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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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{...
- 57
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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 ...
- 94
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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 ...
- 23
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2
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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
$$ | |x + y|^p - |x|^p | \leq \...
- 43
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Decay at infinity of $L^2(\mathbb{R}^n)$ functions
I am trying to justify that a (normalized) solution $\phi$ in $L^2(\mathbb{R}^n)$ of:
$-\Delta\phi+f(x)\phi=K\phi$, with $f(x)=0$ in $\Omega$, $f(x)=M$ in $\Omega^c$
has to vanish outside $\Omega$ ...
- 21
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Dirichlet Problem with $L^p$ Boundary Data
I am seeking a proof of the following result related to the Dirichlet problem with $L^p$ boundary data. I am not quite sure how to approach the proof. Does anyone know where I might find such a proof ...
- 27
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Is $u(x)=\frac{1}{|x|^{\alpha}}$ in $W^{1,p}(B_1(0))$?
Consider a function $$ u(x)=\frac{1}{|x|^{\alpha}} \quad x\in B_1(0) \subset \mathbb{R}^N. $$
I should find condition about $p, N, \alpha$ for $u$ to be in $W^{1,p}(B_1(0))$. Following different books ...
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Proof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated, then $L^p(X,\mathscr{A},\mu)$ $1\leq p<+\infty$ is separable.
Background
I have some trouble understanding a step of the proof of the following proposition:
Proposition$\quad$ Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is $\...
- 2,493
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For $1\le p < +\infty$ $L^p$ is a Banach space: Real and abstract analysis, Hewitt - Stromberg
I have some doubts about the proof of this theorem. From time to time I will put my justification.
For $1\le p < +\infty$, $L^p$ is a Banach space
Let $(f_n)_n$ be a Cauchy sequence in $L^p$, i.e., ...
- 47
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1
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Proving that convergence of norms and convergence a.e. implies strong convergence
I have in my notes the following theorem
Theorem
$(Y,\mathcal{F},\mu)$ $\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that $$\lim_{n\...
- 516
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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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1
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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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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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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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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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