On the Riemann summability of Fourier integrals and real Hardy spaces

We consider the Riemann means of single and multiple Fourier integrals of f unctions belonging to L-1 or the real Hardy. spaces defined on R-n, where n greater than or equal to 1 is an integer. We prove that the maximal Rieman n operator is bounded both from H-1(R) into L-1(R) and from L-1(R) into wea k- L-1(R). We also prove that the double maximal Riemann operator is bounde d from the hybrid Hardy spaces H-(1,H-0)(R-2), H-(0,H-1)(R2) into weak-L-1 (R-2). Hence pointwise Riemann summability of Fourier integrals of function s in H-(1,H- 0) boolean OR H-(0,H- 1)(R-2) follows almost everywhere. The m aximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.