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Evaluation of the limits of maximal means
O. P. Filatov Samara State University
Abstract:
It is proved that the limit
$$
\lim_{\Delta\to\infty}\sup_\gamma\frac 1\Delta
\int_0^\Delta f\bigl(\gamma(t)\bigr)\,dt,
$$
where
$f\colon\mathbb R\to\mathbb R$ is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation
$\dot\gamma\in[\omega_1,\omega_2]$, coincides with the limit
$$
\lim_{T\to\infty}\sup_{c\ge0}\varphi_f(k,T,c),
$$
where
$$
\varphi_f=\frac{(k-1)\overline I_f(T,c)}
{1+(k-1)\overline\lambda_f(T,c)},\qquad
k=\frac{\omega_2}{\omega_1}.
$$
Here
$\overline\lambda_f=\lambda_f/T$,
$\overline I_f=I_f/T$, and
$\lambda_f$ is the Lebesgue measure of the set
$$
\bigl\{\gamma\in[\gamma_0,\gamma_0+T]:
f(\gamma)\ge c\bigr\}=A_f,\qquad
I_f=\int_{A_f}f(\gamma)\,d\gamma.
$$
It is established that this limit always exists for almost-periodic functions
$f$.
UDC:
517.828
Received: 03.11.1994
DOI:
10.4213/mzm1770
Fulltext:
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