Questions tagged [lp-spaces]
For questions about $L^p$ spaces. That is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence classes of measurable functions such that $|f|^p$ is $\mu$-integrable. Questions can be about properties of functions in these spaces, or when the ambient space in a problem is an $L^p$ space.
5,707
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estimation of different $L^p$ norms [closed]
I am wondering if it is possible to find a constant $C=C(p,T)$ such that
$\mathbb E[\int_0^T|Y_t|^p\mathrm{d} t]\le C(p,T) \mathbb E[\sup_{t\in [0,T]}|Y_t|^2],$
where $p>1$, $T$ some finite time ...
4
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1
answer
81
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weak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.
My proof:
Since $f^\pm ...
- 1,575
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1
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71
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How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
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-2
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49
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Limits of functions in $L^p$ spaces and Hölder inequality
I have a severe problem understanding $L^p$ spaces and everything related. For example, see my thoughts on the following exercise:
Let $f_n \in L^1(0,1) \cap L^2(0,1)$ for $n = 1, 2, 3, \ldots$ and ...
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0
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26
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Why are limits of $L^p$ sequences defined almost surely? [closed]
I have heard it said that if a sequence of random variables $\{X_n\}$ converges in $ L^p $, then it converges to a limit $ X $ that is defined almost surely. I am trying to understand the precise ...
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3
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98
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Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is defined as $T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}},$
Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm.
$T$ is defined as $$T(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}...
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2
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Prove that $T$ is not a compact operator.
Let $T:\ell_{2}(\mathbb{Z})\rightarrow \ell_{2}(\mathbb{Z})$ be the operator defined by,
$$T((x_i)_{i\in \mathbb{Z}})=((y_i)_{i\in \mathbb{Z}}).$$
where
$$
y_{j}=\frac{x_{j}+x_{-j}}{2}, \quad j \in \...
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1
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$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $𝐿^𝑝(𝑋) \cap 𝐿^\infty(𝑋)$, $p<\infty$, with the norm $||𝑓||_{𝐿^𝑝 \cap 𝐿^\infty}=||𝑓||_𝑝+||𝑓||_\infty$ is a Banach space. I imagine that if we remove the norm $||𝑓||_\infty$ ...
- 3,004
3
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1
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67
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Eliminating Neumann boundary condition for elliptic PDE
In his PDE book, Evans demonstrates that for elliptic PDEs with Dirichlet boundary condition, the boundary term can be eliminated:
I am now wondering if this also works with Neumann boundary ...
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1
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2
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235
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Is there a smooth function, which is in $L^1$, but not in$L^2$? [closed]
I am studying measure theory. While going over $L^p$-spaces I asked myself, whether there is $f\in C^\infty(\mathbb{R})$ s.t. $f\in L^1(\mathbb{R})\setminus L^2(\mathbb{R})$? I assume there could be ...
- 29
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Spectrum of the laplacian outside of a compact
Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
- 57
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Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$
Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$?
I know how to prove for $p=2$ ...
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Equivalence of Fourier Transform on $\ell_2(\mathbb{Z}_+)$ and $L_2(\mathbb(R)_+)$ via equivalence of $H_p( \mathbb{D})$ and $H_p(\mathbb{C}_+)$?
Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients.
We know that the Fourier Transform is an isometric ...
- 286
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38
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Weighted $L^2$ space on Torus.
I'm studying weighted $L^2$ spaces in the circle $[0,2\pi]$
Definition 1
A weight is a function $w\colon [0,2\pi]\to \mathbb{R}^+$ (non negative)
Definition 2 The weighted $L_w^2([0,2\pi])$ is defined ...
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What justifies the use of global coordinates when computing the $L^p(\mathbb{T}^n)$ norm?
Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:
$$\int_{\mathbb{T}^n}|f|^p < \infty....
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Weighted inequality on torus
In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
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What is wrong with this proof that a linear, bounded, time invariant operator on $L_p$ must be a convolution?
I'm trying to understand if this is true and how to prove it, "If $T$ is a bounded, time invariant operator on $L_p(\mathbb{R})$, then $T$ is a convolution operator.''
Here's an attempt at a ...
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26
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Is there a Hilbert space of Henstock–Kurzweil square-integrable integrable functions?
As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert ...
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2
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92
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Limit depending on parameter and $L^1$ function
What is the $\lim_{n\to\infty} n^a\int_0^1 \frac{f(x)dx}{1+n^2x^2}$ depending on $a\in\mathbb{R}$, if $f\in L^1(0,1)$?
By Banach-Steinhaus theorem I deduced that the limit is zero for $a\leq 0$, but I ...
- 6,515
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27
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Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statement
If $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then $||\...
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2
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$L^{\infty}$ (uniform) decay of Dirichlet heat equation $u_t=\Delta u$
Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^N$. Consider the following initial-boundary value problem for the heat equation:
\begin{equation}
\begin{cases}
u_t=\Delta u\quad\quad\quad\;...
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Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
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Is the set of $L^2([0,1])$ functions $g$ s.t. $g\circ \psi = f\circ \phi$ for fixed $f\in L^2$ and some $\phi_*(dx)=dx, \psi_*(dx)=dx$ closed?
Consider the $L^2$ space for the Lebesgue measure $dx$, i.e., the set of functions $f:[0,1]\to \mathbb{R}$ such that $\int_{0}^{1}|f(x)|^2dx<\infty$. Fix one function $f\in L^2$ and the space of $...
- 1,565
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$L_p$ inequality for measurable sets
Let $(U,\mu)$ a finite and positive measure space, and $1\leq p<\infty$. Suppose that for every $\varepsilon$ and measurable subset $A\subset U$, there exists a measurable subset $B\subset U$ such ...
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Measurability of $\int_\Omega \varphi(x)u(t,x) \mathrm{d}x$ for $\varphi \in L^1(\Omega)$ and $u$ in a Bochner space
I have a function $u \in L^\infty((0,\infty), L^\infty(\Omega))$ where $\Omega$ is a bounded domain. Take $\varphi \in L^1(\Omega)$ and consider
$$f(t) := \int_\Omega \varphi(x)u(t,x) \mathrm{d}x.$$
...
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How to interpret $L^2$ norm for functions from $[0,T]\to\mathbb{R}^n$?
I have a function $\alpha \in L^2(0,T;A)$ where $A\subseteq \mathbb{R}^n$.
I understand what it means when $A= \mathbb{R}$, i.e.
$$\Bigg(\int_0^T|\alpha(t)|^2dt\Bigg)^{1/2}<\infty$$
If $A\subseteq \...
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2
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$L^\infty(\Omega)$ is dense in $L^{p,\infty}(\Omega)$ if $\Omega$ is compact
Given a compact set $\Omega\subset \mathbb{R}^N$, I am wondering if $L^\infty(\Omega)$ is dense in the weak $L^p$ space $L^{p,\infty}(\Omega)$ with $1< p<\infty$ (see here the definition).
I ...
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1
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A problem in L1 space
Problem: Let $(X, \mathcal{A}, \mu)$ be a measure space. Let $f: X \to [0, \infty)$ be measurable. Then define the set $$A_f = \left\{g \in L^1 (\mu)\ |\ |g| \leq f\mbox{ a.e.} \right\}.$$
Prove the ...
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Weakly sequentially closed set in $L^p$
Let $\Omega\subset \mathbb{R}^n$ be bounded and Lebesgue measurable, $p \in [1,\infty]$, and $a,b \in L^p(\Omega)$. Consider the set
$$
K = \big\{ u \in L^p(\Omega):\, a(x) \leq u(x) \leq b(x) \, \...
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1
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A Multiplication operator in a Hilbert space: $M_h$ is bounded and $||M_h|| \leq || h||_{\infty}$ [duplicate]
I'm trying to understand the example below, taken from Axler's Measure Integration and Real Analysis book.
How does one prove that $M_h$ is bounded and that $||M_h|| \leq || h||_{\infty}$?
I was ...
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Dominated convergence theorem for $L^{\infty}$ with additionnal hypothesis of vanishing at infinity
Let $f\in L^{\infty}(\mathbb{R}^n, \mathbb{R})$. Denote $\chi_R$ the characteristic function on $B(0,R)\subset\mathbb{R}^n.$ If $\underset{\|x\|\to \infty}{\text{lim}} f(x) = 0$, then will
$\underset{...
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1
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Problem on density of a subset on $L^2([a,b],\mathbb{R})$: looking for a better solution
I have this problem that the professor gave us:
Let $\gamma ,a,b\in\mathbb{R}$ and $$D_{\gamma}=\{u\in C^2([a,b],\mathbb{R}):\gamma u(a)-u'(a)=0,\gamma u(b)-u'(b)=0\}$$
Prove that $D_\gamma$ is dense ...
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1
answer
45
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Weak Star Convergence of Integral Averages
Suppose $U$ is a Banach space and let $V\subseteq U$ be bounded, convex and (norm-)closed. Consider the Bochner-Lebesgue space $L^r(0,T;U)$ with $T>0$ and $r\in[1,\infty]$ consisting of strongly ...
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votes
2
answers
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$| |x + y|^p - |x|^p | \leq \epsilon |x|^p + C |y|^p$
I want to demonstrate that: Let $1 < p < \infty$; for any $\epsilon > 0$, there exists $C = C(\epsilon) \geq 1$ such that for all $x, y \in \mathbb{R}$, we have
$$ | |x + y|^p - |x|^p | \leq \...
- 43
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40
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Decay at infinity of $L^2(\mathbb{R}^n)$ functions
I am trying to justify that a (normalized) solution $\phi$ in $L^2(\mathbb{R}^n)$ of:
$-\Delta\phi+f(x)\phi=K\phi$, with $f(x)=0$ in $\Omega$, $f(x)=M$ in $\Omega^c$
has to vanish outside $\Omega$ ...
- 21
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0
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25
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Dirichlet Problem with $L^p$ Boundary Data
I am seeking a proof of the following result related to the Dirichlet problem with $L^p$ boundary data. I am not quite sure how to approach the proof. Does anyone know where I might find such a proof ...
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44
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Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$
I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
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1
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Proof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated, then $L^p(X,\mathscr{A},\mu)$ $1\leq p<+\infty$ is separable.
Background
I have some trouble understanding a step of the proof of the following proposition:
Proposition$\quad$ Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is $\...
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votes
1
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62
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For $1\le p < +\infty$ $L^p$ is a Banach space: Real and abstract analysis, Hewitt - Stromberg
I have some doubts about the proof of this theorem. From time to time I will put my justification.
For $1\le p < +\infty$, $L^p$ is a Banach space
Let $(f_n)_n$ be a Cauchy sequence in $L^p$, i.e., ...
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1
answer
38
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Proving that convergence of norms and convergence a.e. implies strong convergence
I have in my notes the following theorem
Theorem
$(Y,\mathcal{F},\mu)$ $\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that $$\lim_{n\...
- 516
4
votes
1
answer
150
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Proving that operator in $L^2[0,1]$ is compact
I need help with some functional analysis:
Let $A$ be a continuous linear operator on $L^2[0,1]$ and for any $f \in L^2[0,1]$ the function $Af$ is Lipschitz continuous. Show that $A$ is compact.
It is ...
- 51
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votes
1
answer
50
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$L_p$ norm estimate of a sum
Let $R>0$, and $(B_k)_{k \in \Bbb{N}}$ a collection of disjoint balls of radius R and let $f$ be a measurable function on $\mathbb{R}^n$ of the form
$$f = \sum_{k=1}^{\infty} a_k X_{B_k}$$
for ...
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votes
1
answer
67
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Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
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2
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96
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Understanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p<+\infty$)
Background
I have some questions when reading the proof of $L^p(X,\mathscr{A},\mu)$ is complete for $1\leq p<+\infty$. The proof is proceeded by showing that each absolutely convergent series in $L^...
- 2,493
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votes
1
answer
99
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Integral function of bounded variation function derivative
Let $f: [a,b] \to \mathbb{R}$ be bounded variation. So $f’$ exists almost everywhere, and let
$g(x):=\int_a^x f’(y)dy$.
(Due to the fact that it is possible that $f\notin AC([a,b])$ it is not ...
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0
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18
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Is weighted $L^p$ space isomorphically $L^p$ spaces?
Let $w:\mathbb{R}^n\to ]0,\infty[$ continuous (a weight)
Is the weighted $L^p$ spaces $L_w^p(\mathbb{R}^n)$ isomorphically to $L^p$ space?
My attempt: Let $L_p(\mathbb{R}^n)\to L_w^p(\mathbb{R}^n)\...
- 3,650
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1
answer
30
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Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?
Context. Throughout this post I will be dealing with the Lebesgue measure over $\mathbb R^n$. Moreover, I denote the measure of a measurable set $E \subset \mathbb R^n$ by $|E|$ and $\Omega \subset \...
- 1,141
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2
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141
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Understanding the proof of $L^{\infty}$ is complete.
I got lost when reading the proof of $L^{\infty}$ is complete. The book proceed the proof as follows:
We show that each absolutely convergent series in $L^{\infty}(X,\mathscr{A},\mu)$ is convergent. ...
- 2,493
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0
answers
40
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"Elementary" $L^2$ inequality for combination of compactly supported functions
I am reading the paper Ondelettes et poids du Muckenhoupt by Lemarié and, at some point, he needs to prove a certain operator is bounded in $L^2(\mathbb R)$ to apply weighted inequalities theory. The $...
- 283
2
votes
0
answers
78
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Convergent subsequences in $L^p(\mathbb R^n)$
This is a particular fact that I ended up proving in the process of attempting one of my recent homeworks, but I don't think I've seen this particular fact online even though it feels like a fairly ...
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