Summability of Multi-Dimensional Fourier Series and Hardy Spaces

On the maximal operator of Walsh-Kaczmarz-Fejér means

“…, n ∈ N) a martingale with respect to (F n , n ∈ N) (for details see, e. g. [19], [20]). The maximal function of a martingale f is defined by…”

The Hardy–Lorentz spaces Hp,q(Rn)

“…This approach combines the concept of p-quasilocal operator of Weisz (see [16]) with the idea of variable dilations of R. Fefferman and Soria (see [10]). Intuitively, since Hörmander's condition implies that T maps H 1 into L 1 , say, for T to be defined in H 1,s , 1 < s ≤ ∞, some strengthening of this condition is required.…”

Sharp Weak Type Inequality for Fractional Integral Operators Associated with d-Dimensional Walsh–Fourier Series

“…As evidenced in numerous papers, such inequalities play an important role in many areas of mathematics, including approximation theory, Fourier analysis, harmonic analysis and probability theory. We refer the interested reader to the works [2,5,20,21,23,24] and references therein.…”

“…For this and closely related topic, consult e.g. the works of Goginava [5,6], Goginava and Weisz [7], Nagy [9], Simon [13,14] and Weisz [21][22][23]. We will study this interplay from a slightly different point of view.…”