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
37 views

Is 'f' belong sobolev?

I was trying to show that the function $$f(x) = \dfrac{x^{1/2}}{1+x^2} \in W^{1,3/2} (0,\infty)$$ that is, have to show that $$f\in L^{3/2}(0,\infty)$$ and $$f_x\in L^{3/2}(0,\infty).$$
0
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1answer
61 views

Isometry from $\ell^1$ to $\ell^\infty$

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
1
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0answers
40 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
1
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1answer
39 views

The derivative of a function is square integrable assuming Fourier transform dominated

I am struggling in solving the second part of this problem. Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove ...
2
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0answers
44 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
1
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1answer
25 views

How to prove $f(x)=e^{\frac{1}{x}}$ is continous in $(0,a), a>0 $ and $\int_{0}^{a}e^{\frac{y}{x}}dx, y>0$ does not exist

I would aprecciate any advice. I'm trying to prove that in the context of a measure space, $(X,B,\lambda)$ , with $X=(0, + \infty) $, $B$ the Borel sigma-algebra and $\lambda$ the Lebesgue measure, ...
2
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1answer
45 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
0
votes
1answer
55 views

Inequality important in $L^p$ space

If$\,\,$ $0<p<\infty$, put$\,\,$ $\gamma_{p}=\max(1,2^{p-1})$, and show that $$|\alpha-\beta|^p \leq \gamma_{p}(|\alpha|^p + |\beta|^p)$$ for arbitrary complex numbers $\alpha$ and $\beta$. ...
1
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2answers
93 views

What does it mean to be an L^1 function?

I am struggling to understand what the space L^1 is, and what it means for a function to be L^1. A friend told me that a function f is $L^1$ if $\int_\mathbb{R} |f|$ is finite. It is $L^2$ if ...
1
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1answer
44 views

I would like to show that $\ell^1$ is separable

So here is my question, I want to prove that $\ell^1$ is separable. So i need to show that there exists a countable dense subset in $\ell^1$. Since I am not sure if my idea was right i hoped ...
1
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1answer
17 views

Relation between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$

As an exercice, I'm looking to find an inclusion or equality relationship between $L^2(\mathbb{R}_+)$ and $L^1_w(\mathbb{R}_+)$ when $w: x \to x^{-1/2}$. Actually, I think that we have the inclusion ...
4
votes
3answers
79 views

How to apply the Hölder's inequality in a clever way?

Here is the problem: Let $f\in L^p(\mathbb R^n)\cap L^q(\mathbb R^n)$ and $s\in[p,q]$. Show that $f\in L^s(\mathbb R^n)$ I'm almost sure that this is a simple exercise on Hölder's inequality yet ...
1
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1answer
38 views

When is this function in Lp?

Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i ...
0
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1answer
38 views

On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
1
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0answers
58 views

Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
2
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1answer
64 views

Use convergence theorem to prove the limit

Suppose $f:\mathbb{R}\to\mathbb{R}$ is $L^1$ and it is continuous at the origin. Let $g_n(x)=\frac{1}{1+x^2}f(\frac{x}{\sqrt{n}})$. The problem is to prove $$\lim_n\int_0^\infty ...
1
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1answer
46 views

Which of the following sets are open (or closed)?

a.) $A:= \{(x_n)_{n\in \mathbb{N}} : x_n \in [0,1] \hspace{2mm}\text{for all}\hspace{2mm} n\in\mathbb{N}\}$ in $(l^\infty, \|\cdot\|_{\infty})$ and b.) $B:= \{f\in C([0,1]) : |f(t)-t|<1 ...
0
votes
0answers
19 views

Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
0
votes
1answer
46 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
2
votes
1answer
48 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
-1
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1answer
81 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
2
votes
1answer
66 views

$\int_E f_n\to\int_E f$ implies $f_n\to f$ pointwise

The problem is to show that for a bounded sequence $\{f_n\}$ in $L^p[a,b]$ that $$ f_n\rightharpoonup f\,\Longleftrightarrow\, \int_E f_n\to\int_E f,\,\, \forall E_\text{(measurable)}\subset [a,b]. ...
3
votes
1answer
46 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
2
votes
1answer
42 views

Bound for $fg\in L^{1}$ in terms of $f,g \in L^{1} $

Let's assume that $fg\in L^{1}$ with $f\in L^{1}$ and $g\in L^{1}$. Is there a common way to bound $\Arrowvert fg \Arrowvert_{1}$ in terms of the norms $\Arrowvert f \Arrowvert_{1}$ and $\Arrowvert ...
1
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1answer
129 views

Approximation of $L^\infty$ functions by $\mathcal{D}$ functions?

Let $Q=(0,T)\times\Omega$ with $\Omega$ a bounded domain. I read this: "the inequality holds for all $f \in \mathcal{D}(Q)$, and by approximation, it holds for all $f \in L^\infty(Q)$ since the ...
1
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2answers
45 views

$f_{n}\to f$ in $L^{1}\implies \hat{f_{n}}\to \hat{f}$ in $L^{1}$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R .$ Suppose there exists $f_{n}, \in L^{1}(\mathbb R)$ such ...
3
votes
0answers
33 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
1
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0answers
23 views

The relationship of L^1(U) and C(U)

Let $U$ be a open set of $\mathbb{R}^n$, C(U) is all continuous functions on U, for example C(0,1), when $U=(0,1)$. And $L^1(U)$ is lp-space where $p=1$. It was said that $L^1(U)$ is the completion ...
2
votes
1answer
37 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
2
votes
2answers
59 views

Subsequence of functions in $L^p$

On a problem sheet we were asked to find a sequence of functions $(f_n)_{n \geqslant 0} \in L^p [0,1]$ such that $\lim_{n \to \infty} ||f_n||_p = 0$ but $\lim_{n \to \infty} f_n (x)$ doesn't exist ...
0
votes
2answers
32 views

1.Convergence in $L^1$

Let f be measurable function such that $||f||_\infty=\infty.$ Show that there exists {${g_ n}$} $\subset L^1$ such that $||fg_n||_1\to\infty$. Anyhelp would be appreciated..
1
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1answer
35 views

Approximation in $L^p$ spaces

Let $X$ denotes the span {$x^n:n \ge1 $}. Is it true that $X $ is dense in $L^1([0,1])?.$ I showed that $X$ is dense in the space of continuous functions that vanishes at zero. I also know space of ...
1
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0answers
65 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
0
votes
1answer
46 views

Norm of operator between two $L^p$ spaces?

In my reading, I've come across notation like $ ||T||_{2 \to \infty} $, where $T$ is an operator defined on every $L^p$ space. What does this mean? Is it simply the norm of $T$ viewed as an operator ...
0
votes
1answer
40 views

Interpolation inequality in $L^p$ space

Let $f \in L^{p}(E) \cap L^{q}(E)$ with $p<q$. How to prove that $f \in L^{h}(E)$ for every $h \in (p,q)$ and the following interpolation inequality: $||f||_{h} \leq ||f||_{p}^{\frac{p}{h}} + ...
1
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1answer
47 views

Functional Analysis, $L^p$ spaces

Can anyone help me finding for which $p$ the function $$f(x)=\frac{1}{x^{\alpha}+x^{\beta}} \in L^{p}(0,\infty)$$ where $0<\alpha \le \beta<\infty$ are given. Thanks a lot.
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0answers
54 views

Inclusion of $L^p$ spaces if $X$ arbitrary

If $X$ is a finite measure space then one can show that if $1 \le p < q$ then $L^q \subseteq L^p$. Is there anything known about the inclusion if $X$ is an arbitrary measure space? Or given some ...
3
votes
1answer
110 views

Show that $L^1\subsetneq (L^\infty)^*$ [duplicate]

How does one show that $L^1\subsetneq (L^\infty)^*$? I am having trouble in this. Any help would be appreciated.
1
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1answer
27 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
2
votes
1answer
37 views

Closed ball in $l_p$ is also closed in $l_q$

I am trying to figure out if a closed unit ball in $l_p$ is also closed in $l_q$ for $1 \le p < q < \infty $. It looks easy at a first glance, but I got stuck pretty soon. I supposed there's a ...
1
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0answers
33 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
1
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1answer
34 views

A compactness argument for small high frequencies

I would like to prove the following statement: Let $N\geq 1$, $1\leq q<\infty$ and let be $E$ a relatively compact subset of $L^q(\mathbb{R}^N)$. Then \begin{equation*} \sup_{u\in ...
6
votes
1answer
101 views

Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
1
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1answer
56 views

Product of weakly convergent sequence and sequence boundedly convergent in measure

Question: Let $\Omega \subset \mathbb{R}^d$ be open and bounded, $f, f_n \in L^2 (\Omega)$ and $f_n \rightarrow f$ boundedly in measure (meaning that $f_n \rightarrow f$ in measure and $sup\ ...
2
votes
1answer
216 views

Convergent sequence in Lp has a subsequence bounded by another Lp function

For $E$ a measurable set and $1\leq p<\infty $, assume $f_n\to f$ in $L^p(E)$. Show that there is a subsequence $\{f_{n_k}\}$ and a function $g\in L^p(E)$ such that $|f_{n_k}|\leq g$ almost ...
0
votes
1answer
131 views

Show that $L^1$ is strictly contained in $(L^\infty)^*$

How does one show that $L^1$ is strictly contained in $(L^\infty)^*$? Here, $(L^\infty)^*$ is the space of linear continuous functionals on $L^\infty$.
0
votes
0answers
159 views

Set of infinitely differentiable functions compactly supported in a domain of $\mathbb{R}^n$ not dense in $L^\infty$

How does one show that the set of infinitely differentiable functions compactly supported in a domain $\Omega\subset\mathbb{R}^n$ is not dense in $L^\infty(\Omega)$? Thanks!
1
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1answer
172 views

Sobolev, Holder, Lp spaces continuous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
2
votes
1answer
208 views

$g, f, \hat {f} \in L^{1}(\mathbb R)\cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R) \implies \widehat{(fg)}= \hat{f} \ast \hat{g} ? $

Let $f, g\in L^{1}(\mathbb R)$ and it Fourier transform of $f$, $\hat{f} (y) = \int _ {\mathbb R} f(x) e^{-2\pi i x \cdot y} dx, \ (y\in \mathbb R)$ and the convolution of $f $ and $g$; $f\ast g ...
2
votes
1answer
39 views

Are the $L^p$ norms ordered by $p$?

A question left over from this post is: Are the $L^p$ norms ordered by $p$ like the power means are?