A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions
on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin
and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means
of these Fourier series is bounded from Hp to Lp (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin,
Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.
It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the
dyadic Hardy-Lorentz spaceHp,q toLp,q (1/2<p<∞, 0<q≤∞) and is of weak type (L1,L1). We define the twodimensional dyadic hybrid Hardy spaceH1‖
and verify that the corresponding maximal operator of a two-dimensional function is of weak type (H1‖
,L1). As a consequence, we obtain that the dyadic integral of a two-dimensional functionfεH1‖
⊃LlogL is dyadically differentiable and its derivative is a.e.f.
It is shown that the maximal operator of the two-parameter dyadic derivative of the dyadic integral is bounded from the two-parameter dyadic Hardy-Lorentz space Hp,q to Lp,q (1/2 < p < 8, 0 < qL is dyadically differentiable and its derivative is f a.e.
ВВОДьтсь ДВА НОВых кл АссА пРОстРАНстВBMO, ИМ ЕУЩИх БОльшОЕ жНАЧЕНИЕ В тЕОРИИ ИНтЕРпОлИРО ВАНИь, И ДОкАжыВАУтсь НЕкОтОРыЕ сООтНОшЕНИь ЁкВИВАл ЕНтНОстИ Их И ОБыЧНых
пРОстРАН стВBMO. ИжУЧАУтсь сООтВЕтстВУУЩИЕ “sharp”Ф УНкцИИ И ОБОБЩАУтсь НЕкОтОР ыЕ тЕОРЕМы гАРсИИ, И ФЕ ФФЕРМАНА И стЕИНА.
We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-θ-means of a tempered distribution is bounded from Hp(ℝ2) to Lp(ℝ2) for all 2/(2 + α) < p ≤ ∞; consequently, it is of weak type (1,1), where 0 < α ≤ 1 is depending only on θ. As a consequence, we obtain that the triangular-θ-means of a function f ∈ L1(ℝ2) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of
the triangular-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski, and Riesz summations.
We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order
(m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the
d-dimensional Ciesielski-Fourier series is bounded from the Hardy space Hp([0, 1)d1×…×[0, 1)dl to Lp ([0, 1)d) if 1/2<p<∞ and mj≥0, ‖kj‖≤mj+1. By an interpolation theorem, we get that the maximal operator is also of weak type (
It is proved that the maximal operator of the triangular Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space to for all 2/(2+α)<p≦∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Cesàro means of a function converge a.e. to f.