Hardy-Sobolev Space Associated With Twisted Convolution

Hardy space and Sobolev space have many important applications in harmonic analysis and the theory of partial differential equations.Sobolev space is defined by the functions that whose derivatives belong to the Lebesgue space Lp for p>1.When p ?(0,1],the naturally substitute space for Lebesgue space is the Hardy space.Therefore,the right substitute space for Sobolev space is the Hardy-Sobolev space defined by the functions whose derivatives belong to a Hardy type space.In recent years,there are already mature theories for Hardy space and Sobolev space associated with twisted convolution.After the fundamental work of predecessors,we further study the Hardy-Sobolev space associated with twisted convolution.We first define the Hardy-Sobolev space associated with twisted convolution and prove that this space is a Banach space.We also give an equivalent characterization of the Hardy-Sobolev space.Then we appropriately define the atom of the Hardy-Sobolev space that similar to the definition of atom in the Hardy space.We will prove the uniform boundedness of the maximal function on the atom by sharp estimate of the integral kernel of Poisson semigroup.Then we give the atomic decomposition of Hardy-Sobolev Space by using the estimate of the gradients of Poisson kernel.Finally,we give the applications of the Hardy-Sobolev space associated with the twisted convolution.