Martingale Hardy Spaces And Their Applications In Fourier Analysis

By E. Mayer. 2016

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Maximal Operators of Vilenkin–Nörlund Means (2015, background): “ˆ Mn, if x∈ In, 0, if x/∈ In. The σ-algebra generated by the intervals {In (x):x∈ Gm} will be denoted by n (n∈ N). Denote by f= f(n),n∈ N a martingale with respect to n (n∈ N).(for details see e.g. [37]). The maximal function of a martingale f is deﬁned by f∗ =sup n∈N f(n) . For 0<p<∞ the Hard”
New martingale inequalities and applications to Fourier analysis (2018, methodology): “For the dyadic stochastic basis, it was proved by Weisz [78].”
Martingale Hardy spaces with variable exponents (2014, methodology): “∈ P and T be the sets of all stopping times relative to {Fn}n≥0. Deﬁne BMOp(·) = n f = (fn)n≥0 : kfkBMO p(·) < ∞ o , where kfkBMO p(·) = sup τ∈T χ {τ<∞} −1 p(·) kf − fτ−1k p(·). Lemma 5.8. (see [27]) If 1 ≤ p < ∞, then kfkBMO1 ≈kfkBMOp. Proposition 5.9. If p(·) ∈ P satisﬁes (1.5) and 1 ≤ p−”
On the partial sums of Vilenkin-Fourier series (2014, background): “λ>0 λµ(f>λ)1/p <+∞. The σ-algebra generated by the intervals {In (x) : x∈ Gm} will be denoted by n (n∈ N).Denote by f= (fn,n∈ N) a martingale with respect to n (n∈ N).(for details see e.g. [21]). The maximal function of a martingale f is defend by f∗ = sup n∈N f(n) . In case f∈ L 1(Gm),the ”