Fourier series and the maximal operator on the weighted special atom spaces (Q1824794)

Summary: For an interval I in [0,2\(\pi\) ] with halves L and R, a weighted special atom looks like \(b(t)=(1/\rho (| I|))[\chi_ L(t)-\chi_ R(t)],\) where \(\rho\) is a nonnegative function satisfying some properties. We consider the weighted special atom space B(\(\rho)\) formed by \(\ell^ 1\) linear combinations of these weighted atoms. We show that if \(f\in B(\rho)\) then its Fourier series converges almost everywhere, using the Carleson-Hunt idea on their famous result about the almost everywhere convergence on \(L_ p\)-spaces.