Quasianalyticity and pluripolarity

We show that the graph <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma Subscript f Baseline equals StartSet left-parenthesis z comma f left-parenthesis z right-parenthesis right-parenthesis element-of double-struck upper C squared colon z element-of upper S EndSet"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>,</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>:</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>z</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>S</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma _f=\{(z,f(z))\in {\mathbb {C}}^2:\,z\in S\}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {C}}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the unit circle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is either continuous and quasianalytic in the sense of Bernstein or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript normal infinity"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">C^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and quasianalytic in the sense of Denjoy is pluripolar.

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