Hardy space

In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the Hardy spaces (or Hardy classes) 
 
 
 
 
 H
 
 p
 
 
 
 
 {\displaystyle H^{p}}
 
 are <a href="/facts/Function_space/lLUE2R1t">spaces</a> of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> on the <a href="/facts/Unit_disk/phrUu7yC">unit disk</a> or <a href="/facts/Upper_half_plane/7fPqrowr">upper half plane</a>. They were introduced by <a href="/facts/Frigyes_Riesz/EN47QveQ">Frigyes Riesz</a> (Riesz 1923), who named them after <a href="/facts/G._H._Hardy/zBkDLb05">G. H. Hardy</a>, because of the paper (Hardy 1915). In <a href="/facts/Real_analysis/12WjlXCW">real analysis</a> Hardy spaces are spaces of <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributions</a> on the real n-space 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
, defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the <a href="/facts/Lp_space/gbad0Rv1">Lp spaces</a>. For 
 
 
 
 1
 ≤
 p
 <
 ∞
 
 
 {\displaystyle 1\leq p<\infty }
 
 these Hardy spaces are <a href="/facts/Subset/SJGERmbA">subsets</a> of <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces</a>, while for 
 
 
 
 0
 <
 p
 <
 1
 
 
 {\displaystyle 0<p<1}
 
 the 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces have some undesirable properties, and the Hardy spaces are much better behaved. Hence, 
 
 
 
 
 H
 
 p
 
 
 
 
 {\displaystyle H^{p}}
 
 spaces can be considered extensions of 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces.
Hardy spaces have a number of applications, both in <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> itself as well as in interdisciplinary areas such as <a href="/facts/Control_theory/aIuERpn6">control theory</a> (e.g. <a href="/facts/H_infinity/kJSBxQ2q">
 
 
 
 
 H
 
 ∞
 
 
 
 
 {\displaystyle H^{\infty }}
 
 methods</a>) and <a href="/facts/Scattering_theory/79Nx76Ub">scattering theory</a>.

Hardy space open-in-new

Hardy space