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 \(L^p(\mathbb {R}^n)\) 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 \(\mathbb {R}^2\) and a similar transform over tubes of fixed diameter in \(\mathbb {R}^3\), are considered.
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Notes
Note that \(k< (n-1)/2\) in the last line is equivalent to \(2n/(n-1)<n/k\), where n/k is the upper bound for p.
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Rubin, B. On the injectivity of integral operators related to the Euler–Poisson–Darboux equation and shifted k-plane transforms. Anal.Math.Phys. 13, 56 (2023). https://doi.org/10.1007/s13324-023-00819-5
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DOI: https://doi.org/10.1007/s13324-023-00819-5