Spectral gaps and Fourier decay for self-conformal measures on the plane

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript omega Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mi> ω </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C^\omega (\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> self-conformal Iterated Function System (IFS) on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay for any such measure. Our technique is based on a refinement of a method of Oh-Winter (2017) where we do not require separation from the IFS or the Federer property for the underlying measure.