On $(H,k)$-summability of multiple trigonometric Fourier series

A theorem is proved from which, in particular, it follows that if $f\in L(\ln^+L)^{N-1}$ on $T^N\equiv[-\pi,\pi]^N$, then the multiple trigonometric Fourier series of $f$ and all conjugate series are $(H,k)$-summable almost everywhere on $T^N$ for every $k>0$.
In the case where $f\in L(\ln^+L)^{N+1}$ this result was obtained by Marcinkiewicz (Collected papers, PWN, Warsaw, 1964).
That it is unimprovable, in a certain sense, follows from a result of Saks (On the strong derivatives of functions of intervals, Fund. Math. 25 (1935), 235–252).
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