Local Musielak-Orlicz Hardy Spaces

In this chapter, we introduce a local Musielak-Orlicz Hardy space h(phi()R(n)) by the local grand maximal function, and a local BMO-type space bmo(phi()R(n)) which is further proved to be the dual space of h(phi()R(n)). As an application, we prove that the class of pointwise multipliers for the local BMO-type space, bmo(phi()R(n)), characterized by E. Nakai and K. Yabuta, is just the dual of L-1(R-n) + h(phi 0)(R-n), where phi is an increasing function on (0,infinity) satisfying some additional growth conditions and phi(0) a Musielak-Orlicz function induced by phi. Characterizations of h(phi()R(n)), including the atoms, the local vertical or the local non-tangential maximal functions, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of h(phi()R(n)), from which, we further deduce some criterions for the boundedness on h(phi()R(n)) of some sublinear operators. Finally, we show that the local Riesz transforms and some pseudo-differential operators are bounded on h(phi()R(n)).