Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets Ak ⊂ ℤ and a sequence of functions λk: Ak → ℂ provide the existence of a number C such that any function f ∈ L1 satisfies the inequality ‖UA,Λ(f)‖p ≤ C‖f‖1 and what is the exact constant in this inequality? Here, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}$\end{document} and cm(f) are Fourier coefficients of the function f ∈ L1. Problem 2: what conditions on a sequence of finite subsets Ak ⊂ ℤ guarantee that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}$\end{document} belongs to Lp for every function h of bounded variation?