Science China Mathematics

, Volume 57, Issue 5, pp 903–962 | Cite as

Orlicz-Hardy spaces and their duals

  • Eiichi Nakai
  • Yoshihiro Sawano


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(ℝ n ) 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(ℝ n ). 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(ℝ n ) = H(log H) q (ℝ n ). 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 (ℝ n ) to L n/(nα)(log L)(ℝ n ) for 0 < α < n.


Hardy space Orlicz space atomic decomposition Campanato space bounded mean oscillation 


42B30 46E30 42B35 


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Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsIbaraki UniversityMito, IbarakiJapan
  2. 2.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachioji-shi, TokyoJapan

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