On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms

Boris Rubin

doi:10.1007/s13324-023-00819-5

On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms

Boris Rubin^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We study injectivity of integral operators which map the Cauchy initial data for the Euler–Poisson–Darboux equation to the fixed time measurement of the solution of this equation. These operators generalize the well-known spherical means and are closely related to the shifted k-plane transforms, which assign to functions in L^p(Rⁿ) their mean values over all k-planes at a fixed distance from the given k-planes. Several generalizations, including the Radon transform over strips of fixed width in R² and a similar transform over tubes of fixed diameter in R³ , are considered.

Original language	English
Article number	56
Journal	Analysis and Mathematical Physics
Volume	13
Issue number	4
DOIs	https://doi.org/10.1007/s13324-023-00819-5
State	Published - Aug 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

Euler–Poisson–Darboux equation
Injectivity
L spaces
Radon transforms

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10.1007/s13324-023-00819-5

Cite this

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abstract = "We study injectivity of integral operators which map the Cauchy initial data for the Euler–Poisson–Darboux equation to the fixed time measurement of the solution of this equation. These operators generalize the well-known spherical means and are closely related to the shifted k-plane transforms, which assign to functions in Lp(Rn) their mean values over all k-planes at a fixed distance from the given k-planes. Several generalizations, including the Radon transform over strips of fixed width in R2 and a similar transform over tubes of fixed diameter in R3 , are considered.",

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N2 - We study injectivity of integral operators which map the Cauchy initial data for the Euler–Poisson–Darboux equation to the fixed time measurement of the solution of this equation. These operators generalize the well-known spherical means and are closely related to the shifted k-plane transforms, which assign to functions in Lp(Rn) their mean values over all k-planes at a fixed distance from the given k-planes. Several generalizations, including the Radon transform over strips of fixed width in R2 and a similar transform over tubes of fixed diameter in R3 , are considered.

AB - We study injectivity of integral operators which map the Cauchy initial data for the Euler–Poisson–Darboux equation to the fixed time measurement of the solution of this equation. These operators generalize the well-known spherical means and are closely related to the shifted k-plane transforms, which assign to functions in Lp(Rn) their mean values over all k-planes at a fixed distance from the given k-planes. Several generalizations, including the Radon transform over strips of fixed width in R2 and a similar transform over tubes of fixed diameter in R3 , are considered.

KW - Euler–Poisson–Darboux equation

KW - Injectivity

KW - L spaces

KW - Radon transforms

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ER -

On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms

Abstract

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Keywords

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