Spherical partial sums of the double Fourier series of functions of bounded generalized variation

The spherical partial sums of the double Fourier series of functions in the Waterman classes are studied. The main result of the paper is as follows.
Theorem 1. {\it Let $\Lambda_\varepsilon =\biggl\{\dfrac{n^{3/4}}{(\ln(n+1))^{1/2+\varepsilon}}\biggr\}_{n=1}^\infty$ for $\varepsilon>0$. Let $f(x,y)\in\Lambda_\varepsilon BV(T^2)$ and let
\begin{align*} I_r(f)&=\sup_{x,y\in T}\sup_{u,v\in[-1,1]}J_r(f) \\ &=\sup_{x,y\in T}\sup_{u,v\in[-1,1]}\sum_{r-1<|(m,n)|\leqslant r+1}|a_{m,n}(\psi_{x,y,u,v})|\leqslant C \end{align*}
for $r\geqslant 1$, where
$$ \psi _{x,y,u,v}(s,t)=\psi (s,t)=f(x+t,y+s)w(t)w(s)e^{-i(tu+sv)}, \quad and\quad w(\tau)=\frac\tau{2\sin(\theta/2)}\,. $$
Then
$$ \sup_{R\geqslant 1}\sup _{(x,y)\in T^2}|S_R(f,x,y)|\leqslant C(f,\varepsilon). $$
for each $R\geqslant 1$.}
Problem of circular convergence of Fourier series of the characteristic function of plane convex sets are also considered.