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
33 views

weak $L_p$ implies bounded integral on finite measure set

Let $(X, \mu)$ be a measure space which is $\sigma$-finite. $ 1 < p < \infty $. $f : X \to \mathbb C$ is a measurable function. If we know $f$ is in the weak $L_p$ space, i.e. $ ||f||_{L^{(p, ...
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0answers
48 views

the dual space of $L^p$ [duplicate]

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
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1answer
33 views

how to show this function doesn't belong to Hilbert space?

I am trying to show $\chi_{B_R(0)}(x) \notin H^1 (\mathbb{R}^n)$ , ∀R>0. since $H^1 (\mathbb{R}^n) := W^{1,2}(\mathbb{R}^n)$ That is, I have to show that $\chi_{B_R(0)} (x) \notin ...
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1answer
105 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
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2answers
110 views

Inequality of strong $L^p$ and weak $L^p$ norm on a finite set with counting measure

If $X$ is a counting finite set with counting measure. Let $f : X \to \mathbb C$ be a complex valued function. For any $ 0 < p < \infty$, show that $$ ||f||_{ L^p } \le C_p (\log (1 + |X| ...
1
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1answer
46 views

Inequality of $L^p$ norm and distribution function

Let $(X,\mu)$ be a measure space and $ f \in L^p$, $ 0 < p < \infty$. Let $ \lambda_f (t) : = \mu ( \{ x \in X : |f(x)| \ge t \} )$. I want to that exists a constant $c_p$ depending only on $p$ ...
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1answer
51 views

Criteria to be in weak $L^{p}$ space

Let $X$ be a $\sigma$-finite measure space. Let $f : X \rightarrow \mathbb{C}$ be a measurable function and $1 < p < \infty$. Suppose for $f$ there is a constant $C$ such that ...
1
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1answer
44 views

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ almost uniformly?

$f_n\rightarrow f$ in $L_2$ and $f_n\rightarrow f$ in measure How to show or give an counterexample: $f_n\rightarrow f$ almost uniformly. We believe it is false. Since both convergences imply there ...
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0answers
28 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
2
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0answers
49 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
0
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1answer
37 views

How do you prove $L^{\infty}$ is a C*-algebra?

If we define on $L^{\infty}$ the essential supremum norm ($\| \|_{\infty}$), then how can I prove this norm is submultiplicative ($\| T_1T_2\|_{\infty}\leq \| T_1\|_{\infty}\|T_2 \|_{\infty}\, \forall ...
2
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1answer
25 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
2
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3answers
59 views

If $f\in L^1$ has a compact support and $0 \leq p \leq1$ then $|f|^p\in L^1$

My text proved that If $f\in L^1$ is bounded and $p \geq1$ then $|f|^p\in L^1$ I wanted to prove the seemingly very similar statement: If $f\in L^1$ has a compact support and $0 \leq p ...
1
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1answer
33 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
60 views

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

Is there $f:\ell^1\to \ell^\infty$ so that $f$ is surjective $\forall x,y\in \ell ^1, \|x-y\|_1=\|f(x)-f(y)\|_\infty$
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0answers
36 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
37 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
41 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
23 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, ...
1
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1answer
37 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
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1answer
53 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
77 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
27 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
71 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
23 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
36 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
57 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
56 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
42 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 ...
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0answers
17 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
43 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
45 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
77 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
61 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
45 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
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1answer
41 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
125 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 ...
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2answers
41 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 ...
10
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2answers
156 views

Finite dimensional subspace of $C([0,1])$

Let $S$ be a subspace of $C([0,1])$, i.e. the continuous real functions on $[0,1]$. Assume that there exists $c>0$ such that $\|f\|_\infty\leq c \|f\|_2$ for all $f\in S$. Then $S$ must be ...
3
votes
0answers
32 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
22 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
36 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
56 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
34 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
57 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
40 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
33 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
vote
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.