Projection and slicing theorems in Heisenberg groups Pertti Mattila - PowerPoint PPT Presentation

Projection and slicing theorems in Heisenberg groups Pertti Mattila Projection and slicing theorems in Heisenberg groups Pertti Mattila University of Helsinki 10.12.2012

papers Projection and slicing theorems in Heisenberg groups Pertti Mattila Z. Balogh, E. Durand Cartagena, K. F¨ assler, P. Mattila, J. Tyson: The effect of projections on dimension in the Heisenberg group, to appear in Revista Math. Iberoamericana Z. Balogh, K. F¨ assler, P. Mattila, J. Tyson: Projection and slicing theorems in Heisenberg groups, Advances in Math. 231 (2012), pp. 569-604

Heisenberg group H n Projection and slicing theorems in Heisenberg groups Pertti Mattila Heisenberg group H n is R 2 n +1 equipped with a non-abelian group structure, with a left invariant metric and with natural dilations.

The first Heisenberg group H 1 Projection and slicing theorems in Heisenberg groups H = C × R , p = ( w , s ) , q = ( z , t ) ∈ H Pertti Mattila p · q = ( w + z , s + t + 2 Im ( w ¯ z )) || p || = ( | z | 4 + t 2 ) 1 / 4 d ( p , q ) = || p − 1 · q || = ( | w − z | 4 + | s − t − 2 Im ( w ¯ z ) | 2 ) 1 / 4 δ r ( p ) = ( rz , r 2 t ) d ( δ r ( p ) , δ r ( q )) = rd ( p , q ) d ( p · q 1 , p · q 2 ) = d ( q 1 , q 2 ) dim H H = 4, Heisenberg Hausdorff dimension

Projections in H 1 Projection and slicing theorems in Heisenberg groups Pertti Mattila V θ = { te θ : t ∈ R } , e θ = (cos θ, sin θ, 0) , 0 ≤ θ < π, horizontal line in H 1 W θ = V ⊥ θ vertical plane in H 1 H 1 = W θ · V θ , that is, for p ∈ H 1 , p = Q θ ( p ) · P θ ( p ) , P θ ( p ) ∈ V θ , Q θ ( p ) ∈ W θ P θ : H 1 → V θ , Q θ : H 1 → W θ , 0 ≤ θ < π , are the group projections

Projections in H 1 Projection and slicing theorems in Heisenberg groups Pertti Mattila p = ( z , t ) = ( x + iy , t ) ∈ H 1 P θ ( p ) = (( x cos θ + y sin θ ) e θ , t ); P θ is the standard linear projection Q θ ( p ) = θ , t − 2(cos θ ) xy + sin(2 θ )( x 2 − y 2 )); (( y cos θ − x sin θ ) e ⊥ Q θ is a non-linear projection

Marstrand’s projection theorem If A ⊂ R 2 is a Borel set, then (dim E is the Euclidean Hausdorff Projection and slicing theorems in dimension) for almost all θ ∈ [0 , π ), Heisenberg groups dim E P θ ( A ) = dim E A for almost all θ ∈ (0 , π ) if dim E A ≤ 1 , Pertti Mattila H 1 ( P θ ( A ) > 0 for almost all θ ∈ (0 , π ) if dim E A > 1 . Kaufman’s proof for the first part: Let 0 < s < dim E A . Then there is a non-trivial Borel measure �� | x − y | − s d µ xd µ y < ∞ . Let P θ µ µ on A such that I s ( µ ) = be the push-forward under P θ : P θ µ ( B ) = µ ( P − 1 θ ( B ). Then � π ��� | P θ ( x − y ) | − s d µ xd µ yd θ I s ( P θ µ ) d θ = 0 � π | θ | − s d θ I s ( µ ) < ∞ . ≈ 0

Horizontal projection theorem Projection and slicing theorems in Theorem Heisenberg groups Let A ⊂ H 1 be a Borel set. Then for almost all θ ∈ [0 , π ) , Pertti Mattila dim H P θ ( A ) ≥ dim H A − 2 if dim H A ≤ 3 , H 1 ( P θ ( A )) > 0 if dim H A > 3 . This is sharp: consider A = { ( x , 0 , t ) : x ∈ C , t ∈ [0 , 1] } , C ⊂ R . Then dim H A = dim E C + 2 and dim H P θ ( A ) = dim E P θ ( A ) = dim E P θ ( C ) = dim E C for all but one θ .

Vertical projection theorem Projection and slicing theorems in Heisenberg groups Pertti Mattila Theorem Let A ⊂ H 1 be a Borel set. If dim H A ≤ 1 , then for almost all θ ∈ [0 , π ) , dim H A ≤ dim H Q θ ( A ) ≤ 2 dim H A . For A with dim H A ≤ 1 this is sharp: if A ⊂ t -axis, dim H Q θ ( A ) = dim H A for all θ , if A ⊂ x -axis, dim H Q θ ( A ) = 2 dim H A for all but one θ .

Vertical projection theorem Projection and slicing theorems in Heisenberg groups p = ( z , t ) , q = ( ζ, τ ) ∈ H 1 , ϕ 1 = arg( z − ζ ) , ϕ 2 = arg( z + ζ ) Pertti Mattila d ( p , q ) 4 = | z − ζ | 4 + ( t − τ + | z 2 − ζ 2 | sin( ϕ 1 − ϕ 2 )) 2 d ( Q θ ( p ) , Q θ ( q )) 4 = | z − ζ | 4 sin 4 ( ϕ 1 − θ ) + ( t − τ − | z 2 − ζ 2 | sin( ϕ 2 + ϕ 1 − 2 θ )) 2 � π 0 d ( Q θ ( p ) , Q θ ( q )) − s d θ � d ( p , q ) − s , To get for 0 < s < 1, one needs for a ∈ R , � π d θ | a + sin θ | s / 2 � 1 0

Vertical projection theorem Projection and slicing theorems in Heisenberg groups Pertti Mattila If dim H A > 1, we have some estimates which quite likely are not sharp. For example, we don’t know if dim H A > 3 implies H 2 ( Q θ ( A )) > 0 for almost all θ ∈ [0 , π ). A related Euclidean question: does dim E A > 2 imply H 2 ( Q θ ( A )) > 0 for almost all θ ∈ [0 , π )?

Higher dimensions Projection and slicing theorems in H n = C n × R , p = ( w , s ) , q = ( z , t ) ∈ H n Heisenberg groups p · q = ( w + z , s + t + ω ( w , z )) , Pertti Mattila ω ( w , z ) = 2 Im ( w · z ) = � n j =1 ( v j x j − u j y j ), w = ( u j + iv j ) , z = ( x j + iy j ) || p || = ( | z | 4 + t 2 ) 1 / 4 d ( p , q ) = || p − 1 · q || = ( | w − z | 4 + | s − t − ω ( w , z ) | 2 ) 1 / 4 δ r ( p ) = ( rz , r 2 t ) d ( δ r ( p ) , δ r ( q )) = rd ( p , q ) d ( p · q 1 , p · q 2 ) = d ( q 1 , q 2 ) dim H H n = 2 n + 2

Projections in H n Projection and slicing theorems in Heisenberg groups G h ( n , m ) = { V ∈ G (2 n , m ) : ω ( w , z ) = 0 ∀ w , z ∈ V } , Pertti Mattila 0 < m ≤ n , isotropic subspaces unitary group U ( n ) ⊂ O (2 n ) acts transitively on G h ( n , m ); g ∈ U ( n ) : ω ( g ( w ) , g ( z )) = ω ( w , z ) ∀ w , z ∈ C n H n = V ⊥ · V , V ⊥ ⊂ R 2 n +1 , V ∈ G h ( n , m ), p = Q V ( p ) · P V ( p ) , P V ( p ) ∈ V , Q V ( p ) ∈ W , for p ∈ H n P V : H n → V is the standard linear projection Q V ( z , t ) = ( P V ⊥ ( z ) , t − ω (( P V ⊥ ( z ) , P V ( z ))) is a non-linear projection, Q V : H n → V ⊥

Horizontal projection theorem in H n Projection and slicing theorems in Theorem Heisenberg Let A ⊂ H n be a Borel set. If dim H A ≤ m + 2 , then groups Pertti Mattila dim P V ( A ) ≥ dim H A − 2 for µ n , m almost all V ∈ G h ( n , m ) . Furthermore, if dim H A > m + 2 , then H m ( P V ( A )) > 0 for µ n , m almost V ∈ G h ( n , m ) . This is again sharp. Above µ n , m is the unique U ( n )-invariant Borel probability measure on G h ( n , m ).

Vertical projection theorem in H n Projection and slicing theorems in Heisenberg groups Pertti Mattila Theorem Let A ⊂ H n be a Borel subset with dim H A ≤ 1 . Then for µ n , m almost V ∈ G h ( n , m ) , dim H A ≤ dim H Q V A ≤ 2 dim H A . This is again sharp when dim H A ≤ 1. Some, probably rather weak, partial results are known when dim H A > 1.

Vertical projection theorems in H n Projection and slicing theorems in � | z − ζ | 4 + ( t − τ − 2 ω ( ζ, z )) 2 Heisenberg 4 d H ( p , q ) = groups Pertti Mattila d H ( Q V ( p ) , Q V ( q )) 4 = | P V ⊥ ( z − ζ ) | 4 + ( t − τ − 2 ω ( P V ⊥ ( z ) , P V ( z )) + 2 ω ( P V ⊥ ( ζ ) , P V ( ζ )) − 2 ω ( P V ⊥ ( ζ ) , P V ⊥ ( z ))) 2 . The key estimate in the proof is � | a − 2 ω ( v , P V ( w )) | − s / 2 d µ n , m V � 1 G h ( n , m ) for all 0 < s < 1 , a ∈ R and v , w ∈ S 2 n − 1 . This estimate is false for s ≥ 1.

Slicing theorems in H n Projection and slicing theorems in Heisenberg groups Pertti Mattila Theorem Let A ⊂ H n be a Borel set with dim H A > m + 2 . Then for µ n , m almost V ∈ G h ( n , m ) , H m ( { v ∈ V : dim H ( A ∩ ( V ⊥ · v )) = dim H A − m } ) > 0 . The assumption dim H A > m + 2 is necessary.

Slicing theorems in H n Projection and slicing theorems in Heisenberg groups Pertti Mattila Theorem Let A ⊂ H n be a Borel set with 0 < H s H ( A ) < ∞ for some s > m + 2 . Then for H s H almost all p ∈ A we have dim H ( A ∩ ( V ⊥ · p )) = s − m for µ n , m almost all V ∈ G h ( n , m ) .

Thank you Projection and slicing theorems in Heisenberg groups Pertti Mattila Thank you Ka-Sing, De-Jun and all others