On (H_pq, L_pq)-type inequality of maximal operator of Marcinkiewicz-Fejér means of double Fourier series with respect to the Kaczmarz system (original) (raw)

On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

2009

The main aim of this paper is to prove that the maximal operator of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system is bounded from the dyadic Hardy-Lorentz space H pq into Lorentz space L pq for every p > 2/3 and 0 < q ≤ ∞. As a consequence, we obtain the a.e. convergence of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system. That is, σ n (f, x 1 , x 2) → f (x 1 , x 2) a.e. as n → ∞. lim sup n→∞ D κ n (x) log n ≥ C > 0 holds a.e. Consequently, it is harder to obtain pointwise convergence results for Walsh-Kaczmarz-Fourier series than for Walsh-Fourier series. In 1974 Schipp [14] and Young [18] proved that the Walsh-Kaczmarz system is a convergence system. Skvorcov in 1981 [16]

Almost everywhere strong summability of Marcinkiewicz means of double Walsh-Fourier series

Analysis Mathematica, 2014

In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation (1 n n−1 m=0 |Smmf − f | p) 1/p → 0 for every two-dimensional functions belonging to L log L and 0 < p ≤ 2. From the theorem of Getsadze [6] it follows that the space L log L can not be enlarged with preserving this strong summability property.