In this note we give an alternative proof of a theorem due to Bourgain [2] concerning the growth of the constant in the Littlewood-Paley inequality on T as p Ñ 1`. Our argument is based on the endpoint mapping properties of Marcinkiewicz multiplier operators, obtained by Tao and Wright in [10], and on Tao's converse extrapolation theorem [9]. Our method also establishes the growth of the constant in the Littlewood-Paley inequality on T n as p Ñ 1`. Furthermore, we obtain sharp weak-type inequalities for the Littlewood-Paley square function on T n , but when n ě 2 the weak-type endpoint estimate on the product Hardy space over the n-torus fails, contrary to what happens when n " 1.
Extending work of Pichorides and Zygmund to the d-dimensional setting, we show that the supremum of L p -norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces H p A pT d q blows up like pp´1q´d as p Ñ 1`. Furthermore, we obtain an L log d L-estimate for square functions on H 1 A pT d q. Euclidean variants of Pichorides's theorem are also obtained. S T d pf qpxq "`ÿ k1,¨¨¨,k d PZ |∆ k1,¨¨¨,k d pf qpxq| 2˘1 {2 , 2010 Mathematics Subject Classification. 42B15, 42B25, 42B30.
In this note we prove a variant of Yano's classical extrapolation theorem for sublinear operators acting on analytic Hardy spaces over the torus.2010 Mathematics Subject Classification. Primary 30H10, 42B35, 46B70; Secondary 42B25.
We examine versions of the classical inequalities of Paley and Zygmund for functions of several variables. A sharp multiplier inclusion theorem and variants on the real line are obtained.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.