THE REFINED AREA FORMULA FOR SOBOLEV MAPPINGS $$W^{k,p}$$

Abstract We establish the area formula for change-of-variable mappings in the Sobolev space $$W^{k,p}_{\text {loc}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mtext>loc</mml:mtext> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> . Our approach relies on constructing Lipschitz approximations of Sobolev functions that agree with the original functions outside a set of Riesz capacity zero.