$f\in L^2(0,1)$ if and only if $f\in L^1(0,1)$ and ere exists an increasing function $g:[0,1]\rightarrow \mathbb{R}$ such that $$\left|\int_a^b f(x) dx \right|^2 \leq (g(b)-g(a))(b-a)\quad\quad (*)$$ ...
2
votes
1answer
43 views

Need help with application of Hardy-Littlewood inequality (Marcinkiewicz space and distribution functions)

I am going over this work here. I couldn't understand the equality where the Hardy-Littlewood inequality is used. I think $\delta$ here is a weight so we can take it to be $1$ for simplicity. Would ...
1
vote
0answers
20 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f_{r}(t-y)- f_{r}(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $ r\to \infty $?

Let $f\in \mathcal{S}(\mathbb R)$ with $\hat{f}$ has a compact support. For $r>0,$ put $f_{r}(x)= r^{-1}f(x/r), (x\in \mathbb R).$ We note that, $\int_{\mathbb R} |f_{r}(x)| dx = r^{-1} ...
5
votes
1answer
58 views

$xf''(x) , xf', f \in L^{2}$ is $f' \in L^{1}$?

I am stuck on the following problem. I have a function $f$ such that $f$ is bounded on $(0,1)$, $xf'(x)$ is bounded on $(0,1)$, $f \in L^{2}(0,1)$, $xf' \in L^{2}(0,1)$, and $xf'' \in L^{2}(0,1)$. ...
0
votes
0answers
18 views

Duality set for $L~p$ spaces, $1<p<\infty$.

I need to show that, given $f \in L^p$, $1<p<\infty$, the duality set $F(f)$ is equal to the point $$\|f\|_p^{2-p}|f|^{p-2}\overline{f}.$$ I have a hint: this is a consequence of convexity of ...
1
vote
1answer
26 views

Don't understand a $L^\infty$ bound argument involving measure of set

I'm trying to understand the proof of Proposition 2.2, part 2 of this paper. this is where I am stuck. For any $k > 0$, we have $$k^{\frac{2(N+1)}{N}}|\{|u|^m > k\}| \leq ...
5
votes
0answers
43 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
1
vote
3answers
46 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
1
vote
1answer
53 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
3
votes
2answers
36 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
1
vote
1answer
18 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
1
vote
1answer
42 views

Banach valued sequence spaces $\ell^p(X)$

Let $X$ be a Banach space and $\ell^p(X)$ denote the space of sequences $x_i\in X$ for which the norm $\big(\sum_{i=1}^\infty\|x_i\|^p\big)^\frac1p$ is finite, when $X=\mathbb{R}$ we get the usual ...
3
votes
1answer
67 views

$f$ is in $L^p$ iff sum is finite

Let $p\in [1,\infty)$.Prove that $f\in L^p(\mu)$ if and only if $\sum_{n=1}^\infty(2^n)^p\mu (\{x:|f(x)|\gt2^n\})\lt \infty.$ My idea, I assume measure is finite, I wrote ...
3
votes
1answer
18 views

$L^p$ integral on every measurable subset of $\Bbb R$

Suppose $f:\Bbb R \to \Bbb R$ is in $L^p$ for some $p>1$ and also in $L^1$. Prove there exist constants $c>0$ an $\alpha \in (0,1)$ such that $\int_A|f(x)|dx\le cm(A)^{\alpha}$, for every ...
1
vote
1answer
23 views

On finite measurable space $X$, the whole of $L^p(X)$ is closed in $L^1(X)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f \in L^p(X)$

On finite measurable space $(X, \mathcal{M}, \mu)$, the whole of $L^p(X, \mu)(p>1)$ is closed in $L^1(X,\mu)$ iff there is a const $C$ st $||f||_p\leq C||f||_1, \forall f\in L^p(X)$, iff both ...
3
votes
0answers
50 views

$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with ...
1
vote
1answer
29 views

Given $u \in L^1$, is there approximating sequence $u_n \in L^\infty$ uniformly bounded in $L^p$?

Let $u \in L^1(U)$ where $U$ is a bounded domain. Is it possible to find a sequence $u_n \in L^\infty $ converging to $u$ in $L^1$ such that the $u_n$ are uniformly bounded for all $n$ in some $L^p$ ...