二维Vilenkin型系统的Dirichlet核的加权平均

Weighted average of Dirichlet kernels for two-dimensional Vilenkin-like systems

对二维Vilenkin型系统,我们定义加权平均极大算子T(i.e.Tf:=supn=(n1,n2)∈P2,β-1n1/n2β|Hnf|),并证明此算子是弱(1,1)型、强(p,p)型(1<p∞)以及(H,L)型,其中Hnf表示部分和的加权平均,H表示Hardy空间.借用此结果得到序列Hnf是几乎处处收敛于可积函数f.

For two-dimensional Vilenkin-like systems we define the weighted maximal operator TT（i.e. Tf：=supn=（n1,n2）∈p^2,β^-1≤n1/n2≤β｜Hnf｜）, where Hnf is the weighted average for partial sums and prove the operator T is of weak type （1, 1） and of type （p,p） for 1 〈 p ≤ ∞. As a consequence, we prove the a.e. convergence of sequence Hnf provided the quotient of the indices is bounded. Moreover the operator T is of type （H, L）, where H is the Hardy space.