Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

Abstract

Abstract. Let ϕ be a Musielak-Orlicz function satisfying that, for any (x, t) ∈ R n × (0, ∞), ϕ(·, t) belongs to the Muckenhoupt weight class A ∞ (R n ) with the critical weight exponent q(ϕ) ∈ [1, ∞) and ϕ(x, ·) is an Orlicz function with 0 < i(ϕ) ≤ I(ϕ) ≤ 1 which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces H ϕ (R n ) which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the authors characterize H ϕ (R n ) via all the first order Riesz transforms whenn , and via all the Riesz transforms with the order not more than m ∈ N when i(ϕ) q(ϕ) > n−1 n+m−1 . Moreover, the authors also establish the Riesz transform characterizations of H ϕ (R n ), respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when ϕ(x, t) := tw(x) for all x ∈ R n and t ∈ [0, ∞), these results also widen the range of weights in the known Riesz characterization of the classical weighted Hardy space H 1 w (R n ) obtained by R. L. Wheeden from w ∈ A 1 (R n ) into w ∈ A ∞ (R n ) with the sharp range q(w) ∈ [1, n n−1 ), where q(w) denotes the critical index of the weight w.