Analytic holonomicity of real C^{_rj}-class distributions

We introduce a notion of distributions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , called distributions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript normal e normal x normal p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">x</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathscr {C}^{\mathrm {exp}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -class, based on wavelet transforms of distributions and the theory of Cluckers, Comte, Miller, Rolin, and Servi [Duke Math. J. 167 (2018), pp. 1239–1309] about <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript normal e normal x normal p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">x</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathscr {C}^{\mathrm {exp}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -class functions. We prove that the framework of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript normal e normal x normal p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">x</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathscr {C}^{\mathrm {exp}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript normal e normal x normal p"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">x</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathscr {C}^{\mathrm {exp}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -class.