On the convergence of Fourier–Laplace series

In the present paper we prove the following theorem. For any $\varepsilon>0$ there exists a measurable set $G\subset S^3$ with measure $mes G>4\pi-\varepsilon$, such that for each $f(x)\in L^1(S^3)$ there is a function $g(x)\in L^1(S^3)$, coinciding with $f(x)$ on $G$ with the following properties. Its Fourier–Laplace series converges to $g(x)$ in metrics $L^1(S^3)$ and the inequality holds $\displaystyle\sup_N||\sum_{n=1}^N Y_n[g,(\theta, \varphi)]||_{L^1(S^3)}\ll 3||g||_{L^1(S^3)}\leq12||f||$.