International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 4, Pages 817-820
On the convergence of Fourier series
Department of Mathematics, Auburn University, 36849, Alabama, USA
Received 5 March 1984
Copyright © 1984 Geraldo Soares de Souza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We define the space . Each is a special -atom, that is, a real valued function, defined on , which is either or , where is an interval in , is the left half of and is the right half. denotes the length of and the characteristic function of . is endowed with the norm , where the infimum is taken over all possible representations of . is a Banach space for . is continuously contained in for , but different. We have THEOREM. Let . If then the maximal operator maps into the Lorentz space boundedly, where is the -sum of the Fourier Series of .