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Transactions of the American Mathematical Society

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Surjectivity of Euler type differential operators on spaces of smooth functions


Authors: Paweł Domański and Michael Langenbruch
Journal: Trans. Amer. Math. Soc. 372 (2019), 6017-6086
MSC (2010): Primary 44A15, 35A01; Secondary 35A09, 35A22, 45E10
DOI: https://doi.org/10.1090/tran/7367
Published electronically: August 5, 2019
MathSciNet review: 4024514
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Abstract: We develop a (global) solvability theory for Euler type linear partial differential equations $ P(\theta )$ on $ C^\infty (\Omega )$, with $ \Omega $ an open subset of $ \mathbb{R}^d$, i.e., for a special type of linear partial differential equation with polynomial coefficients. There is a natural closed upper bound $ C^\infty _{I(P)}(\Omega )$ for the range of $ P(\theta )$ on $ C^\infty (\Omega )$. We characterize by $ P(\theta )$-convexity type conditions those $ \Omega $ such that $ P(\theta )$ is surjective on $ C^\infty _{I(P)}(\Omega )$. We also clarify when all shifted operators $ P(c+\theta )$ are surjective on $ C^\infty _{I(P(c+\ \cdot \ ))}(\Omega )$. We classify in geometric terms those $ \Omega $ with $ 0\in \Omega $ such that every Euler operator $ P(\theta )$ is surjective on $ C^\infty _{I(P)}(\Omega )$. Moreover, we determine the operators $ P(\theta )$ which are surjective onto $ C^\infty _{I(P)}(\Omega )$ for every open set $ \Omega \subseteq \mathbb{R}^d$. Under some mild assumptions on $ \Omega $, we characterize those Euler operators which are invertible on $ C^\infty (\Omega )$. Under the same assumptions we also calculate the spectrum of $ P(\theta )$ on $ C^\infty (\Omega )$. The results follow from the solvability theory for Hadamard type operators on the space of smooth functions and from a new general Mellin transform, both developed in this paper.


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Additional Information

Paweł Domański
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland

Michael Langenbruch
Affiliation: Department of Mathematics, University of Oldenburg, D–26111 Oldenburg, Germany
Email: michael.langenbruch@uni-oldenburg.de

DOI: https://doi.org/10.1090/tran/7367
Keywords: Hadamard type operators, linear Euler type partial differential operators, linear partial differential operators with polynomial coefficients, smooth functions, Mellin transform, global solvability, invertibility.
Received by editor(s): July 21, 2017
Published electronically: August 5, 2019
Additional Notes: This research was supported by the National Center of Science (Poland), grant no. UMO-2013/10/A/ST1/00091.
Dedicated: Dedicated to the memory of Paweł Domański, a great friend and mathematician who left us far too early
Article copyright: © Copyright 2019 American Mathematical Society