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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8 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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0answers
12 views

Hilbert Space, Fourier Transform, convergence of partial sum problem

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
2
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0answers
26 views

tough lp inequalities

Let $1<p<\infty$. If possible, find a positive decreasing sequence $w_1>w_2>\cdots$ such that $\lim w_i=0$, and a (uniform) constant $K>0$, such that ...
1
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0answers
15 views

Wiener Algebra, absolute convergence of fourier series

So how do you prove if $f, g\in L^2(\mathbb{T})$, then $f*g\in \mathbb{A}(\mathbb{T})$. $\mathbb{T}$ denote $[0,1)$ and $\mathbb{A}(\mathbb{T})$ denote the Wiener algebra such that if $f\in ...
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1answer
45 views

The proof of a Sobolev embedding inequality by a compactness argument

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
2
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1answer
34 views

Properties of $L^p$ spaces [duplicate]

Let $\Omega$ $\subseteq$ $\Bbb R^{n}$ be bounded and $u$ be a measurable function on $\Omega$ Such that $|u|^{p} \in {L^{1}_{\mathrm{loc}}(\Omega)}$ for some $p \in \Bbb R$. Then how to show the ...
2
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1answer
28 views

Supremum of integrals of products of functions in $L^p$ space

Here is the problem I'm dealing with I'm not having success with...well, anything. Any hits on how I could get started and where I would go? edit: information on $L^p$ space
2
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2answers
71 views

If $\int u\varphi = 0$ for all $\varphi \in C_c^\infty(M)$ with $\int_M \varphi =0$, is $u=0$ a.e.?

Let $M$ be a compact Riemannian manifold and $u \in L^2(M)$. we know that if for all $\varphi \in C_c^\infty(M)$, $$\int_M \varphi u = 0,$$ then $u=0$ a.e. Suppose $$\int_M \varphi u =0$$ for all ...
2
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2answers
34 views

Is $f(x) \in L^p(\mathbb R)$ always bounded for $x\longrightarrow\pm\infty$?

I need to prove the following result on the derivative of an Hilbert transform for $f,f'\in L^p(\mathbb R)$ $$\mathcal H\bigg\{\frac{df(x)}{dx}\bigg\}=\frac{d}{dx}\mathcal Hf(x) $$ In particular ...
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1answer
13 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
4
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1answer
23 views

BMO functions are $L^p$ Loc for all $1<p<\infty$

In order to motivate my question, I'd like to remember that if $\Omega$ is a bounded domain and $f \in L^q(\Omega)$ for some $q>1$, by Hölder inequality $f \in L^p(\Omega)$ for $p \in (1,q]$ with ...
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votes
3answers
55 views

$f, f'\in L^{1}(\mathbb R) \implies \lim_{x\to \infty} f(x)=0 ?$

Suppose $f\in L^{1}(\mathbb R) $ and $f'\in L^{1}(\mathbb R).$ My Question is: Can we show, $\lim_{x\to \infty} f(x)=0$ ? Thanks,
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1answer
38 views

Convergence in $l^p$ space?

We know that $\lim_{k \to \infty} a_{n}^{k} = b_n$. All $(a_{n}^{k}) \in l^p$ space. How to show that $(\lim_{k \to \infty} a_{1}^{k},\lim_{k \to \infty} a_{2}^{k},...,\lim_{k \to \infty} ...
4
votes
1answer
40 views

Do $\mathbb{R}^n$ and $\mathbb{C}^n$ valued ordinarily measureable functions form a Banach space under p-norm?

By measureable function I mean an "ordinarily" measureable function, that is measureable in a sense of this definition: a function between measurable spaces is said to be measurable if the preimage of ...
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1answer
21 views

If a function $F$ belongs to mixed Lebesgue space $L^{p,1}$, does its reflection $G(x,y):=F(y,x)$ also lie in the space?

Let $F:\mathbb R^{2}\to \mathbb C$ be a function. Suppose $F\in L^{p,1}(\mathbb R \times \mathbb R); (1<p< \infty).$ Define $G:\mathbb R^{2}\to \mathbb C$ as follows: $$G(x,y):=F(y,x)$$ My ...
5
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1answer
53 views

When is an analytic function in $L^2(\Bbb R)$?

Suppose $f:\Bbb R\to\Bbb C$ is real analytic. In order for $f$ to be in $L^2(\Bbb R)$, clearly all terms in the power series cannot be positive since $f$ would diverge at $\pm\infty$. Likewise, the ...
2
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1answer
31 views

composition of $L^{p}$ functions

Suppose $f, g\in L^{p}(\mathbb R), (1\leq p < \infty).$ For simplicity, let us assume that, $g,f:\mathbb R\to \mathbb R$ so that composition of $f$ and $g$, namely, $f\circ g(x)= f(g(x)); (x\in ...
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1answer
21 views

How is the L$_p$ norm the maximum of its arguments

I am not able to wrap my head around this result. $\lim_{p\to\infty}(x_1^p + x_2^p + x_3^p .. + x_n^p)^{1/p} = \max [x_1, x_2, .. x_n]$ ?
2
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0answers
44 views

Prove that $l^{\infty}(\mathbb{Z^+})$ is not separable.

Let $l^{\infty}(\mathbb{Z^+})$ be the set of all bounded complex functions on $\mathbb{Z^+}$. Then prove that $l^{\infty}(\mathbb{Z^+})$ is not separable. My attempt: Suppose $E\subset ...
1
vote
1answer
50 views

Uniform Boundedness Principle for $L^p( \mathbb{R})$

Suppose $\{f_n\}$ is a sequence in $L^p$ such that for each $g\in > L^q$, the sequence $\{\int f_n g\}$ is bounded. Then $\{f_n\}$ is bounded in $L^p$. $(1\leq p<\infty)$ Proof: Argue by ...
0
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0answers
36 views

weak convergent sequence in $L^p(\mathbb{R})$ with $(1\leq p < \infty)$ implies norm is bounded

$f_n \rightharpoonup f$ in $L^p(\mathbb{R})$ with $(1\leq p <\infty)$ implies $||f_n||_p$ are bounded. And for $p = \infty$, if $f_n \xrightarrow{w^*} f$, then $||f_n||_\infty$ are ...
1
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1answer
18 views

Most general type of $L^p(X,\ V)$ space where compactly-supported continuous functions are dense

Let $(X,\ \tau)$ be a topological (locally compact?) space, $(X,\ \mathcal{F},\ \mu)$ a measure space, $(V,\ \|\cdot\|)$ a Banach space, $1\leq p < \infty$ and $\|\cdot\|_p$ a function defined for ...
4
votes
2answers
59 views

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$

$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$ My attempt: Set $f_n(x) = \sin(nx)$. Argue by contradiction, suppose there exists a Cauchy ...
9
votes
1answer
63 views

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$

There is no continuous mapping from $L^1([0,1])$ onto $L^\infty([0,1])$. Proof: suppose $T:L^1 \rightarrow L^\infty$ continuous and onto. $L^1$ is separable, let $\{f_n\}$ be a countable dense ...
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2answers
46 views

$L^{\infty} (X, \mu)$ is not separable? [closed]

Show that $L^{\infty} (X,\mu)$ is not separable if $X$ contains sequences of disjoint sets of strictly positive measure?
6
votes
1answer
51 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
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0answers
28 views

About convergence in L^1 [duplicate]

Let $(f_n)_{n \in \mathbb{N}} \subset L^1$ and $f \in L^1$ such $f_n \longrightarrow f \ a.e. $ and $ ||f_n||_1 \longrightarrow ||f||_1 $; then $ ||f_n - f||_1 \longrightarrow 0$. Why?
2
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3answers
64 views

Compactness of $L^p$ inclusion into $L^q$

Let $ i \colon L^p (0, 1) \longrightarrow L^q (0, 1) $, when $ p \ge q $ the canonical inclusion. It is clearly continuos but never compact: I cannot succeed in showing the last point. Thanks for the ...
9
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2answers
530 views

Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
3
votes
1answer
29 views

The isomorphism of the Morrey space $L^{2, n} (\Omega)$ and $L^{\infty} \cap L^2 (\Omega)$

The Morrey space $L^{2, \nu} (\Omega)$ for an open set $\Omega \subset \mathbb{R}^n$ and for $1 \leq \nu \leq n$ is defined as $$L^{2, n} (\Omega) = \{ f \in L^2(\Omega); \hspace{2mm} [f]^2_{L^{2, ...
3
votes
1answer
59 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 compact 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
65 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
26 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
29 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
33 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
47 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
38 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
49 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
56 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
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0answers
49 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
84 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
45 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
39 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
35 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
32 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
33 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
29 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
71 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
58 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 ...
2
votes
1answer
32 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 ...