Estimates for sums of moduli of blocks from trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets $A_k\subset\mathbb Z$ and a sequence of functions $\lambda_k\colon A_k\to\mathbb C$ provide the existence of a number $C$ such that any function $f\in L_1$ satisfies the inequality $\|U_{\mathcal A,\Lambda}(f)\|_p\le C\|f\|_1,$and what is the exact constant in this inequality? Here, $U_{\mathcal A,\Lambda}(f)(x)=\sum_{k=1}^\infty\big|\sum_{m\in A_k}\lambda_k(m)c_m(f)e^{imx}\big|$, and $c_m(f)$ are Fourier coefficients of the function $f\in L_1$. Problem 2: what conditions on a sequence of finite subsets $A_k\subset\mathbb Z$ guarantee that the a function $\sum_{k=1}^\infty\big|\sum_{m\in A_k}c_m(h)e^{imx}\big|$ belongs to $L_p$ for every function $h$ of bounded variation?