Vector-valued q -variational inequalities for averaging operators and the Hilbert transform
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Archiv der Mathematik
Vector-valued q-variational inequalities for averaging operators and the Hilbert transform Guixiang Hong, Wei Liu, and Tao Ma
Abstract. Recently, the authors have established Lp -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vectorvalued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform. Mathematics Subject Classification. Primary 42B20, 42B25; Secondary 46E30. Keywords. Variational inequalities, Averaging operators, Hilbert transform, Martingale cotype q, UMD.
1. Introduction. It is well known that in the setting of vector-valued harmonic analysis, many results are related to the properties of the geometry of Banach spaces. In particular, the so-called UMD, and convexity properties of Banach spaces have become the most fundamental tools in the study of vectorvalued harmonic analysis. The Lp -boundedness of vector-valued singular integral operators is an important problem in this area of research. Burkholder [4] first proved that whenever a Banach space X has the UMD property, the Hilbert transform is bounded on Lp (Rd ; X) for any p ∈ (1, ∞). Later, Bourgain [2] pointed out that the UMD property is indeed necessary for the boundedness of the Hilbert transform. Hyt¨ onen [12] proved that the two-sided Littlewood–Paley–Stein g-function estimate is equivalent to the UMD property of a Banach space X. Moreover, the martingale cotype q plays an important role in vector-valued analysis. The martingale cotype q was introduced by Pisier [25] to characterize the properties of uniformly convex Banach spaces.
G. Hong et al.
Arch. Math.
In [27], Xu provided a characterization of the martingale cotype q property via the vector-valued Littlewood–Paley theory associated to the Poisson kernel on the unit circle. Later on, Mart´ınez et al. [24] further characterized this property through general Littlewood–Paley–Stein theory. We refer the reader to [15] for an extensive study on vector-valued harmonic analysis. On the other hand, variational inequalities have received a lot of attention in the past decades for many reasons, one of which is that we can measure the speed of convergence for the family of operators under consideration. In 1976, L´epingle [21] applied the regularity of Brownian motion to obtain the first scalar-valued variational inequality for martingales. Ten years later, motivated by the development of Banach space geometry, Pisier and Xu [26] provided another proof of L´epingle’s result based on the stopping time argument. Almost at the same time Bourgain [3] used L´epingle’s results to establish variational inequalities for the erg
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