Several dimensional theta-summability and Hardy spaces

The d-dimensional Hardy spaces H-p (T-d1 x ... x T-dk) (d = d(1) + ... + d( k)) and a general summability method of Fourier series and Fourier transfor ms are introduced with the help of integrable functions theta (j) having in tegrable Fourier transforms. Under some conditions on theta (j) we show tha t the maximal operator of the theta -means of a distribution is bounded fro m H-p (T-d1 x ... x T-dk) to L-p (T-d) where p(0) < p < infinity and p(0) < 1 is depending only on the functions theta (j). By an interpolation theore m we get that the maximal operator is also of weak type (H-1(#i), L-1) (i = 1, ... , k) where the Hardy space H-1(#i) is defined by a hybrid maximal f unction and H-1(#i) = L-1 if k = 1. As a consequence we obtain that the the ta -means of a function integral is an element of H-1(#i) superset of L(log L)(k-1) converge a. e. to the function in question. If k = 1 then we get t his convergence result for all integral is an element of L-1. Moreover, we prove that the theta -means are uniformly bounded on the spaces H-p (T-d1 x ... x T-dk) whenever p(0) < p < infinity, thus the theta -means converge t o integral in H-p (T-d1 x ... x T-dk) norm. The same results are proved for the conjugate theta -means and for d-dimensional Fourier transforms, too. Some special cases of the theta -summation are considered, such as the Weie rstrass, Picar, Bessel, Fejer, Riemann, de La Vallee-Poussin, Rogosinski an d Riesz summations.