Relations between functions from some Lorentz type spaces and summability of their Fourier coefficients

Abstract: This Licentiate Thesis is devoted to the study of summability of the Fourier coefficients for functions from some Lorentz type spaces and contains three papers (papers A - C) together with an introduction, which put these papers into a general frame.

Let $Lambda_p(omega),;; p>0,$ denote the Lorentz spaces equipped with the (quasi) norm
$$
|f|_{Lambda_p(omega)}:=left(int_0^1left(f^'(t)omega(t) ight)^pfrac{dt}{t} ight)^{frac1p}
$$
for a function $f$ on [0,1] and with $omega$ positive and equipped with some additional growth properties.
In paper A some relations between this quantity and some corresponding sums of Fourier coefficients are proved for the case with a general orthonormal bounded system. Under certain circumstances even two-sided estimates are obtained.

In paper B we study relations between summability of Fourier coefficients and integrability of the corresponding functions for the generalized spaces $Lambda_p(omega)$ in the case of a regular system. For example, all trigonometrical systems, the Walsh system and Prise's system are special cases of regular systems. Some new inequalities of Hardy-Littlewood-P'{o}lya type with respect to a regular system for the generalized Lorentz spaces $Lambda_p(omega)$ are obtained. It is also proved that the obtained results are in a sense sharp.

The following inequalities are well-known:
egin{equation}label{f--}
c_1left|overline{f} ight|_{L_pleft[0,1 ight]}^pleq sum_{k=1}^{infty}k^{p-2}|a_k|^{p}leq
c_2left|tf' ight|_{L_pleft[0,1 ight]}^p,;;; ext{for};1end{equation}
where $overline{f(t)}=frac1tleft|int_0^tf(s)ds ight|$ and $f'(t)$ is the derivative of the function $f(t).$ (Here ${a_k}_{k=1}^infty $ are the Fourier coefficients of the function $f$). In paper C we prove some analogues Hardy-Littlewood-P'{o}lya type inequalities eqref{f--} with respect to the regular system for generalized Lorentz spaces $Lambda_{p}(omega).$