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.
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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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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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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 ...
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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 $||\...
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$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 $A\subseteq \...
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$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 ...
- 433
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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 ...
- 596
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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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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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