Characterization of parabolic <i>H</i> ¹ Sobolev spaces by square functions

Abstract We consider <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H^{1}} -Sobolev spaces associated with the Calderón–Torchinsky Hardy spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {H^{1}} and characterize these spaces by means of certain square functions. This characterization generalizes and, in fact, improves upon previously known results in the Euclidean setting.