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
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0answers
43 views

Proving a trivial bound on $L_2$ norm of the error in a sparse approximation of a vector

Trying to understand this supposedly 'trivial' bound from a paper: If $\theta_N$ denotes the vector $\theta$ with everything except $N$ largest coefficients set to $0$ then we have $$ || \theta - \...
0
votes
2answers
31 views

Mixed up with hierarchy of $L_p$ spaces

Consider the interval $[0,1]$ and define $$ X_1:= \left[0, \frac{1}{2}\right], ~~~X_2 := \left[\frac{1}{2}, \frac{3}{4}\right], ~~ X_3 := \left[\frac{3}{4}, \frac{7}{8}\right], ...$$ Define a ...
0
votes
2answers
62 views

Monotone increasing sequence in Lp convergent a.e.

I´m having trouble with the proof of the following theorem in measure theory: Consider the sequence $(f_n)_{\,n \in \Bbb N}\subset L^p(X)$ with $\;0\le f_n(x)\le f_{n+1}(x) \;\;\;\forall x\in X\;\...
3
votes
1answer
65 views

Does the convergence of integrals against $H^1_0$ functions imply boundedness in $L^2(\Omega)$?

In order to show something, I would like to have this strange side result, which seems obvious yet surprisingly I cannot find a way to show it "rigoriously". Suppose we have a sequence $(u_n) \subset ...
0
votes
1answer
20 views

$L^p(\Bbb R^n)$ spaces decomposition

Let us fix $1\le p<r<q$; fix an $l>0$ and take an $f\in L^r(\Bbb R^n)$; then we split the function $$ f=\underbrace{f\chi_{\{|f|>l\}}}_{=:f_1}+\underbrace{f\chi_{\{|f|\le l\}}}_{=:f_2} $$ ...
1
vote
1answer
40 views

Problem with $L^p(\Bbb R^n)$ spaces (Marcinkiewicz Theorem)

Marcinkiewicz Theorem says that, if $1\le p<q$ given a bounded operator $T$ from $L^p(\Bbb R^n)$ to $L^{p,w}(\Bbb R^n)$ (the last one is the $L^p$ weak space) AND EVEN from $L^q(\Bbb R^n)$ to $L^{q,...
3
votes
0answers
33 views

$L^p$-bounding inequality

Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$. ...
5
votes
2answers
65 views

Weak convergence in different $L^p$ spaces

Consider $p \ge \alpha \ge 1.$ If a sequence converges weakly in $L^p,$ say $u_n \rightharpoonup u$, is it true that: $$u_n^{\alpha} \rightharpoonup u^{\alpha} \text{ in $L^{p/ \alpha}$}$$ This ...
0
votes
1answer
26 views

From convergence in $L^2$ to convergence of the squares in $L^1$

Given a function $f\in L^2(X,\mu)$ for some measure space $(X,\mu)$, we of course have that $|f|^2$ is a positive function in $L^1(X,\mu)$. I now came across the following. Suppose $\int_X |f_n(x)-g_n(...
1
vote
1answer
51 views

Poincaré's Inequality on Sobolev Spaces in One Dimension

The following is a version of Poincaré's inequality: Let $I$ be a bounded interval, then there exists a constant $C$ dependent on $I$ such that $$\|u\|_{W^{1,p}(I)} \leq C\|u'\|_{L^p(I)} \ \ \ \ \...
1
vote
2answers
68 views

Jensen's inequality; what's the need for the probability measure?

Jensen's inequality states that if: $\mu$ is a probability measure on $\Omega$, $f$ is integrable function (on $\Omega$) and $\phi$ is convex on the range of f then: $\phi \left( \int_{\Omega}g(x)\,d\...
6
votes
5answers
261 views

Why define norm in $L_p$ in that way?

Who first defined the norm in $L_p$ space as $$\left(\int{\lvert f(x) \rvert^p}\right)^{1/p}$$ Is there any reference for this? Is it just an simple extension from $L_2$? $L_p$ space has some ...
3
votes
1answer
48 views

Prove (possibly using the Uniform Boundedness Principle) that g in L^q

The question is Prove that, if $(X, M, \mu)$ is a $\sigma$-finite measure space, $1\leq p\leq\infty$, and $\frac{1}{p} + \frac{1}{q} = 1$ then: If $f: X\rightarrow\mathbb{C}$ is a ...
4
votes
1answer
48 views

Unbounded operator, when is it dense?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$My two ...
3
votes
0answers
27 views

Characterization of the function $u(x) = (1 + |\log x|)^{-1}$ [closed]

Let $u(x) = (1 + |\log x|)^{-1}$. Are the following all true? $u \in W^{1, 1}(0, 1)$ $u(0) = 0$ ${{u(x)}\over x} \in L^1(0, 1)$
1
vote
1answer
44 views

In which sense is the Fourier transform the continuous analogue of the discrete Fourier transform? A mathematical point of view.

I try to understand in which sense the Fourier transform is the continuous analogue of the discrete Fourier transform. I know, there are many books and many questions on this site concerning this ...
2
votes
2answers
74 views

Is the product $f_ng_n$ of weak-star convergence sequences $(f_n)$ and $(g_n)$ in $L^\infty$ also weak-star convergent?

Suppose $X$ is a finite measure space, and $f_n$ is uniformly bounded and converges to $f$ in the weak-star topology of $L^\infty(X)$. This means $\int f_n\phi \to \int f\phi$ for all $\phi\in L^1(X)$....
1
vote
0answers
24 views

Notation: meaning of $L^{p}(\mathbb{R}^{N},\mathcal{A})$

Let's say $\mathcal{A}$ is some subset of $\mathcal{M}_{N}(\mathbb{R})$ (square matrices of order $N$ with real coefficients). What does $L^{p}(\mathbb{R}^{N},\mathcal{A})$ mean (where $p\in\mathbb{N}...
1
vote
2answers
37 views

Showing that the following set is closed in $L^p$

Question: Let $1\le p < \infty$ and $1\le q \le \infty$. Prove that the following set is closed in $L^p$. $$ \{f \in L^p \cap L^q : |f|_q \le 1 \} $$ My try: Let $f_n$ be a sequence in the above ...
4
votes
2answers
57 views

Is $D(A)$ necessarily dense in $E$? Is $G(A)$ necessarily closed in $E \times E$?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$I have two ...
1
vote
1answer
47 views

Strangely defined ball compact in $L^p(I)$ or not?

Let $I = (0, 1)$ and $1 \le p \le \infty$. Set$$B_p = \{u \in W^{1, p}(I) : \|u\|_{L^p(I)} + \|u'\|_{L^p(I)} \le 1\}.$$When $1 < p \le \infty$, does it necessarily follow that $B_p$ is compact in $...
4
votes
1answer
33 views

Approximation of $L^1$ function with compactly supported smooth function with same mass and same uniform bounds

Recently, I have asked this question. Now, I even want to make this better. Given $f\in L^1(\mathbb{R})$ with $0\leq f\leq 1$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such ...
0
votes
2answers
40 views

When we raise $f$ to a positive power, what happens to the norm?

I am missing something in the identity $(1.18)$ below. What does that identity have to do with the fact that the $L^p$ norm is a non-negative number? The identity $(1.18)$ says that when we raise $f$...
0
votes
0answers
20 views

Applying homogeneity for a norm to prove triangle inequality

I am hoping to have someone help explain to me the steps of this proof of the triangle inequality for $L^p$ spaces, $p\ge 1$. Specifically the first two applications of the homogeneity property don't ...
1
vote
1answer
27 views

$L_p$ norm $\leq L_2$ norm for $1\leq p\leq2$ for Random Variables

Let {$X_i;i\geq0$} be a sequence of random variables defined on the probability space ($\Omega,\mathcal{F},P$). If $||.||_p$ is the $L^p$ norm defined as $||X_i||_p=(E[|X_i|^p])^{1/p}$, how should I ...
0
votes
1answer
31 views

Show that $l^{\infty}$ is a normed linear space

Show that $l^{ \infty}$ is a normed linear space. where $l^{ \infty}$ is define as $$||{a_k}||=Sup_{1 \leq k \leq \infty} |a_k|$$. There are three properties that i need to check in order for this to ...
3
votes
1answer
41 views

Approximation of $L^1$ function with compactly supported smooth function with same mass

Given $f\in L^1(\mathbb{R})$, I can find for any $\epsilon>0$ a $g\in C_c^\infty(\mathbb{R})$ such that $\|f-g\|_{L^1}\leq \epsilon$. Can I assume wlog that $\|f\|_{L^1} = \|g\|_{L^1}$?
2
votes
1answer
43 views

Proving weak compactness of $L^p$ when $p \neq 1$ or $p \neq \infty$

I want to prove weak compactness of $L^p$ when $p \neq 1,\infty$ and if $\sup_n \|f_n\| < \infty$, which is a problem in Stein and Shakarchi's Functional Analysis book. I've done the part where ...
2
votes
1answer
61 views

Proving that dual space of $L^\infty(\mathbb{R})$ is not separable

This is the second part of a problem in Stein and Shakarchi's Functional Analysis. I proved in the first part that $L^\infty(\mathbb{R})$ is not separable by constructing for each $a \in \mathbb{R}$ ...
3
votes
1answer
50 views

Trace of multiplication operator on $L^2(\mathbb{T})$

Let $H=L^2(\mathbb{T})$, where $\mathbb{T}$ is the Torus. Consider a multiplication operator with a sufficiently nice function $f$. Is there somehow a formula like $$\mathrm{tr} M_f = C \int_{\mathbb{...
6
votes
1answer
105 views

With regards to a comment on Problem 5.14 from Evans PDE, absolute values.

See here. Problem 5.14, Evans. Let $U$ be bounded with a $C^1$ boundary. Show that a ''typical'' function $u \in L^p(U) \ (1 \leq p < \infty)$ does not have a trace on $\partial U$. More ...
1
vote
1answer
63 views

Almost everywhere continuous functions

A function that is almost everywhere continuous is in $L_2$; however, the converse might not be true. I couldn't find any example to show this, could you help me with this? $$L_2= \left \{ f:\mathbb{...
5
votes
2answers
181 views

If $u \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $u \in L^\infty(\mathbb{R}^n)$?

How do I use the Fourier transform to see that if $u \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $u \in L^\infty(\mathbb{R}^n)$, with the bound$$\|u\|_{L^\infty(\mathbb{R}^n)} \le C\|u\|_{H^s(\...
0
votes
1answer
20 views

What can we say about the bound of a function given its $L^1$ bounds of its partial derivatives [closed]

Suppose $f \in C_0 ^\infty (\mathbb R^d ) $ be the space of smooth function which vanishes at infinity and we know that $\| \partial_i f\|_{L^1 (\mathbb R^d ) } \le M_i $ for all $1\le i\le d$. Can we ...
0
votes
0answers
14 views

Increasing integrability by decreasing domain of integration?

I am interested in knowing if the following is true, but sincerely I have no clue of how to prove it or disprove it. Help would be appreciated: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and ...
4
votes
1answer
29 views

If $0 < r < p < \infty$, then $\|f\|_r \leq (\frac{p}{p-r})^{1/r} \mu(X)^{1/r - 1/p} \|f\|_{p, w}$ (weak $L^p$ norm)

Let $(X, \mu)$ be a finite measure space. If $0 < r < p < \infty$, prove that $||f||_r \leq (\frac{p}{p-r})^{1/r} \mu(X)^{1/r - 1/p} \|f\|_{p, w}$. Here $\|f\|_{p, w}$ is the "weak $L^p$ ...
0
votes
0answers
27 views

Continuous extension of a linear operator

Fixed two measure space $X,Y$. Suppose that $T$ is a linear operator that maps simple functions of finite measure support on $X$ to measurable functions on $Y$ (modulo almost everywhere equivalence), ...
1
vote
0answers
25 views

An application of the Closed Graph Theorem [duplicate]

Let $T:L^2([0,1]) \to L^2([0,1])$ be a bounded linear map of Hilbert spaces such that if $f\in L^2([0,1])$ is continuous then so is $Tf$. Show that there is a positive constant C such that $$sup_{x\...
0
votes
0answers
32 views

Relation of convergence in $L^\infty$ and $L^p$

Let $(\Omega, \mathcal{A}, \mu)$ be a measurable space with $\mu(\Omega) < \infty$. Now, let $f_n$ be a sequence of functions such that $f_n \rightarrow 0$ in $L^\infty(\mu)$. Does this imply that $...
8
votes
1answer
122 views

Inequalities involving Sobolev spaces.

Let $I = (0, 1)$. I have two questions. Let $p > 1$. For all $\epsilon > 0$, does there necessarily exist $C = C(\epsilon, m, p)$ such that$$\sum_{j = 0}^{m-1} \|D^j u\|_{L^\infty(I)} \le \...
1
vote
1answer
35 views

Is the subspace of $L^2([0,1])$ of all “functions” vanishing on $[0,1/2]$ closed?

I am trying to understand the following example from my lecture notes: $\mathcal{H} := L^2([0,1],\lambda)$, then $$K := \{f \in \mathcal{H} \colon f(x) = 0, \text{ for } 0 \leq x \leq \frac{1}...
4
votes
2answers
50 views

Followup question, does $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ still hold when $p = 1$?

This is a followup to this question. Let $p > 1$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, p)$ such that$$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(...
4
votes
1answer
61 views

Exists $C = C(\epsilon, p)$ where $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $u \in W^{1, p}(0, 1)$?

Let $p > 1$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, p)$ such that$$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$$for all $u \in W^{1, p}(0, 1)$?
3
votes
0answers
51 views

If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?

The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4. However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...
2
votes
0answers
62 views

Using wavelets to capture the $L^2$ norm of $f''$

The goal I have in mind is to express the $L^2$ norm of the second derivative of a quite regular function as a sum of some coefficients. My idea was that such coefficients involved some system of ...
1
vote
1answer
62 views

$T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. [closed]

Let $T\colon L^2[0,1] \to L^2[0,1]$ be a bounded linear map of Hilbert spaces such that if $f\in L^2[0,1]$ is continuous then so is $Tf$. Show that there is a constant $C$ such that $$\sup_{x\in[0,1]}|...
1
vote
1answer
26 views

Convergence in $L_p$ with $\sigma$-finite measure

I have a problem about convergence in $L_p$ spaces, and I really not understand why I need of a specific hypothesis. Let be $L_p(X,\Omega,\mu)$ a measure space, where $\mu$ is $\sigma$-finite, $...
2
votes
1answer
26 views

The convergence of re-scaled function with mollification

Let $u\in L^p_{\operatorname{loc}}(\mathbb R^N)$. Let $\Omega\subset \mathbb R^N$ be open bounded. Let $\eta_\epsilon$ be the standard mollification function and we define $u_\epsilon:=u\ast \eta_\...
2
votes
0answers
42 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuously differentiable functions on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)...
0
votes
1answer
33 views

Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?

Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and $n$. ...