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Международная конференция по функциональным пространствам и теории приближения функций, посвященная 110-летию со дня рождения академика С. М. Никольского
25 мая 2015 г. 18:20, Функциональные пространства, г. Москва, МИАН
 


Boas theorem for Lorentz spaces $\Lambda_q(\omega)$

A. N. Kopezhanova

L. N. Gumilyov Eurasian National University
Материалы:
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Аннотация: Let $0<q\leq\infty$ and $\omega$ be a nonnegative function on $[0,1]$. The generalized Lorentz spaces $ \Lambda_q(\omega)$ consists of the functions $f$ on $[0,1]$ such that $\|f\|_{\Lambda_q(\omega)}<\infty$, where
$$ \|f\|_{\Lambda_q(\omega)}:=\begin{cases} \displaystyle (\int_0^1(f^*(t)\omega(t))^q\frac{dt}{t})^{\frac1q} & for 0<q<\infty,
[4mm] \displaystyle \sup_{0\leq t\leq 1}f^*(t)\omega(t) & for q=\infty. \end{cases} $$

These spaces $\Lambda_q(\omega)$ coincide to the classical spaces $L_{pq}$ in the case $\omega(t)=t^{\frac1p},$ $1<p<\infty$ (see [N389:1]).
Let $\mu=\{\mu(k)\}_{k\in {\mathbb N}}$ be a sequence of positive number and the space $\lambda_q(\mu)$ consists of all sequences $a=\{a_k\}_{k=1}^\infty$ such that $\|a\|_{\lambda_q(\mu)}<\infty$, where
$$ \|a\|_{\lambda_q(\mu)}:= \begin{cases} \displaystyle (\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k)^{\frac1q} & for 0<q<\infty,
[4mm] \displaystyle \sup_{k}a_k^*\mu(k) &for q=\infty. \end{cases} $$

Here $\{a_k^*\}_{k=1}^\infty$ is the nonincreasing rearrangement of the sequence $\{|a_k|\}_{k=1}^\infty$. Boas theorem was generalized also for more general Lorentz spaces $\Lambda_q(\omega)$ in 1974 by L.-E. Persson for the case when $\Phi=\{e^{2\pi ikx}\}_{k=-\infty}^{+\infty}$ is the trigonometric system (see [N389:2]).
Let the function $f$ be periodic with period 1 and integrable on $[0,1]$ and let $\Phi=\{\varphi_k\}_{k=1}^\infty $ be an orthonormal system on $[0,1]$. The numbers
$$ a_k=a_k(f)=\int_0^1f(x)\overline{\varphi_k(x)}dx, \qquad k\in {\mathbb N} $$
are called the Fourier coefficients of the functions $f$ with respect to the system $\Phi=\{\varphi_k\}_{k=1}^\infty$.
We say that the orthonormal system $\Phi=\{\varphi_k\}_{k=1}^\infty$ is regular if there exists a constant $B$, such that
  • 1) for every segment $e$ from $[0,1]$ and $k\in{\mathbb N}$ it yields that
    $$ |\int_e\varphi_k(x)dx|\leq B \min(|e |, 1/k), $$
  • 2) for every segment $w$ from ${\mathbb N}$ and $t\in (0,1]$ we have that
    $$ (\sum_{k\in w}\varphi_k( \cdot ))^*(t)\leq B\min(|w |, 1/t), $$
    where $ (\sum_{k\in w}\varphi_k(\cdot))^*(t)$ as usual denotes the nonincreasing rerrangement of the function $\sum_{k\in w}\varphi_k(x)$.

Examples of regular systems are all trigonometric systems, the Walsh system and Price's system. In [N389:3], [N389:4], [N389:5] some results were obtained with respect to the regular system using network space.
Let $\delta>0$ be a fixed parameter. Consider a nonnegative function $\omega(t)$ on $[0,1]$. We define the following classes:
$$ \begin{aligned} A_{\delta}&:=\{\omega(t): \omega(t)t^{-\frac12-\delta} is an increasing function and \omega(t)t^{-1+\delta } is a decreasing function\},
B_{\delta}&:=\{\omega(t): \omega(t)t^{-\delta} is an increasing function and \omega(t)t^{-1+\delta} is a decreasing function\}, \end{aligned} $$
Then the classes $A,$ $B$ can be defined as follows: $A=\bigcup_{\delta>0}A_{\delta}$, $B=\bigcup_{\delta>0}B_{\delta}$.
The main results of this work are the following generalizations of the Boas theorem.
\begin{etheorem} Let $1\leq q\leq\infty$ and $\omega\in B.$ Let $\Phi=\{\varphi_k\}_{k=1}^\infty$ be a regular system and let $f\stackrel{a.e.}{=}\sum_{k=1}^\infty a_k\varphi_k$. If $f$ is a nonnegative and a nonincreasing function, then
$$ (\int_0^1 (f(t)\omega (t ) )^q\frac{dt}{t})^{\frac1q} \approx (\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k)^{\frac1q}, $$
where $\mu(k)=k\omega(1/k)$. \end{etheorem}
We say that a function $f$ on $[0,1]$ is generalized monotone if there exists some constant $M>0$ such that
$$ |f(x)|\leq M\frac{1}{x}|\int_{0}^xf(t) dt|,\qquad x>0. $$

Our next main result reads.
Теорема. Let $1\leq q\leq\infty$ and $\omega\in A$. Let $\Phi=\{\varphi_k\}_{k=1}^\infty$ be a regular system and let $f\stackrel{a.e.}{=}\sum_{k=1}^\infty a_k\varphi_k$. If $f$ is a nonnegative and a generalized monotone function, then
$$ \|f\|_{\Lambda_q(\omega, [0,1])}\approx (\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k)^{\frac1q}, $$
where $\mu(k)=k\omega(1/k)$.

Материалы: abstract.pdf (176.7 Kb)

Язык доклада: английский

Список литературы
  1. J. Bergh, J. Löfström, Interpolation spaces. An Introduction, Springer Verlag, Berlin, 1976  zmath
  2. L.-E. Persson, Relation between regularity of periodic functions and their Fourier series, Ph.D thesis, Dept. of Math., Umeå University, 1974  zmath
  3. E. D. Nursultanov, “On the coefficients of multiple Fourier series from $L_p$-spaces”, Izv. Math., 64:1 (2000), 93–120  mathnet  crossref  mathscinet  zmath  isi  scopus
  4. E. D. Nursultanov, “Net spaces and the Fourier transform”, Dokl. Akad. Nauk, 361:5 (1998), 597–599  mathnet  mathscinet  zmath  isi  scopus
  5. E. D. Nursultanov, “Network spaces and inequalities of Hardy–Littlewood type”, Sb. Math., 189:3-4 (1998), 399–419  mathnet  crossref  mathscinet  zmath  isi  scopus


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