Multiplication between Hardy spaces and their dual spaces

For any p is an element of (0, 1) and alpha = 1/p - 1, let H-p(R-n) and C-alpha (R-n) be the Hardy and the Campanato spaces on the R-n-dimensional Euclidean space R-n, respectively. In this article, the authors find suitable Musielak-Orlicz functions Phi(p), defined by setting, for any x is an element of R-n and t is an element of [0, infinity), Phi(p)(x, t) := {t/1+[t(1+vertical bar x vertical bar)(n)](1-p) when n(1/p-1) is not an element of N, T/1+[t(1+vertical bar x vertical bar)(n)](1-p)[log(1+vertical bar x vertical bar)](p) when n(1/p - 1) is an element of N, and then establish a bilinear decomposition theorem for multiplications of functions in H-p(R-n) and its dual space C-alpha(R-n). To be precise, for any f is an element of H-p(R-n) and g is an element of C-alpha(R-n), the authors prove that the product of f and g, viewed as a distribution, can be decomposed into S(f, g)+T(f, g), where S is a bilinear operator bounded from H-p(R-n) x C-alpha(R-n) to L-1(R-n) and T a bilinear operator bounded from H-p(R-n) x C-alpha(R-n) to the Musielak-Orlicz Hardy space H-Phi p(R-n) associated with the above Musielak-Orlicz function Phi(p). Such a bilinear decomposition is sharp when na N, in the sense that any other vector space Y subset of H-Phi p(R-n) adapted to the above bilinear decomposition should satisfy (L-1(R-n)+Y)* = (L-1(R-n) + H-Phi p(R-n)*. To obtain the sharpness, the authors establish a characterization of the class of pointwise multipliers of C-alpha(R-n) by means of the dual space of H-Phi p(R-n) which is of independent interest. As an application, an estimate of the div-curl product involving the space H-Phi p(R-n) is discussed. This article naturally extends the known sharp bilinear decomposition of H-1(R-n) x BMO (R-n). (C) 2019 Elsevier Masson SAS. All rights reserved.