Almost Everywhere Convergence And Weak-type Inequalities

In 1913,Luzin thought out a conjecture:the Fourier series of a continuous function in T1 was convergent almost everywhere. The conjecture was neither proved nor negated in the next fifty years.In the year 1965,Carleson proved the conjecture.He proved that the Fourier series of the function in L2(T) was convergent almost everywhere. Then Hunt spread the conclusion to LP(T)(1< p<∞). In this paper,we will go a step further to study the relationship between the weak-type inequality and convergence a.e. First, the paper discusses that the partial sums of the Fourier series of the function in L2(T) is equivalent to the weak (2,2) type inequality of the corresponding maximal operator which is called Carlderon theorem. Then the paper discusses that in L1(T) there exists a Fourier series which is divergent a.e.,which is called Kolmogorov theorem.At the same time, I will structure an a.e. divergence Fourier series.The main work of this paper is to make the proving process of the two theorems precise and simplify the proving process.