Page "Hardy space" Paragraph 0

In complex analysis, the Hardy spaces ( or Hardy classes ) H < sup > p </ sup > are certain spaces of holomorphic functions on the unit disk or upper half plane.

They were introduced by Frigyes Riesz, who named them after G. H. Hardy, because of the paper.

In real analysis Hardy spaces are certain spaces of distributions on the real line, which are ( in the sense of distributions ) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the L < sup > p </ sup > spaces of functional analysis.

For 1 ≤ p ≤ ∞ these real Hardy spaces H < sup > p </ sup > are certain subsets of L < sup > p </ sup >, while for p < 1 the L < sup > p </ sup > spaces have some undesirable properties, and the Hardy spaces are much better behaved.