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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16 views

If $\nabla \cdot (|\nabla u|^{p-2}\nabla u) \in L^2$ what space is $u$ in?

Define $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2}\nabla u)$. I want to know, if $\Delta_p u \in L^2(\Omega)$, then what space is $u$ in? I am having trouble figuring it out. Take $p=2$. Then ...
5
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1answer
54 views

An example of a function not in $L^2$ but such that $\int_{E} f dm\leq \sqrt{m(E)}$ for every set $E$

I am thinking about this problem: Let $f\in L^1 [0,1]$ to be a nonnegative function satisfied: $$\int_{E} f dm\leq \sqrt{m(E)}$$ for every measurable set $E\subset [0,1]$, Prove that $f\in ...
2
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1answer
42 views

A basic question on $L^p$ norm

Consider a probability space and $f_m$ be sequence of measurable functions a.s. converging to $f$. What can be said about the limit $$ \lim_{m\to \infty} \|f_m\|_m$$ where $\|.\|_p$ stands for the ...
3
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1answer
38 views

Showing inequalities for $l^p$ sequences

If I show that an inequality (e.g. Holder or Minkowski) holds for the $L^p$ space, then can I automatically conclude that the inequality also holds for $\ell^p$ sequences, just by integrating wrt. the ...
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0answers
44 views

Particular $L^p$ space

I am confusing some definitions. Suppose we have a Cauchy sequence $(f_n) \subset L^2(\Omega,C^0([0,1],\mathbb{R}))$, where $\Omega$ is a measurable space with measure $\mu$ and ...
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0answers
45 views

Equivalence of weak $L^p$ norms

I'm kind of new to the subject of weak $L^p$ spaces. The definition of the (quasi-)norm in weak $L^p$ ($p\in(0; \infty)\,$) over $\sigma$-finite measure space $(X, \mu)$ I use is $||f||_{L^{p, ...
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1answer
34 views

Show that $L^p(\mathbb{R}^d)$ spaces are not comparable one another.

I have to show that, in $\mathbb{R}^d$ with Lebesgue measure, the $L^p$ spaces are not comparable one another. More precisely, I want to show that given $p$ and $q$ such that $1\leq p<q\leq\infty$, ...
2
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1answer
53 views

Convergence of truncation in $L^{p}$

If you have a truncation $T_{k}u$ defined as: $$ T_{k}u := \begin{cases} u,& \text{ if }~ |u(x)| \leq 1\\ k\frac{u}{|u(x)|}, & \text{ if }~|u(x)| > k \end{cases} $$ If you consider ...
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1answer
55 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
2
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1answer
54 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
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1answer
47 views

Riesz Representation Theorem for $l_p$

Let $ 1 \leq p < \infty$, with $q$ the conjugate of $p$, and let $T \in l^{p*}$. Then for some sequence $g \in l^q,$ $T(f)=\sum_{\mathbb{N}} fg$ for all $f \in l^p$. I am trying to prove this ...
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2answers
42 views

If $F$ is the distribution function of an $L^p$ function, then $\lambda^p F(\lambda)\to0$ as $\lambda\to0$

Let $1\leq p<\infty$, $f\in L^p (\mathbb{R}^n)$. Let $F(\lambda)=m\{|f(x)|>\lambda\}$, show that: $$\lim_{\lambda\to 0} \lambda^{p}F(\lambda)=0$$ What I only know about distribution function ...
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1answer
155 views

Proving the Riemann-Lebesgue Lemma in $L^1(\mathbb{R}^n)$

$\mathbf{Riemann-Lebesgue \ Lemma \ in \ L^1(\mathbb{R}^n)}$. Suppose that $f \in L^1(\mathbb{R}^n)$. Then $\hat{f}(k) \rightarrow 0$ as $|k| \rightarrow \infty$. I cannot understand any of the ...
2
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1answer
29 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
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1answer
33 views

Closure of $l^1(\mathbb{N})$ in $l^2(\mathbb{N})$

I am trying to understand $l^p$ spaces better and I got stuck. I showed that $l^1(\mathbb{N})$ is a subspace of $l^2(\mathbb{N})$. I also found a counterexample which shows that $l^1(\mathbb{N})$ is ...
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1answer
27 views

Showing that $L^{\infty}([0,1])$ is not strictly convex

Can somebody give an example that shows that $L^{\infty}([0,1])$ (regarding $|| \cdot ||_{\infty}$) is not strictly convex? Thanks in advance!
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0answers
22 views

Estimate bounds on Minkowski distance from point to a segment in Lp space

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...
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1answer
44 views

Example of a sequence in L1 with these conditions

Is there an example of a sequence $\{f_n\}$ in $L^1(\mathbb{R})$, such that: $\{||f_n||_1\}$ is bounded. There's a convergent subsequence $f_{\phi(n)}$, i.e. $\exists f \in L^1(\mathbb{R})$ such ...
1
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1answer
36 views

$\mathcal L^{\infty}$ space properties

Can anybody give an example that for $1 \leq p < \infty$ neither $\mathcal L^p (\mathbb R) \subseteq \mathcal L^{\infty} (\mathbb R)$ nor $\mathcal L^{\infty} (\mathbb R) \subseteq \mathcal L^p ...
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1answer
46 views

Does an integral operator with a symmetric integrable kernel have to be bounded on $L^2$?

Suppose $K(x,y)$ is a symmetric kernel. Let $\phi\in L^2(\Omega)$, where $\Omega$ everywhere is a domain in $R^n$. Can $\int_{\Omega}K(x,y)\,\phi(y)\,dy$ belong to $L^2$? In other words can an ...
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1answer
21 views

The norm of dual operator over $L^p(\Bbb R^N)\times L^p(\Bbb R^N)$

Let $1<p<\infty$ and $E:=L^p(\Bbb R^N)\times L^p(\Bbb R^N)$. Let $\Phi\in E^*$, i.e., the dual of $E$. Hence by Riesz representation we have there exist $u_0$, $u_1\in L^{p'}(\Bbb R^N)$ such ...
3
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1answer
66 views

Question on $L_p$ spaces involving $\lambda^n$-measure on $\mathbb{R}^n$

Q/ Consider $L_p=L_p(\lambda^n)$ with the Lebesgue measure on $\mathbb{R}^n$ and $1\leq p<\infty$. Let $f_0=|x|^{-\alpha}$ for $|x|<1$ and $0$ otherwise. Show $f_0\in L_p$ iff $p\alpha < n$. ...
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0answers
34 views

$L^2$ inequality for derivatives of polynomials on triangles

I'm reading a paper which states the following inequality, but the (presumably) elementary proof is cited to be in a document, which is too old to get access to. Let $p: \mathbb{R}^2 \to \mathbb{R}$ ...
3
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3answers
87 views

An example of a function in $L^1[0,1]$ which is not in $L^p[0,1]$ for any $p>1$

Title says most of it. Could you help me find an example? It is easy obviously to show a function that would not be in $L^p[0,1]$ for a specific $p$ (say $(1/x)^{1/p}$, but I can't see how it would ...
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1answer
42 views

Uniform lower bound on convex functions bounded in $L^2$ norm

Consider a class of (proper closed) convex function on $[0,1]^d$, which we shall denote $\mathcal{F}$. If every element of $\mathcal{F}$ is bounded in $L_2$, say $$\int_{[0,1]^d} |f(x)|^2\ dx\leq 1,$$ ...
2
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1answer
31 views

Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset ...
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0answers
13 views

$L^p$ is a quasi normed space for $0<p<1$ [duplicate]

I know that $L^p$ is a vector space for $p>0$ and a normed space for $p \geqslant 1$ now I need show that for $ 0<p<1$ and $f,g \in L^p$ exist $K \in \mathbb{R}$ such that $||f+g||_p ...
3
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1answer
29 views

$L^\infty(S^1)$ is not separable

Let $S^1$ be the unit circle and $L^\infty(S^1)$ the space of measurable functions $f:S^1\to\mathbb{C}$ such that $\|f\|_\infty<\infty$. (In fact $L^\infty(S^1)$ consists of equivalence classes of ...
0
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2answers
70 views

Interpolation inequality in sobolev space

Let $U$ be a bounded, connected open subset of $\mathbb R^n$ with $C^1$ boundary $\partial U$. Asume $|\beta| \leq k-1$ and $k$ is a integer. Show that for each $\epsilon >0$ there exists a ...
2
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1answer
83 views

$f \in L^1$, but $f \not\in L^p$ for all $p > 1$

"Find an $f \in [0,1]$ such that $f \in L^1$ but $f \not\in L^p$ for any $p > 1$." I've thought about doing something like $$f(x) = \frac{1}{x}$$ where $|f|^p = \frac{1}{x^p}$ doesn't converge ...
0
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1answer
60 views

Is $L^\infty$ compactly embedded in $L^1$?

I'm trying to find a contraction example to show that the space $L^\infty$ is not compactly embedded in $L^1$ with the Lebesgue measure. Please help me!
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0answers
48 views

Question about Young's Inequality

I came across Young's Inequality for convolutions, stated as: Let $f\in L^p, g\in L^q$, where $p,q,r \in [1,\infty]$ and $p^{-1}+q^{-1}=1+r^{-1}$. Then $f*g$ is defined $m$-a.e. on $\mathbb{R}^d$, ...
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0answers
40 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
5
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0answers
89 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
3
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0answers
21 views

Simple $L^p$ space question [duplicate]

Let $X$ be a measure space with positive measure $\mu$. If $\|f_n-f\|_p\to 0$ (i.e. $f_n$ converges to $f$ in the $L^p(\mu)$ norm) does it follow that $f_n(x)\to f(x)$ almost everywhere?
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1answer
76 views

Proving this set is dense in $\ell^2$

I found this weirdest question and was wondering how could this be proved. This question is a part of a beautiful semi-constructive built of two dense disjoint convex sets in $\ell^2$, which I find ...
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0answers
19 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
5
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1answer
96 views

orthonormal sequence in $L^2[0,1]$ - how to prove these following equivalent terms?

I've been asked this following very interesting question and would like some help figuring out why it is true :) Let $u_n$ be an orthonormal sequence in $L^2[0,1]$ Prove that the following are ...
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1answer
25 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
5
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2answers
97 views

$e_n \to 0$ weakly in $l^\infty$

Given the the sequence $(e_n)_n$ in $l^\infty$, I want to show that that $e_n$ converges weakly to $0$ in $l^\infty$, i.e. $$e_n\rightharpoonup 0 \text{ as } n\to \infty.$$ By $e_n\in l^\infty$, ...
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1answer
34 views

Hardy-Littlewood maximal function $Mf$ is greater than $f$

Suppose that $f\in L^{1}(\mathbb R)$ and $x\in \mathbb R.$ We denote $B_{x}$ by the ball in $\mathbb R$ with centred $x,$ and $|B_{x}|=$ length(Lebsgue measure) of $B_{x}.$ Put, $Mf(x)=\sup ...
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0answers
19 views

Basic question about $L_1$ integrable functions [duplicate]

I need to show that if a function is in $L_1$, i.e., the integral of $|f|$ over the real line (Lesbesgue measure) is finite, then the limit as $n$ goes to infinity of the integral of $f$ (from $n$ to ...
3
votes
1answer
29 views

A function in $L^p(E)$ for only $2 \leq p < 3$

Give a set $E$ and a function $f$ on $E$ such that $\{p:f \in L^p(E)\} =[2,3)$. This was a homework problem (whose due date has passed) that my brightest fellow classmates AND the class TA ...
0
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1answer
114 views

Determine if these two norms are equivalent

Let we have the space $C[a,b]$ (the space of all functions that are continuous on closed interval $[a,b]$). And we have two norms on this space: $$\|X\|_1= \max_{t\in [a,b]} | x(t) |$$ $$\|X\|_2 = ...
3
votes
1answer
28 views

Is the $\sigma$-finiteness condition necessary to ensure that $L^p(\mu)$ is reflexive?

Suppose $(X,\mu)$ is a measure space, and $p,q>0$ such that $1/p+1/q=1$. We know that if $\mu$ is $\sigma$-finite, then $L^p(\mu)$, $L^q(\mu)$ are reflexive and dual to each other. The proof could ...
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0answers
255 views

Proof Riesz Representation Theorem (bounded linear functional in Lp)

I have a little problem with this proof (I'm using Royden), can you help me? Let $F$ be a bounded linear functional on $L^p$, $1 \leqslant p \leqslant \infty$. Then there is a function $ge \in L^q$ ...
2
votes
1answer
50 views

$L^\infty$, $L^p$ question

Suppose $f \in L^0$. I read that for a general measure space, if $\mu(X)<\infty$, then we cannot have that both $||f||_\infty< \infty$ and $||f||_p=\infty$ for every $p\in (0,\infty)$, but if ...
3
votes
1answer
66 views

Weak convergence in $l_p$ implies pointwise convergence?

Could someone please share their thoughts on this one: Consider at $l_p(Y)$, for $1<p<\infty$ with the counting measure on $Y$. Show that if a sequence weakly converges in $l_p(Y)$ then it ...
1
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2answers
63 views

Translation operator and continuity

I came across a text that proves that translation operator $T_a(f):=f(x-a)$ where $a\in\mathbb{R}^n$ and $f\in L^p(\mathbb{R}^n)$ is continuous. The proof follows: ...
1
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1answer
24 views

Formulation of Fourier transform

I would like to know about Fourier transform more. I attended a standard lecture of mathematics, but we did not talk about Fourier transform on $L^2$ much, nor the theory of $L^2$. We only defined it ...