Fourier Series,Equivalent Formulations Of Convergence In Norm And Almost Everywhere

In Fourier analysis,it is necessary to study the convergent problem of Fourier series.In 1966,Carleson proved Luzin conjecture.For (?)f L2(T),the Fourier series is convergence almost everywhere in T. U.S. mathematician R.Hunt proved that Fourier series of functions in LP(1< p< oo) space are convergent almost everywhere.There are multiple forms of convergence.Fourier series is also convergence in norm,this is a difficult problem.This paper will take advantage of operator norm bounded to solve this problem. Calderon proved that Fourier series is convergent almost everywhere for (?)f L2(T) is equivalent to s* is weak (2,2) inequality.This paper will prove for (?)f L1(T) is equivalent to s* is weak (1,1) inequality. Kolmogorov once over take advantage of constructing a integrable function to prove that exist f ∈ L1(T) of Fourier series is divergence almost everywhere. The prove in this paper is contrary to Kolmogorov.It will take advantage of a new way to give that Fourier series partical sum s* is not unmatched weak (1,1) for (?)f L1(T). Thereupon we can get that exist Fourier series is divergence almost everywhere in T for f ∈ L(T).