Unique Continuation for Locally Uniformly Distributed Measures

Abstract In this note we show that the support of a locally k -uniform measure in $${\mathbb {R}}^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math Scand 90(1): 152-160, 2002) and David, Kenig and Toro (Comm Pure Appl Math 54(4): 385-449, 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J Reine Angew Math 379: 115-151, 1987) that each connected component of the support of a locally n -uniform measure in $${\mathbb {R}}^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> is contained in the zero set of a quadratic polynomial.