Promotion Of Calderon Theorem In L~p (T~n)

In thirties of last century, Hardy and Littlewood use the maximal function to study Fourier series.It is large the function itself, and it can be used to control some operators of Fourier analysis.While the maximal operator is the operator of the function which reflects to it's maximal function.For the general sublinear operator series, if i t's a.e. convergent in the dense subspace of a Lp space, and it's corresponding maximal operator is weak (p,q), then it is a.e. convergent in this LP space.Otherwise, if the operator series is a.e. convergent in a LP space, then the type of it's corresponding maximal operator is worth to study.For every function in LP space (p≥1, n≥1), Let SNf(x)=∑[k]≤N cke2πik·x be the square summability of Fourier series,andk= (k1,k2,...,kn), [k]=max(|k1|,|k2|,...,|kn|). A.P.Calderon first proved that when p= 2,n= 1, if SNf(x) is a.e. convergent,then it's corresponding maximal operator S*f(x)= supN|SNf(x)|is weak (p,p).This conclusion was presented in Zygmund's book in 1965. This paper is devoted to prove that when p≥1, n≥1, Thorough the specificity of the Fourier series of trigonometric polynomials, the a.e. convergence of SNf(x) and it's corresponding maximal operator S*f(x) are equivalent.