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Orlicz-Hardy spaces and their duals

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Abstract

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions. The class will be wider than the class of all the N-functions. In particular, we consider the non-smooth atomic decomposition. The relation between Orlicz-Hardy spaces and their duals is also studied. As an application, duality of Hardy spaces with variable exponents is revisited. This work is different from earlier works about Orlicz-Hardy spaces H Φ(ℝn) in that the class of admissible functions ϕ is largely widened. We can deal with, for example,

$$\Phi (r) \equiv \left\{ \begin{gathered} r^{p1} \left( {\log \left( {e + {1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} \right)} \right)^{q1} , 0 < r \leqslant 1, \hfill \\ r^{p2} \left( {\log \left( {e + r} \right)} \right)^{q2} , r > 1, \hfill \\ \end{gathered} \right.$$

with p 1, p 2 ∈ (0,∞) and q 1, q 2 ∈ (−∞,∞), where we shall establish the boundedness of the Riesz transforms on H Φ(ℝn). In particular, ϕ is neither convex nor concave when 0 < p 1 < 1 < p 2 < ∞, 0 < p 2 < 1 < p 1 < ∞ or p 1 = p 2 = 1 and q 1, q 2 > 0. If Φ(r) ≡ r(log(e+r))q, then H Φ(ℝn) = H(log H)q(ℝn). We shall also establish the boundedness of the fractional integral operators I α of order α ∈ (0,∞). For example, I α is shown to be bounded from H(log H)1 − α/n (ℝn) to L n/(n − α)(log L)(ℝn) for 0 < α < n.

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Authors and Affiliations

  1. Department of Mathematics, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

    Eiichi Nakai

  2. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo, 192-0397, Japan

    Yoshihiro Sawano

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  1. Eiichi Nakai
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Nakai, E., Sawano, Y. Orlicz-Hardy spaces and their duals. Sci. China Math. 57, 903–962 (2014). https://doi.org/10.1007/s11425-014-4798-y

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