2023/03/22 by Akemi Iwahashi, Yu Kitabeppu, Akari Yonekura
Mathematics · #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Point processes and geometric inequalities
paper · doi:10.1090/proc/16477
Ambrosio, Honda, and Tewodrose proved that the regular Weyl’s law is equivalent to a mild condition related to the infinitesimal behavior of the measure of balls in compact finite dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf RCD</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces. Though that condition is seemed to always hold for any such spaces, however, Dai, Honda, Pan, and Wei recently showed that for any integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at least 2, there exists a compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf RCD</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dimension fails to satisfy the regular Weyl’s law. In this short article we prove that one dimensional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper D"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf RCD</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces always satisfy the regular Weyl’s law.