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On the Density Problem in the Parabolic Space
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In this work we establish relationships between densities, tangents, and rectifiability in parabolic spaces, namely
R
n
+
1
\mathbb {R}^{n+1}
equipped with parabolic dilations. In particular, we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of
1
1
-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant density condition for a
1
1
-codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma.
American Mathematical Society (AMS)
Title: On the Density Problem in the Parabolic Space
Description:
In this work we establish relationships between densities, tangents, and rectifiability in parabolic spaces, namely
R
n
+
1
\mathbb {R}^{n+1}
equipped with parabolic dilations.
In particular, we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of
1
1
-codimensional uniform measures.
Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant density condition for a
1
1
-codimensional Ahlfors-regular measure implies the bilateral weak geometric lemma.
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