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 (I broken vertical bar)(a"e (n) ) in that the class of admissible functions I center dot is largely widened. We can deal with, for example, with p (1), p (2) a (0,a) and q (1), q (2) a (-a,a), where we shall establish the boundedness of the Riesz transforms on H (I broken vertical bar)(a"e (n) ). In particular, I center dot is neither convex nor concave when 0 < p (1) < 1 < p (2) < a, 0 < p (2) < 1 < p (1) < a or p (1) = p (2) = 1 and q (1), q (2) > 0. If I broken vertical bar(r) a parts per thousand r(log(e+r)) (q) , then H (I broken vertical bar)(a"e (n) ) = H(log H) (q) (a"e (n) ). We shall also establish the boundedness of the fractional integral operators I (alpha) of order alpha a (0,a). For example, I (alpha) is shown to be bounded from H(log H)(1 - alpha/n) (a"e (n) ) to L (n/(n - alpha))(log L)(a"e (n) ) for 0 < alpha < n.