2
votes
0answers
54 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
0
votes
0answers
30 views

rising sun inequality [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
4
votes
3answers
95 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
2
votes
1answer
40 views

Koopman is not surjective

I want to prove that the Koopman operator $U_f : L^2 (\mu) \rightarrow L^2 (\mu)$ such that $U_f(\phi) =\phi \circ f$ is not surjective. Where $ \mu $ is a measure preserving mapping $f$. I was ...
1
vote
0answers
65 views

Means of sequence of functions

Let $(\Omega_n)$ be a sequence of subsets of $\mathbb{R}^d$ with $\Omega_n\uparrow\mathbb{R}^d$ where the Lebesgue measure $\lambda^d(\Omega_n)$ is finite for every $n$. Let $(f_n)$ be a sequence of ...