An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions

Abstract We consider the weighted Radon transforms R W along hyperplanes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>3</mml:mn> </mml:mstyle> </mml:math> , with strictly positive weights <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>θ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mstyle> </mml:math> . We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions. In addition, the related weight W is infinitely smooth almost everywhere and is bounded. Our construction is based on the famous example of non-uniqueness of Boman (1993 J. d’Anal. Math . 61 395–401) for the weighted Radon transforms in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mstyle> </mml:math> and on a recent result of Goncharov and Novikov (2016 Eurasian J. Math. Comput. Appl . 4 23–32).

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