On semigroups associated with the Dunkl operators Joint work with - PowerPoint PPT Presentation

On semigroups associated with the Dunkl operators Joint work with Jacek Dziubański Agnieszka Hejna Instytut Matematyczny Uniwersytet Wrocławski Będlewo, 21.05.2019 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 1 / 43

Table of Contents 1. Introduction Fourier analysis in the rational Dunkl setting 2. Dunkl translations 3. Semigroups of operators Radial case - heat semigroup associated with the Dunkl Laplacian Nonradial cases 4. Idea of the proofs Convolution with radial function Support of τ x f ( −· ) A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 2 / 43

Some classical semigroups Classical heat semigroup Generator: ∆ = � N j = 1 ∂ 2 j Associated multiplier: e −| ξ | 2 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 3 / 43

Some classical semigroups Classical heat semigroup Generator: ∆ = � N Upper heat kernel estimate ( t = 1) j = 1 ∂ 2 j 1 Associated multiplier: ( 4 π ) N / 2 e − 1 4 | x − y | 2 e −| ξ | 2 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 3 / 43

Some classical semigroups Classical heat semigroup Generator: ∆ = � N Upper heat kernel estimate ( t = 1) j = 1 ∂ 2 j 1 Associated multiplier: ( 4 π ) N / 2 e − 1 4 | x − y | 2 e −| ξ | 2 Semigroups associated with higher order derivatives Generator: L = ( − 1 ) ℓ + 1 � N j = 1 ∂ 2 ℓ j Associated multiplier: e − � N j = 1 | ξ j | 2 ℓ A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 3 / 43

Some classical semigroups Classical heat semigroup Generator: ∆ = � N Upper heat kernel estimate ( t = 1) j = 1 ∂ 2 j 1 Associated multiplier: ( 4 π ) N / 2 e − 1 4 | x − y | 2 e −| ξ | 2 Semigroups associated with higher order derivatives Generator: L = ( − 1 ) ℓ + 1 � N Upper integral kernel estimate (t=1) j = 1 ∂ 2 ℓ j 2 ℓ Associated multiplier: Ce − c | x − y | 2 ℓ − 1 e − � N j = 1 | ξ j | 2 ℓ A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 3 / 43

Some Dunkl semigroups Dunkl heat semigroup Generator: ∆ = � N j = 1 T 2 Upper heat kernel estimate ( t = 1) j Associated multiplier: w ( B ( x , 1 )) − 1 e − cd ( x , y ) 2 e −| ξ | 2 Semigroups associated with higher order Dunkl operators Generator: L = ( − 1 ) ℓ + 1 � N Upper integral kernel estimate (t=1) j = 1 T 2 ℓ j 2 ℓ Associated multiplier: w ( B ( x , 1 )) − 1 e − cd ( x , y ) 2 ℓ − 1 e − � N j = 1 | ξ j | 2 ℓ A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 4 / 43

H¨ ormander’s multiplier theorem Theorem (H¨ ormander) Let ψ be a smooth radial function such that supp ψ ⊆ { ξ : 1 4 � � ξ � � 4 } and ψ ( ξ ) ≡ 1 for { ξ : 1 2 � � ξ � � 2 } . If m satisfies M = sup � ψ ( · ) m ( t · ) � W s 2 < ∞ t > 0 for some s > N / 2, then T m f = ( m ˆ � f ) , is (A) of weak type ( 1 , 1 ) , (B) of strong type ( p , p ) for 1 < p < ∞ , (C) bounded on the Hardy space H 1 atom . A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 5 / 43

H¨ ormander’s multiplier theorem Theorem (J. Dziubański, A.H.) Let ψ be a smooth radial function such that supp ψ ⊆ { ξ : 1 4 � � ξ � � 4 } and ψ ( ξ ) ≡ 1 for { ξ : 1 2 � � ξ � � 2 } . If m satisfies M = sup � ψ ( · ) m ( t · ) � W s 2 < ∞ t > 0 for some s > N , then T m f = F − 1 ( m F f ) , is (A) of weak type ( 1 , 1 ) , (B) of strong type ( p , p ) for 1 < p < ∞ , (C) bounded on the Hardy space H 1 atom . A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 6 / 43

Reflections We consider the Euclidean space R N with the scalar product � x , y � = � N j = 1 x j y j , x = ( x 1 , ..., x N ) , y = ( y 1 , ..., y N ) . Reflection For a nonzero vector α ∈ R N the reflection σ α with respect to the orthogonal hyperplane α ⊥ orthogonal to a nonzero vector α is given by σ α x = x − 2 � x , α � � α � 2 α. A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 7 / 43

Root system and Weyl group Root system A finite set R ⊂ R N \ { 0 } is called a root system if σ α ( R ) = R for every α ∈ R . A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 8 / 43

Root system and Weyl group Root system A finite set R ⊂ R N \ { 0 } is called a root system if σ α ( R ) = R for every α ∈ R . Weyl group The finite group G generated by the reflections σ α is called the Weyl group ( reflection group ) of the root system. A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 8 / 43

Examples - product root systems A 1 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 9 / 43

Examples - product root systems A 1 × A 1 A 1 × A 1 × A 1 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 10 / 43

Examples of root systems A 2 B 2 A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 11 / 43

Examples of root systems G 2 I 2 ( 5 ) A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 12 / 43

Multiplicity function Multiplicity function A multiplicity function is a G -invariant function k : R → C which will be fixed and � 0. A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 13 / 43

Measure Let � N = N + k ( α ) α ∈ R ( N is the homogeneous dimension ). A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 14 / 43

Measure Let � N = N + k ( α ) α ∈ R ( N is the homogeneous dimension ). We define the measure � |� α, x �| k ( α ) . w ( x ) = α ∈ R A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 14 / 43

Measure Let � N = N + k ( α ) α ∈ R ( N is the homogeneous dimension ). We define the measure � |� α, x �| k ( α ) . w ( x ) = α ∈ R We have w ( B ( x , r )) ∼ r N � ( |� x , α �| + r ) k ( α ) , α ∈ R so dw ( x ) is doubling. A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 14 / 43

Dunkl operators Dunkl operators Given a root system R and multiplicity function k ( α ) the Dunkl operator T ξ is the following k -deformation of the directional derivative ∂ ξ by a difference operator: T ξ f ( x ) = ∂ ξ f ( x ) A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 15 / 43

Dunkl operators Dunkl operators Given a root system R and multiplicity function k ( α ) the Dunkl operator T ξ is the following k -deformation of the directional derivative ∂ ξ by a difference operator: � k ( α ) � α, ξ � f ( x ) − f ( σ α x ) T ξ f ( x ) = ∂ ξ f ( x ) + . 2 � α, x � α ∈ R A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 15 / 43

Dunkl operators Dunkl operators Given a root system R and multiplicity function k ( α ) the Dunkl operator T ξ is the following k -deformation of the directional derivative ∂ ξ by a difference operator: � k ( α ) � α, ξ � f ( x ) − f ( σ α x ) T ξ f ( x ) = ∂ ξ f ( x ) + . 2 � α, x � α ∈ R Example for N = 1 Tf ( x ) = ∂ f ( x ) + k ( α ) f ( x ) − f ( − x ) . x A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 15 / 43

Dunkl operators Dunkl operators Given a root system R and multiplicity function k ( α ) the Dunkl operator T ξ is the following k -deformation of the directional derivative ∂ ξ by a difference operator: � k ( α ) � α, ξ � f ( x ) − f ( σ α x ) T ξ f ( x ) = ∂ ξ f ( x ) + . 2 � α, x � α ∈ R Example for N = 1 Tf ( x ) = ∂ f ( x ) + k ( α ) f ( x ) − f ( − x ) . x Difference No Leibniz rule! A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 15 / 43

Dunkl kernel and Dunkl transform Dunkl kernel For fixed y ∈ R N the Dunkl kernel E ( x , y ) is the unique solution of the system T ξ f = � ξ, y � f , f ( 0 ) = 1 . In particular, T j , x E ( x , y ) = T e j , x E ( x , y ) = y j E ( x , y ) . A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 16 / 43

Dunkl kernel and Dunkl transform Dunkl kernel For fixed y ∈ R N the Dunkl kernel E ( x , y ) is the unique solution of the system T ξ f = � ξ, y � f , f ( 0 ) = 1 . In particular, T j , x E ( x , y ) = T e j , x E ( x , y ) = y j E ( x , y ) . E ( x , y ) is a generalization of exp( � x , y � ) . A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 16 / 43

Dunkl kernel and Dunkl transform Dunkl kernel For fixed y ∈ R N the Dunkl kernel E ( x , y ) is the unique solution of the system T ξ f = � ξ, y � f , f ( 0 ) = 1 . In particular, T j , x E ( x , y ) = T e j , x E ( x , y ) = y j E ( x , y ) . E ( x , y ) is a generalization of exp( � x , y � ) . Dunkl transform =generalization of Fourier transform The Dunkl transform is defined on L 1 ( dw ) by � F f ( ξ ) = c − 1 R N f ( x ) E ( x , − i ξ ) dw ( x ) . k A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 16 / 43

1. Introduction Fourier analysis in the rational Dunkl setting 2. Dunkl translations 3. Semigroups of operators Radial case - heat semigroup associated with the Dunkl Laplacian Nonradial cases 4. Idea of the proofs Convolution with radial function Support of τ x f ( −· ) A. Hejna (IM UWr.) Dunkl semigroups Będlewo, 21.05.2019 17 / 43