Note on estimates for the growth order of sequences of multiple rectangular Fourier sums

Let $\{S_{\mathbf n_k}(f,\mathbf x)\}_{k=1}^\infty$ be some sequence of rectangular partial sums of the multiple trigonometric Fourier series of a function $f$, and let $\{\lambda_k\}_{k=1}^\infty$ be a nondecreasing sequence of positive numbers. We investigate conditions on the belonging of the function $f$ to the classes $\varphi (L)$ under which estimates of the following form are possible:
$$ S_{\mathbf n_k}(f,\mathbf x)=o(\lambda_k )\quad\text{a.e.} $$
where the right-hand side depends on $k$ only.