The theory of Hardy-Sobolev spaces over the upper half-plane

We mainly investigate some results of Hardy-Sobolev spaces on the upper half-plane.In this paper, the Paley-Wiener theorem of Hardy-Sobolev spaces is established.The theorem asserts that the reproducing kernel of the Hardy-Sobolev spaces can be found.On account of this, the estimation of the reproducing kernel is produced.We also consider the multiplication operators on the spaces, and the spectrum of multiplication operators is characterized.In addition, the condition for the boundedness of weighted composition operators can be founded by applying the property of self-maps and the reproducing kernel.