A kernel approach to the local solvability of the tangential Cauchy Riemann equations

Abstract

An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Y(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper C Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {C}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Y(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not satisfied.