Abstract. For a Lebesgue integrable complex-valued function $f$ defined over the $m$-dimensional torus ${\msbm T}^m:=[0,2\pi )^m$, let $\hat f({\bf n})$ denote the Fourier coefficient of $f$, where ${\bf n}=(n^{(1)},\ldots,n^{(m)})\in{\msbm Z}^m$. Recently, in [{\it Acta Math. Hungar.}, \bf128 \rm(2010), 328--343], we have defined the notion of bounded $p$-variation ($p\ge1$) for a complex-valued function on a rectangle $[a_1,b_1]\times\cdots \times[a_m,b_m]$ and studied the order of magnitude of Fourier coefficients of such functions on $[0,2\pi ]^m$. In this paper, the order of magnitude of Fourier coefficients of a function of bounded $p$-variation ($p\ge1$) from $[0,2\pi ]^m$ to ${\msbm C}$ and having lacunary Fourier series with certain gaps is studied and a result analogous to Theorem 2 in [\it Acta Math. Hungar.\rm, \bf104 \rm(2004), 95--104] and Theorem 2 in [\it Acta Math. Hungar.\rm, \bf128 \rm(2010), 328--343] is proved.

AMS Subject Classification (1991): 42B05, 26B30, 26D15

Keyword(s): multiple Fourier coefficient, function of bounded $p$-variation in several variables, order of magnitude

Received April 1, 2011, and in revised form June 26, 2011. (Registered under 17/2011.)