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

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
$$ \|f\|_{\Lambda_q(\omega)}:=\begin{cases} \displaystyle \biggl(\int_0^1(f^*(t)\omega(t))^q\frac{dt}{t}\biggr)^{\frac1q} & \text{for } 0<q<\infty, \\[4mm] \displaystyle \sup_{0\leq t\leq 1}f^*(t)\omega(t) & \text{for } q=\infty. \end{cases} $$

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
$$ \|a\|_{\lambda_q(\mu)}:= \begin{cases} \displaystyle \biggl(\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k\biggr)^{\frac1q} & \text{for } 0<q<\infty, \\[4mm] \displaystyle \sup_{k}a_k^*\mu(k) &\text{for } q=\infty. \end{cases} $$

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
  $$ \biggl|\int_e\varphi_k(x)dx\biggr|\leq B \min(|e |, 1/k), $$
  
• 2) for every segment $w$ from ${\mathbb N}$ and $t\in (0,1]$ we have that
  $$ \biggl(\sum_{k\in w}\varphi_k(\,\cdot\,)\biggr)^*(t)\leq B\min(|w |, 1/t), $$
  where $ \left(\sum_{k\in w}\varphi_k(\cdot)\right)^*(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:
$$ \begin{aligned} A_{\delta}&:=\{\omega(t): \omega(t)t^{-\frac12-\delta} \text{ is an increasing function and } \omega(t)t^{-1+\delta } \text{ is a decreasing function}\}, \\ B_{\delta}&:=\{\omega(t): \omega(t)t^{-\delta} \text{ is an increasing function and } \omega(t)t^{-1+\delta}\text{ is a decreasing function}\}, \end{aligned} $$
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 $f\stackrel{\text{a.e.}}{=}\sum_{k=1}^\infty a_k\varphi_k$. If $f$ is a nonnegative and a nonincreasing function, then
$$ \biggl(\int_0^1 (f(t)\omega (t ) )^q\frac{dt}{t}\biggr)^{\frac1q} \approx \biggl (\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k\biggr)^{\frac1q}, $$
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
$$ |f(x)|\leq M\frac{1}{x}\biggl|\int_{0}^xf(t)\,dt\biggr|,\qquad x>0. $$

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 $f\stackrel{\text{a.e.}}{=}\sum_{k=1}^\infty a_k\varphi_k$. If $f$ is a nonnegative and a generalized monotone function, then
$$ \|f\|_{\Lambda_q(\omega, [0,1])}\approx \biggl(\sum_{k=1}^\infty(a_k^*\mu(k))^q\frac1k\biggr)^{\frac1q}, $$
where $\mu(k)=k\omega(1/k)$.