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
1answer
28 views

Why isn't $\ell^p$ locally convex for $0<p<1$?

I believe we have to distinguish the finite-dimensional from the infinite dimensional case. Regardless, if $0<p<1$, $\|x\|_p := (\sum |x_i|^p)^{\frac 1 p}$ is not a norm as it fails to satisfy ...
0
votes
1answer
12 views

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is $f \in L^p(0,T)$?

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is it necessarily true that $f \in L^p(0,T)$? I don't see why not since the only point we have a problem may be at 0, but that is a null set. ...
0
votes
2answers
46 views

Bound on $f_h(t) := \frac 1h \int_t^{t+h}f$

Given $f \in L^2(0,T;L^2(\Omega))$ define $$f_h(t) = \frac 1h \int_t^{t+h}f(s)\;\mathrm{d}s$$ for $t \in (0,T-h)$ and $f_h(t) = 0$ for $t > T-h$. In this paper (http://dml.cz/bitstream/handle/...
1
vote
1answer
45 views

Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$. I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.
4
votes
1answer
57 views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
1
vote
0answers
34 views

Show $N_p[f]=(\frac{1}{|E|}\int_{E}|f|^p)^{\frac{1}{p}}$ is monotone in $p$

For $0<p\leq \infty$ and $0<|E|<\infty$ ($|E|$ is the lebesgue measure of $E$), define $$ N_p[f]= \left( \frac{1}{|E|} \int_E |f|^p \right)^{1/p}, $$ where $N_\infty[f]$ means $\|f\|_\infty=...
0
votes
1answer
27 views

For what value $p$ is $l_p$ not a norm or a metric? [closed]

Can someone please remind me for which values of $p \in [0, \infty)$ is the little $l_p$ norm or $l_p$ metric not a norm or a metric I vaguely remember that $l_0$ norm is a not a norm. Could someone ...
3
votes
2answers
47 views

L^1 convergence and limsup of convergent sequence

I have to solve this exercise: let $f_n$ be a sequence of positive real function defined on a measure space $(X,M,\mu)$ such that $f_n\in L^1(\mu)$ $\forall n\in \mathbb{N}$ and $f_n$ is convergent in ...
-1
votes
1answer
45 views

How to prove that an $L^2$ function is also an $L^1$ function? [closed]

I have a function $f(t)$ defined on $[-a,a]$ that belongs to $L^2$. How do I prove that $f$ also belongs to $L^1$? In general this fact is not true. Is the cauchy-schwarz inequality the only way? If a ...
4
votes
2answers
37 views

Find all $\alpha$ such that $n^\alpha\chi_{[n,n+1]}$ converges weakly to 0 in $L^p$.

Edit: $1 < p < \infty$ Let $f_n(x) = n^\alpha \chi_{[n,n+1]}.$ Then $$ \begin{align} \left|\int_{-\infty}^{\infty} n^\alpha \chi_{[n,n+1]}g(x)dx\right| &\le n^\alpha \lvert\lvert\chi_{[n,n+...
1
vote
4answers
52 views

$C_c^\infty(\Omega)\subseteq L^p(\Omega)$ for any open $\Omega$?

Let $d\in\mathbb N$ and $\Omega\subseteq\mathbb R^d$. Can we show that $$C_c^\infty(\Omega)\subseteq L^p(\Omega)\tag 1$$ for all $p\in [1,\infty]$? It's clear that $(1)$ holds if $\Omega$ has finite ...
1
vote
2answers
30 views

How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0?

In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the ...
1
vote
1answer
24 views

Convergence of composition of functions in $L^p$

I am dealing with the proof of proposition $9.5$ given in Haim Brezis' Functional analysis, Sobolev Spaces and Partial differential equations. I quote it here: How does one conclude $G \circ ...
1
vote
1answer
59 views

A general version of Gronwall's inequality

For the following $$|u(t)|^p\le C_1 \int_0^t |u(s)|^p\,ds+C_2$$ using Gronwall inequality, we have $$|u(t)|^p\le C_2(1+C_1 te^{C_1 t})$$ Now, my question is, for $$|u(t)|^p\le K_1 \int_0^t(1+|u(s)|^2)...
0
votes
1answer
37 views

Volterra Operator on Sobolev Space

I stumpled over the following result in a script: Let $1 \leq p < \infty$ and $f \in L_p[a,b]$. Define the Volterra operator as $$Vf(t) = \int_a^t f(s) ds.$$ Then we have $Vf \in W^{1, p}[a,b]$ ...
1
vote
1answer
36 views

$\lim_{p \to \infty} \|f\|_p = \|f\|_{\infty}$: is convergence monotone when $\mu(X) \leq 1$?

This question is related to Exercise 3.3.7(b) in Cohn, Measure Theory, 2nd edition, which reads as follows: Let $(X, \mathcal A, \mu)$ be a finite measure space, and let $f$ be an $\mathcal A$-...
2
votes
1answer
47 views

For which $p$ is $\frac{1}{x^a+x^b}$ in $\cal{L}^p$?

Let $f(x)=\frac{1}{x^a+x^b}$ with $x,a,b>0$. For which $p\ge1$ is $f$ in $\cal{L}^p(\lambda)$ over the interval $(0,\infty)$? Here $\lambda$ is the one dimensional Lebesgue measure. Attempt: We ...
1
vote
1answer
25 views

Gradient of the solution for Poisson equation

Let $f(x)=(\nabla N *g)(x)$, where $N(x)=\frac{1}{|x|^{n-2}}$ for $n\geq 3$ is the Newtonian kernel, and $g\in L^1(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R}^n)$. Then we can have that $f\in L^{\infty}$ ...
2
votes
1answer
21 views

How can we show that $\int_{|\alpha |\leq N}\hat f(\alpha )e^{2i\pi x\alpha }d\alpha $ converges to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$.

I am a Ph.D. student in statistic, and I wanted to know if there is an easy way to show that the Fourier inversion converge to $f$ in $L^p(\mathbb R)$ for $1<p\leq 2$. In other word, that $$\lim_{N\...
0
votes
2answers
42 views

The convolution of two functions is L1

I have to proof the following corollary: $ \text{Let } 1 \leq p \leq \infty \text{, } f \in L^{1} \text{, } g\in L^{p} \text{. Then } f \ast g \in L^p \text{ and } \Vert f \ast g \Vert _{L^{p}} \...
2
votes
1answer
47 views

How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether ...
2
votes
0answers
23 views

Convolution is continuous

Convolution is continuous Let $f,g\in L^2\left(\mathbb T,\mathbb C\right)$ (Hilbert space of $1$-periodic functions) then $f*g$ should be continuous by Young's inequality (the map is $(f,g)\mapsto ...
1
vote
0answers
15 views

Is this sequence uniformly bounded in $L^\infty(\Omega)$?

Let $u \in L^\infty(\Omega)$ on a bounded domain $\Omega$. Let $w_j$ be the eigenfuntions of the Neumann Laplacian. Is it true that $$a_n := \sum_{i=1}^n (u,w_j)w_j$$ is such that $\lVert a_n\rVert_{...
3
votes
2answers
37 views

If the limit of a $L^2$ sequence is in $L^\infty$, is the sequence bounded in $L^\infty$?

Let $f_n \to f$ in $L^2$ on a bounded domain. We know that $f \in L^\infty$. Does it follow that $\lVert f_n \rVert \leq A$ for a constant $A$ independent of $n$, for a subsequence if necessary? I ...
0
votes
1answer
15 views

Sequence of Functions on $[0,1]$ with Derivatives Bounded by $L^1$ Function

I'm stuck on the last step of a real analysis/advanced calculus problem and could really use some help. The problem is as follows: Let $f_n$ be continuously differentiable on $[0,1]$ satisfying, for ...
0
votes
0answers
17 views

Estimate Sobolev function by its derivative on a ball

If $f$ is a smooth function and $f(0)=0$, it is clear that $\|f\|_{L^\infty(B)}$ can be estimated (for any Ball $B$) if $\|\nabla f\|_\infty$ is known. My question is whether something similar can be ...
2
votes
0answers
47 views

Show that $L^2(0,1)=\operatorname{span}\{e^{2\pi inx}\}_{n\in \mathbb Z}$

Q1) In a course, it's written that $L^2(0,1)$ is spanned by $\{e^{2\pi inx}\}_{n\in\mathbb Z}$. How can I show it ? Q2) Let $f\in L^2(0,1)$. Then we have Parseval equality, i.e. $$\left\|f\right\|^2=...
0
votes
0answers
15 views

Characterization of a set occurring in the Helmholtz-Hodge decomposition

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 2$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
5
votes
1answer
75 views

Finding $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$

I heard there were functions $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$. Is there a concrete example of such functions ? Thanks in advance !
4
votes
0answers
56 views

Proof of the Helmholtz-Hodge decomposition

Let $\Omega\subseteq\mathbb R^3$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ Let $$G^2(\Omega):=\left\{\nabla p:p\in L^2_{\text{loc}}(\Omega)\text{ with }\nabla p\in L^2(\Omega)^3\right\}$$...
0
votes
0answers
22 views

Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
0
votes
1answer
46 views

If $p$ is a distribution, what is the meaning of the claim $\nabla p\in L^p(\Omega)^d$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 1$ I've seen the following Lemma (without a proof) in a paper and don't understand how I ...
1
vote
0answers
38 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
0
votes
1answer
37 views

Convergence in $L^2(\Bbb R)$ implies convergence of the norms [closed]

If $||f_n-f||_{L^2(\mathbb{R})}\to 0$ is it always true that $||f||_{L^2(\mathbb{R})}=\lim_{n\to\infty}||f_n||_{L^2(\mathbb{R})}$?
2
votes
0answers
21 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
0
votes
1answer
51 views

Counter Example for Limit of $\|f\|_p$ in infinity convergence, When Measure space is not finite [closed]

I found a proof for this fact that limit of $\|f\|_p$ when $p \to \infty $ is $\|f\|_{\infty}$ in here when $f:X \to R $ and $X \in L^p$ measure space is finite. But I need a counter example for ...
-1
votes
2answers
61 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
1
vote
1answer
30 views

Integration of periodic function $f \in L^1([0, 2\pi])$

While studying trigonometric series and $L^p$ spaces I was wondering the following: Let's say we have a $2\pi$-periodic function $f \in L^1([0, 2\pi])$ satisfying $\int_{0}^{2\pi}f(x) \, dx = 0$. Is ...
0
votes
1answer
48 views

Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
3
votes
2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
2
votes
1answer
44 views

Characterization of measures such that $\frac{1}{x} \in L^1(H)$

Let $H$ be a finite measure on $(0,1)$. What conditions must $H$ fulfill, such that \begin{equation*} \frac{1}{x} \in L^1(H),\ \ \ \frac{1}{1 - x} \in L^1(H) \end{equation*} I'm trying to characterize ...
4
votes
1answer
71 views

Explanation of spaces of functions in PDE

Let's consider following equation: The problem $$ \begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \...
2
votes
0answers
26 views

Show linearity of a functional if it holds for nonnegatives

Consider a functional $G^+:L_p \to \mathbb{R}$. Here $L_p = L_p (X,\textbf{X}, \mu)$ is the collection of all integrable fns (f s.t. $\int \vert f \vert^p d \mu < \infty$ on the measure space $(X,\...
0
votes
0answers
24 views

$L^p$ convergence of smooth compactly supported functions

I already checked the similar question here, but want to check slightly different argument. Given $f(x)\in L^p(\mathbb{R}^n)$, can I find $f_n(x) \in C^\infty_0$ s.t. $f_n \rightarrow f$ almost ...
1
vote
1answer
63 views

Why does Rudin say that $f$ is in $L^\infty$?

The setting is that we have $L^\infty([0,1],m)$, $m$ Lebesgue measure, and we have shown the Gel'fand transform is an isometry onto $\mathcal{C}(\Delta)$, $\Delta$ being the space of maximal ideals / ...
2
votes
1answer
69 views

Proof verification: Something similar to Riesz-Fischer Theorem

Question: Suppose $\{f_n\}$ converges to $f$ in $L^p(\mathbb{R})$, $1\leq p<\infty$. Prove that there is a subsequence $\{f_{n_k}\}$ and $g\in L^p(\mathbb{R})$ so that $f_{n_k}\to f$ a.e. and $|f_{...
5
votes
1answer
84 views

Why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite?

As the question title suggests, why are there no finitely additive measures on $\ell_\infty$ for which the measure of every ball is positive and finite? Here, we do not assume that the measure is ...
2
votes
1answer
29 views

Question on Inequality from Bartle's Elements of Integration: Riesz Fischer Theorem

I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)? I tried using Triangle Inequality and ended up with one extra term: $$\begin{align*} |g_k|&=|g_k-...
4
votes
2answers
90 views

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?
0
votes
1answer
32 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...