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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1answer
78 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
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2answers
40 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
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1answer
20 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
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1answer
47 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
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1answer
69 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
19 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 ...
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1answer
25 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
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0answers
65 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
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1answer
31 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 $L^p$ ...
1
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0answers
96 views

Fast Convergence of marginal distribtution

Let $(q_n)$ be sequence of probability density functions of the couple $(x,y)\in \mathbb R^2$, $p_n$ is the marginal density of $q_n$, i.e. $p_n(x):=\int q_n(x,y)dy$. Another sequence of functions ...
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1answer
34 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
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0answers
38 views

Weak star convergent sequence and $L^\infty(0,T;L^\infty(\Omega))$

Suppose $u_n$ is uniformly bounded in $L^\infty(0,T;L^\infty(\Omega))$ where $\Omega$ is a bounded domain. From this answer, we know that the dual space of $(L^1(0,T;L^1))^* = ...
2
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1answer
44 views

Weak-star lower semicontinuity in $L^\infty$

Let $u_n \rightharpoonup^* u$ in $L^\infty(\Omega)$. Do we get something like $$\lVert u \rVert_{L^\infty} \leq \liminf_{n \to \infty} \lVert u_n \rVert_{L^\infty}$$ i.e. a weak-star lower ...
3
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3answers
52 views

Not a basis for $ l^\infty$ then what is it?

We know that $ l^\infty$ has not a Schauder basis and its Hamel basis is uncountably infinite. Let $e_n=(e_{n1}, e_{n2},...)$ (for each $n\in \mathbb{N}$) s.t. $e_{nj}=0$ when $n\neq j$ and ...
4
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2answers
62 views

Reflexivity of $\ell^p$

I'm having bad difficulties in understanding how to prove that $\ell^p$ with $1<p<\infty$ are reflexive spaces. Every text I have consulted give that as a trivial result because "observing that ...
4
votes
2answers
54 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
2
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0answers
44 views

Convergence of Step Function Defined by Averages

For a function $f \in L^2[0,T]$, and a uniform partition $P = \{0=t_0, t_1, \ldots, t_n = T\}$ of the domain, we can define a step function approximation as the average value over each interval in the ...
1
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0answers
53 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
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0answers
61 views

How to prove $\gamma$ is continuous?

In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in ...
1
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1answer
63 views

Strong Convergence in L1 Implies Weak Convergence in L2?

If I have $f_n \to f$ in $L^1(D)$, where $D \subset \mathbb{R}$ is compact, is it accurate to say $f_n \rightharpoonup f$ in $L^2(D)$? The argument is as follows: consider a simple function $\phi = ...
0
votes
0answers
31 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
5
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1answer
82 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
7
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1answer
58 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
0
votes
1answer
30 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
0
votes
1answer
38 views

When does the convergence of the regularization of a function is decreasing?

Hi everyone: Let $\theta(x)$ equal $k\exp\left(-\frac{1}{1-\|x\|^2} \right)$ if $\|x\|<1$, and equal $0$ if $\|x\|\geq1.$ Here $\|\cdot\|$ designates the Euclidean norm in $\mathbb{R}^n$, and the ...
2
votes
2answers
69 views

Help with closed graph theorem

I come across this exercise to use the closed graph theorem, if anyone can give and idea about it, thanks Let $M$ be a closed subspace of $L^2([0,1])$ that is included on $C([0,1])$(with the ...
1
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1answer
37 views

$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in C_{c}^{\infty}(\mathbb R), \|f\|_{L^{\infty}(\mathbb R)}=1\}} ?$

I learn the following from the book: Fact: If $g\in L^{1}(\mathbb R),$ then $$\|g\|_{L^{1}(\mathbb R)}=\sup \{ {|\int_{\mathbb R} fg|: f\in L^{\infty}(\mathbb R), \|f\|_{L^{\infty}}}=1\}.$$ We put ...
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0answers
26 views

Show that the Fourier transform of f is in $L^p(R)$ for every $2 \leq p \leq \infty$.

Let $$f(x)=\sum_{n=1}^\infty \sqrt{n} \chi_{(\frac{1}{n+1},\frac{1}{n})}(x)$$. The Fourier transformation of f is $$\hat{f}(y)=\sum_{n=1}^\infty ...
4
votes
2answers
57 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
2
votes
1answer
34 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
0
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1answer
34 views

An application of Holder's inequality to show one norm is smaller than another

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed. Let $\text{Vol}(\Omega) = 1$. Then can we show that $$\lVert u \rVert_{L^{r(s)}(\Omega)} \leq ...
1
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1answer
41 views

Proving that weak limit in $L^p$ and strong limit in $H^{-1}$ are the same

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Let $p \geq 1$ and suppose that $u_n \rightharpoonup u$ in $L^p(\Omega)$ and $u_n \to v$ in $H^{-1}(\Omega)$. How to show that $u=v$? I can do ...
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1answer
43 views

If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Let $\Omega$ be a bounded domain. Suppose that $u_n^p \rightharpoonup v$ in $L^1(\Omega)$. Does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p(\Omega)$?
2
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2answers
182 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
2
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0answers
53 views

Prove that $f\in L^p[0,1]$ for all $p\in[1,2)$:

Given that $f:[0,1]\to[0,\infty)$ in $L^1$ such that $\int_E f$ $dm\leq\sqrt{m(E)}$ for every $E\subseteq[0,1]$ measurable, prove that $f\in L^p[0,1]$ for all $p\in[1,2)$. This is a qualifying ...
3
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1answer
38 views

need to prove an inequality with absolute value to the power of positive number

I need help to prove the inequalities in the following cases $ ||x|^p-|y|^p|\leq \begin{cases} |x-y|^p & \mathrm{if} \, 0<p<1\\ p|x-y|(x^{p-1}+y^{p-1}) & \mathrm{if} \, 1\leq p<\infty ...
2
votes
2answers
41 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
2
votes
1answer
70 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
1
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0answers
32 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
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1answer
70 views

Minkowski inequality for $l_p$ norm.

I'm trying to prove the Minkowski inequality for the $l_p$ norm: $$ \| f + g\|_p \le \|f\|_p + \|g\|_p $$ where $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are Lebesgue measurable functions and $p ...
0
votes
1answer
50 views

Prove an identity without using Hölder's inequality

How to prove the following without using Hölder's inequality : $$ \|f\|_{p} = \sup_{\|g\|_q =1} \int |fg| d\mu ; \frac{1}{p} + \frac{1}{q} =1$$
3
votes
1answer
47 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
7
votes
1answer
96 views

What information is contained in the function $p\mapsto ||f||_p$?

Given a measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$, we obtain a function $\nu_f:(0,\infty)\rightarrow [0,\infty]$ defined by $\nu_f(p):=||f||_p$ This function $\nu_f$ will not ...
3
votes
0answers
58 views

Is $L^p$ separable?

Whether a $L^p(X,\mu)$ space is separable? I understand that the answer depends on $p$ and $X$. It seems to me that it is separable when $1\leq p < \infty, X=\mathbb{R}^n$ or $X=\mathbb{N}$. ...
1
vote
1answer
39 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
1
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0answers
43 views

Dense subsets in $L^1(\mathbb{R})$

Which of the following are dense subsets in metrical space $L^1(\mathbb{R})$? set of smooth functions $C_0^{\infty}(\mathbb{R})$ with compact supports; set of above-mentioned functions' derivatives ...
3
votes
1answer
91 views

Prove that $L^1(\mathbb{N})$ is a Banach space.

I'm trying to prove that $L^1(\mathbb{N}) := \left\{ (x_n)_{n=1}^{\infty} : \sum\limits_{n=1}^{\infty}\left|x_n\right| < \infty \right\} $, the space of all sequences over the field $\mathbb{C}$ ...
2
votes
2answers
61 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
2
votes
2answers
58 views

Weak convergence of partial sums

I recently came across an interesting problem on weak convergence in $\ell^2 (\Bbb N)$. Suppose that we have canonical basis $\{e_i\}$ in $\ell^2 (\Bbb N)$. We need to prove that the sequence ...
0
votes
2answers
26 views

Let $u_n \to u$ in $L^1(\Omega)$. Does $u_n^p \to u^p$ in $L^1(\Omega)$ if we know $u_n^p \in L^1(\Omega)$?

Suppose $u_n \to u$ in $L^1(\Omega)$ where $\Omega$ is a bounded domain. Suppose that $u_n^p \in L^1(\Omega)$ (actually $L^\infty(\Omega)$ for each $n$). Fix $p \in [1,\infty)$. So $u_n(x) \to u(x)$ ...