C 1 solutions of a general cohomological equation on the plane

<jats:p> Cohomological equation is of special interest because it concerns the study of time change for flows, topological stability and topological conjugacy in dynamical systems, which is also an auxiliary equation to study the problem of linearization. In this paper, we consider a general form of cohomological equation for planar contractions. By using the ideas of invariant manifold and estimations in [W. Zhang and W. Zhang, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" content-type="simple" xlink:href="S0013091526101400_inline2.png"> <jats:alt-text content-type="machine-generated">dollar sign upper C 1 dollar sign</jats:alt-text> </jats:inline-graphic> <jats:tex-math>$C^1$</jats:tex-math> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> linearization for planar contractions, <jats:italic>J. Funct. Anal.</jats:italic> <jats:bold>260</jats:bold> (2011), 2043–2063.], we present new criteria on eigenvalues of the linear parts for the existence of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" content-type="simple" xlink:href="S0013091526101400_inline3.png"> <jats:alt-text content-type="machine-generated">dollar sign upper C 1 dollar sign</jats:alt-text> </jats:inline-graphic> <jats:tex-math>$C^1$</jats:tex-math> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> solutions in the Poincaré domain. Our results are a generalization of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" content-type="simple" xlink:href="S0013091526101400_inline4.png"> <jats:alt-text content-type="machine-generated">dollar sign upper C 1 dollar sign</jats:alt-text> </jats:inline-graphic> <jats:tex-math>$C^1$</jats:tex-math> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> linearization for contractions. </jats:p>