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.
5,902 questions
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Evaluate the norm of an integral operator $J$ defined on $L^{2}([0,1])$
I am trying to solve the following question (this question is taken from an old functional analysis exam)
Let $J$ be an operator on $L^{2}([0,1])$ defined by
$$Jf(s) = \int_{0}^{1} K(s,t) \cdot f(t) ...
- 894
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0
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Reverse Minkowski inequality
I'm working at the 2019 József Wildt International Mathematical Competition,
problem W33, Let $0<\frac{1}{q}\le\frac{1}{p}<1$, and
$$
\frac{1}{p}+\frac{1}{q}=1.
$$
Let $u_{k},v_{k},a_{k}$ and $...
0
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0
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Construction of Conditional Expectation: The reason why a bounded Random Variable $X \in L^1$ implies it is in $L^2$.
I have been revising measure theory from Capinski and Kopp's "Measure, Integral and Probability." The final section of Chapter $5$ deals with Conditional Expectation (Page $153$, Section $5....
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Smoothness and dimensionality of the zero set of two weighted $p$-norms?
Given $s_1,s_2 \in \mathbb R^n$, $s_1 \neq s_2$, $w_1,w_2 \gt 0$, and $p \gt 1$.
Let
$$
\begin{align}
f_i(x)&= w_i\|x-s_i\|_p,\\
h(x)&=f_1(x)-f_2(x),\\
S&=\{x : h(x)=0\}
\end{align}
$$ ...
- 1,570
1
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2
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45
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Bounded continuous functions on $(\mathbb{N},d_{disc})$
I am following the MIT course on functional analysis and have a hard time understanding the following construction:
“The set of all bounded sequences, $\ell^\infty$
, can be identified with $C_\infty(\...
- 17
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0
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66
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Why are local spaces more natural for hyperbolic systems?
I'm studying hyperbolic PDEs and I've noticed that local spaces such as $L^p_{\text{loc}}(\mathbb{R}^n)$ are often used instead of $L^p(\mathbb{R}^n)$. Why is this? Of course local spaces avoid ...
- 1,173
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1
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If $(g_n)$ is bounded in $L^\infty_{\rm loc}$ there is $g\in L^\infty_{\rm loc}$ such that $g_n\to g$ weakly in $L^p_{\rm loc}$ for $1<p<\infty$
I would like to show the following result.
Let $(g_n)_{n \ge 0}$ be a bounded sequence in $L^{\infty}_{{\rm loc}}(U)$, where $U$ is an open set of $\mathbb{R}^n$. Then there exists $g \in L^{\infty}_{...
0
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1
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Bounding the $L^p$ norm of a tensor product in terms of their individual $L^p$ norms
Let $f \in L^p(\mathbb{R}^n; \mathbb{R}^n)$ and $g \in L^q(\mathbb{R}^n; \mathbb{R}^n)$. Then $(f\otimes g)$ is an $n \times n$ matrix whose $ij$-th component is $f_i(x)g_j(x)$. Is it possible to get ...
- 1,173
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$f_n(x) := 1+\sin(n \pi x)$, $f_n^k$ converges weakly to a constant $c_k$. [closed]
I have the following exercise:
Let $p \in (1, +\infty)$. Consider the sequence $\left\{ f_n \right\}_{n \in \mathbb N}$ in $L^p ([0, 1])$ where $f_n (x) := 1 + \sin(n \pi x)$. Let $k \in \mathbb N$
...
- 664
2
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0
answers
157
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Symmetric graph norm
Consider the following problem: let $T$ be an operator $T:L^p(\mathbb C)\to L^p(\mathbb C)$,
which we assume to be continuous for all $p\in(1,\infty)$ (e.g., a CZ operator). Let $C(p)$ denote its norm,...
- 1,282
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0
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Conditions under which the potential energy is weakly continuous
This question is concerned with the following result in Lieb & Loss' Analysis:
More precisely, condition 11.3(14) is stated as follows:
My question is concerned with the case $n \geq 3$. The ...
- 692
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89
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Convolution with "good kernels" is $L^p$ convergent?
Question. Let $K_\delta:\mathbb{R}^d\to \mathbb{R}$ be a family of good kernel as usually defined: $K_\delta>0$; $\int_{\mathbb{R}^d}K_\delta =1$; and $\int_{|x|\geq\eta}K_\delta \to 0$ as $\delta\...
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Does uniform convergence on compact subsets of $\mathbb{R}$ imply weak convergence in $L_p(\mathbb{R})$?
Let $f_n$ be a sequence in $L^p(\mathbb{R})$ with $1 < p < \infty$ so that $f_n$ converges uniformly to $f$ on every compact subset of the real line. Find whether or not $f_n$ converges weakly ...
- 171
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$(X, \Sigma, \mu)$ finite measure space and $(f_n)_{n\in\mathbb{N}}$ is sequence in $L^p(X)$. If $f_n \to f$ uniformly on $X$, then $f \in L^p(X)$.
Help in understanding a proof written by a teacher on the following theorem.
Let $(X, \Sigma, \mu)$ be a finite measure space and let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions in $L^p(X)$.
...
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Step in Brezis' proof of chain rule for Sobolev spaces
I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and ...
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