It is proved a BMO -estimation for quadratic partial sums of two-dimensional Fourier series from which it is derived an almost everywhere exponential summability of quadratic partial sums of double Fourier series. n k=0 S k (x, f ) be the (C, 1) means of (1). Fejér [1] proved that σ n (f ) converges to f uniformly for any 2π-periodic continuous function. Lebesgue in [15] established 0 2010 Mathematics Subject Classification: 40F05, 42B08

Let G be a stratified, nilpotent Lie group and let L be a homogeneous sublaplacian on G. Let E(X) denote the spectral resolution of L on L2(G). Given a function K on R+ , define the operator TK on L2(G) by Tkf= j$> K(\) dE(\)f. Sufficient conditions on K to imply that TK is bounded on L\G) and the maximal operator K*(p(x) = supr>01 TKq>(x) | (where K,(X) = K(tXJ) is of weak type (1,1) are given. Picking a basis e0, e,,.

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In this paper we study the exponential uniform strong approximation of two-dimensional Walsh-Fourier series. In particular, it is proved that the two-dimensional Walsh-Fourier series of the continuous function f is uniformly strong summable to the function f exponentially in the power 1/2. Moreover, it is proved that this result is best possible.

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