Orlicz-Hardy spaces and their duals

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(? ⁿ ) in that the class of admissible functions ? 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(? ⁿ ). In particular, ? 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(? ⁿ ) = H(log H) ^q (? 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} (? ⁿ ) to L ^{n/(n ? α)}(log L)(? ⁿ ) for 0 < α < n.