Besicovitch covering property on graded groups and applications to measure differentiation Séverine Rigot (Université Nice Sophia Antipolis) based on a joint work with E. Le Donne ( University of Jyväskylä ) Geometric Measure Theory, University of Warwick, July 10-14, 2017
Besicovitch covering property A family B of closed balls in a metric space ( X , d ) is said to be a Besicovitch family of balls if for every B ∈ B , x B �∈ � B ′ ∈B\{ B } B ′ � B ∈B B � = ∅ . Definition (Besicovitch covering property) We say that the Besicovitch covering property (BCP) holds on ( X , d ) if there is C ≥ 1 such that Card B ≤ C for every Besicovitch family of balls B .
Besicovitch covering property A family B of closed balls in a metric space ( X , d ) is said to be a Besicovitch family of balls if for every B ∈ B , x B �∈ � B ′ ∈B\{ B } B ′ � B ∈B B � = ∅ . Definition (Besicovitch covering property) We say that the Besicovitch covering property (BCP) holds on ( X , d ) if there is C ≥ 1 such that Card B ≤ C for every Besicovitch family of balls B . An application to differentiation of measures: Theorem (based on Besicovitch’s ideas) Let ( X , d ) be a separable metric space. Assume that BCP holds on ( X , d ) . Then the differentiation theorem holds for every locally finite Borel regular measure over ( X , d ) . The differentiation theorem holds for a measure µ over ( X , d ) if � 1 ∀ f ∈ L 1 loc ( µ ) , lim f d µ = f ( x ) µ -a.e. µ ( B ( x , r )) r ↓ 0 + B ( x , r )
Besicovitch covering property Examples: Finite dimensional normed vector spaces satisfy BCP . Infinite dimensional normed vector spaces do not satisfy BCP .
Besicovitch covering property The validity or non validity of BCP is sensitive to perturbations of the metric.
Besicovitch covering property The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in ( X , d ) . For every 0 < c < 1, there is a distance d c on X such that cd ≤ d c ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss).
Besicovitch covering property The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in ( X , d ) . For every 0 < c < 1, there is a distance d c on X such that cd ≤ d c ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces ( X , d ) on which BCP does not hold but are such that for every ε > 0, there is a distance d ε such that d ≤ d ε ≤ ( 1 + ε ) d and for which BCP holds.
Besicovitch covering property The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in ( X , d ) . For every 0 < c < 1, there is a distance d c on X such that cd ≤ d c ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces ( X , d ) on which BCP does not hold but are such that for every ε > 0, there is a distance d ε such that d ≤ d ε ≤ ( 1 + ε ) d and for which BCP holds. In particular the validity or non validity of BCP is not preserved by a bi-Lipschitz change of distance.
Besicovitch covering property The validity or non validity of BCP is sensitive to perturbations of the metric. Assume that there exists an accumulation point in ( X , d ) . For every 0 < c < 1, there is a distance d c on X such that cd ≤ d c ≤ d for which BCP does not hold ([Le Donne, R] following a slightly different statement given by D. Preiss). There are examples of metric spaces ( X , d ) on which BCP does not hold but are such that for every ε > 0, there is a distance d ε such that d ≤ d ε ≤ ( 1 + ε ) d and for which BCP holds. In particular the validity or non validity of BCP is not preserved by a bi-Lipschitz change of distance. On the other hand, if BCP holds on ( X , d ) then BCP holds on the snowflakes ( X , d α ) for every 0 < α < 1.
Graded Lie algebras and Lie groups A Lie algebra g is positively graduable if it can be written as g = ⊕ t > 0 V t where [ V t , V s ] ⊂ V t + s for all t , s > 0. A Lie algebra is said to be graded if it is positively graduable and endowed with a given positive grading. Associated dilations ( δ λ ) λ> 0 on a graded Lie algebra g = ⊕ t > 0 V t are defined as the unique linear maps δ λ : g → g such that δ λ ( X ) = λ t X for X ∈ V t . The family ( δ λ ) λ> 0 is a one-parameter group of Lie algebra automorphisms.
Graded Lie algebras and Lie groups A Lie algebra g is positively graduable if it can be written as g = ⊕ t > 0 V t where [ V t , V s ] ⊂ V t + s for all t , s > 0. A Lie algebra is said to be graded if it is positively graduable and endowed with a given positive grading. Associated dilations ( δ λ ) λ> 0 on a graded Lie algebra g = ⊕ t > 0 V t are defined as the unique linear maps δ λ : g → g such that δ λ ( X ) = λ t X for X ∈ V t . The family ( δ λ ) λ> 0 is a one-parameter group of Lie algebra automorphisms. A graded group G is a connected and simply connected Lie group whose Lie algebra g is graded. Associated dilations on a graded group G are defined as the unique Lie group automorphisms δ λ : G → G such that δ λ ◦ exp = exp ◦ δ λ . We may identify a graded group with its Lie algebra, the group law being given by the Baker-Campbell-Hausdorff formula, x · y = x + y + [ x , y ] / 2 + · · · , and the exponential map from g to G being the identity.
Homogeneous distances and homogeneous groups A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d ( x · y , x · y ′ ) = d ( y , y ′ ) d ( δ λ ( x ) , δ λ ( y )) = λ d ( x , y ) for every x , y , y ′ ∈ G and every λ > 0.
Homogeneous distances and homogeneous groups A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d ( x · y , x · y ′ ) = d ( y , y ′ ) d ( δ λ ( x ) , δ λ ( y )) = λ d ( x , y ) for every x , y , y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if V t = { 0 } for all 0 < t < 1. We call such groups homogeneous . On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups.
Homogeneous distances and homogeneous groups A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d ( x · y , x · y ′ ) = d ( y , y ′ ) d ( δ λ ( x ) , δ λ ( y )) = λ d ( x , y ) for every x , y , y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if V t = { 0 } for all 0 < t < 1. We call such groups homogeneous . On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups. A homogeneous distance on a homogeneous group induces the manifold topology on G .
Homogeneous distances and homogeneous groups A distance d on a graded group G is called homogeneous if it is left-invariant and one-homogeneous w.r.t. the family of associated dilations, that is, d ( x · y , x · y ′ ) = d ( y , y ′ ) d ( δ λ ( x ) , δ λ ( y )) = λ d ( x , y ) for every x , y , y ′ ∈ G and every λ > 0. Homogeneous distances do exist on a graded group if and only if V t = { 0 } for all 0 < t < 1. We call such groups homogeneous . On general graded groups, one can look at homogeneous quasi-distances. BCP makes sense for quasi-distances. For simplicity, we restrict ourselves in this talk to the study of the validity of BCP for homogeneous distances on homogeneous groups. A homogeneous distance on a homogeneous group induces the manifold topology on G . Any two homogeneous distances on a homogeneous group are bi-Lipschitz equivalent.
Examples Finite dimensional normed vector spaces correspond to Abelian homogeneous groups with Lie algebra endowed with the positive grading g = V 1 and equipped with a homogeneous distance.
Examples Finite dimensional normed vector spaces correspond to Abelian homogeneous groups with Lie algebra endowed with the positive grading g = V 1 and equipped with a homogeneous distance. Snowflakes of finite dimensional normed vector spaces can be recovered as Abelian homogeneous groups with Lie algebra endowed with the positive grading g = V α for some α > 1. Associated dilations are given by δ λ ( X ) = λ α X . Distances d α ( x , y ) = � x − y � 1 /α are left-invariant and one-homogeneous w.r.t. the associated dilations.
t -power of a positive grading Let t > 0. The t -power of a positive grading g = ⊕ s > 0 V s is the positive grading g = ⊕ s > 0 W s where W ts = V s . If ( x 1 , . . . , x n ) �→ ( λ s 1 x 1 , . . . , λ s n x n ) are the dilations associated to the initial grading, then dilations associated to its t -power are given by ( x 1 , . . . , x n ) �→ ( λ ts 1 x 1 , . . . , λ ts n x n ) .
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