Area integral estimates for higher order elliptic equations and systems

Abstract

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> norm between the maximal function and the square function of solutions to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> in Lipschitz domains. Several applications of this result are discussed.