A variant of Yano’s extrapolation theorem on Hardy spaces

Abstract

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.

On a Problem of Pichorides

“…3.1, one can show that the exponent r = 1/2 in (ρ − 1) −1/2 in (3.23) cannot be improved in general. We remark that (3.23) may also be regarded as a direct consequence of (1.3) via extrapolation for analytic Hardy spaces; see [5,6].…”

Extrapolation on Hardy spaces and applications

“…Since by (3.12) one has | r f n | À 2 n for all n P N 0 , by using, as in [3], the elementary inequality t pn`1q{pn`2q ď e r`2 t `pn `1q ´pr`2q , which is valid for all t ě 0 and n P N, one can easily deduce that (3.16) }T pf q}…”

“…Such versions of Yano's theorem can be obtained as consequences of the abstract extrapolation theories for compatible couples of Banach spaces developed by Jawerth and Milman [19] (see also Jawerth and Milman [20] and Milman [35]) and by M. J. Carro and J. Martín [15] combined with available results concerning real interpolation between Hardy spaces. However, a direct approach that combines Yano's original argument with techniques due to S. V. Kislyakov [24] (see also Kislyakov and Q. Xu [27,28]) was presented in [3] for the one-dimensional periodic case. The main purpose of this survey paper is to present some variants of Yano's theorem for Hardy spaces and explain how they can be deduced by using already existing results on real interpolation between Hardy spaces.…”

“…In [3], an elementary proof of Theorem 2 was presented, based on Yano's original approach [56] combined with arguments of Kislyakov and Xu [24,27,28]. A direct proof of Theorem 2 can also be given by using the method of Bourgain [9] concerning Marcinkiewicz-type decompositions of functions in H 1 pTq.…”

“…where the implied constant depends only on C 0 and r. Since (3.17) holds for all analytic trigonometric polynomials, (3.10) can easily be obtained by using the scaling invariance of T , followed by a simple density argument involving [31, Proposition 3.4] and finally, using the fact that the Orlicz space pΦ r pTq, } ¨}ΦrpTq q can be identified with the Lorentz-Zygmund space pL log r LpTq, } ¨}L log r LpTq q; see [3].…”