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On semigroups associated with the Dunkl operators Joint work with Jacek Dziubaski Agnieszka Hejna Instytut Matematyczny Uniwersytet Wrocawski Bdlewo, 21.05.2019 A. Hejna (IM UWr.) Dunkl semigroups Bdlewo, 21.05.2019 1 / 43 Table

slide-1
SLIDE 1

On semigroups associated with the Dunkl

  • perators

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

slide-2
SLIDE 2

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 τxf (−·)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 2 / 43

slide-3
SLIDE 3

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

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SLIDE 4

Some classical semigroups

Classical heat semigroup

Generator: ∆ = N

j=1 ∂2 j

Associated multiplier: e−|ξ|2

Upper heat kernel estimate (t = 1)

1 (4π)N/2 e− 1

4 |x−y|2

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 3 / 43

slide-5
SLIDE 5

Some classical semigroups

Classical heat semigroup

Generator: ∆ = N

j=1 ∂2 j

Associated multiplier: e−|ξ|2

Upper heat kernel estimate (t = 1)

1 (4π)N/2 e− 1

4 |x−y|2

Semigroups associated with higher order derivatives

Generator: L = (−1)ℓ+1 N

j=1 ∂2ℓ j

Associated multiplier: e−N

j=1 |ξj|2ℓ

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 3 / 43

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SLIDE 6

Some classical semigroups

Classical heat semigroup

Generator: ∆ = N

j=1 ∂2 j

Associated multiplier: e−|ξ|2

Upper heat kernel estimate (t = 1)

1 (4π)N/2 e− 1

4 |x−y|2

Semigroups associated with higher order derivatives

Generator: L = (−1)ℓ+1 N

j=1 ∂2ℓ j

Associated multiplier: e−N

j=1 |ξj|2ℓ

Upper integral kernel estimate (t=1)

Ce−c|x−y|

2ℓ 2ℓ−1

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 3 / 43

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SLIDE 7

Some Dunkl semigroups

Dunkl heat semigroup

Generator: ∆ = N

j=1 T 2 j

Associated multiplier: e−|ξ|2

Upper heat kernel estimate (t = 1)

w(B(x, 1))−1e−cd(x,y)2

Semigroups associated with higher order Dunkl operators

Generator: L = (−1)ℓ+1 N

j=1 T 2ℓ j

Associated multiplier: e−N

j=1 |ξj|2ℓ

Upper integral kernel estimate (t=1)

w(B(x, 1))−1e−cd(x,y)

2ℓ 2ℓ−1

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 4 / 43

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SLIDE 8

H¨

  • rmander’s multiplier theorem

Theorem (H¨

  • rmander)

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

t>0

ψ(·)m(t·)W s

2 < ∞

for some s > N/2, then

  • Tmf = (mˆ

f ), is (A) of weak type (1, 1), (B) of strong type (p, p) for 1 < p < ∞, (C) bounded on the Hardy space H1

atom.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 5 / 43

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SLIDE 9

H¨

  • rmander’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

t>0

ψ(·)m(t·)W s

2 < ∞

for some s > N, then Tmf = F−1(mFf ), is (A) of weak type (1, 1), (B) of strong type (p, p) for 1 < p < ∞, (C) bounded on the Hardy space H1

atom.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 6 / 43

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SLIDE 10

Reflections

We consider the Euclidean space RN with the scalar product x, y = N

j=1 xjyj, x = (x1, ..., xN), y = (y1, ..., yN).

Reflection

For a nonzero vector α ∈ RN the reflection σα with respect to the

  • rthogonal hyperplane α⊥
  • rthogonal to a nonzero vector α

is given by σαx = x − 2x, α α2 α.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 7 / 43

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SLIDE 11

Root system and Weyl group

Root system

A finite set R ⊂ RN \ {0} is called a root system if σα(R) = R for every α ∈ R.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 8 / 43

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SLIDE 12

Root system and Weyl group

Root system

A finite set R ⊂ RN \ {0} is called a root system if σα(R) = R for every α ∈ R.

Weyl group

The finite group G generated by the reflections σα is called the Weyl group (reflection group) of the root system.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 8 / 43

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SLIDE 13

Examples - product root systems

A1

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 9 / 43

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SLIDE 14

Examples - product root systems

A1 × A1 A1 × A1 × A1

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 10 / 43

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SLIDE 15

Examples of root systems

A2 B2

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 11 / 43

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SLIDE 16

Examples of root systems

G2 I2(5)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 12 / 43

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SLIDE 17

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

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SLIDE 18

Measure

Let N = N +

  • α∈R

k(α) (N is the homogeneous dimension).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 14 / 43

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SLIDE 19

Measure

Let N = N +

  • α∈R

k(α) (N is the homogeneous dimension). We define the measure w(x) =

  • α∈R

|α, x|k(α).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 14 / 43

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SLIDE 20

Measure

Let N = N +

  • α∈R

k(α) (N is the homogeneous dimension). We define the measure w(x) =

  • α∈R

|α, x|k(α). We have w(B(x, r)) ∼ rN

α∈R

(|x, α| + r)k(α), so dw(x) is doubling.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 14 / 43

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SLIDE 21

Dunkl operators

Dunkl operators

Given a root system R and multiplicity function k(α) the Dunkl operator Tξ is the following k-deformation of the directional derivative ∂ξ by a difference operator: Tξf (x) = ∂ξf (x)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 15 / 43

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SLIDE 22

Dunkl operators

Dunkl operators

Given a root system R and multiplicity function k(α) the Dunkl operator Tξ is the following k-deformation of the directional derivative ∂ξ by a difference operator: Tξf (x) = ∂ξf (x) +

  • α∈R

k(α) 2 α, ξf (x) − f (σαx) α, x .

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 15 / 43

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SLIDE 23

Dunkl operators

Dunkl operators

Given a root system R and multiplicity function k(α) the Dunkl operator Tξ is the following k-deformation of the directional derivative ∂ξ by a difference operator: Tξf (x) = ∂ξf (x) +

  • α∈R

k(α) 2 α, ξf (x) − f (σαx) α, x .

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

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SLIDE 24

Dunkl operators

Dunkl operators

Given a root system R and multiplicity function k(α) the Dunkl operator Tξ is the following k-deformation of the directional derivative ∂ξ by a difference operator: Tξf (x) = ∂ξf (x) +

  • α∈R

k(α) 2 α, ξf (x) − f (σαx) α, x .

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

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SLIDE 25

Dunkl kernel and Dunkl transform

Dunkl kernel

For fixed y ∈ RN the Dunkl kernel E(x, y) is the unique solution of the system Tξf = ξ, yf , f (0) = 1. In particular, Tj,xE(x, y) = Tej,xE(x, y) = yjE(x, y).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 16 / 43

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SLIDE 26

Dunkl kernel and Dunkl transform

Dunkl kernel

For fixed y ∈ RN the Dunkl kernel E(x, y) is the unique solution of the system Tξf = ξ, yf , f (0) = 1. In particular, Tj,xE(x, y) = Tej,xE(x, y) = yjE(x, y). E(x, y) is a generalization of exp(x, y).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 16 / 43

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SLIDE 27

Dunkl kernel and Dunkl transform

Dunkl kernel

For fixed y ∈ RN the Dunkl kernel E(x, y) is the unique solution of the system Tξf = ξ, yf , f (0) = 1. In particular, Tj,xE(x, y) = Tej,xE(x, y) = yjE(x, y). E(x, y) is a generalization of exp(x, y).

Dunkl transform =generalization of Fourier transform

The Dunkl transform is defined on L1(dw) by Ff (ξ) = c−1

k

  • RN f (x)E(x, −iξ) dw(x).
  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 16 / 43

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SLIDE 28
  • 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 τxf (−·)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 17 / 43

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SLIDE 29

Dunkl translations

Dunkl translations =generalization of translations

The Dunkl translation τxf of f ∈ S(RN) by x ∈ RN is defined by τxf (y) = c−1

k

  • RN E(iξ, x) E(iξ, y) Ff (ξ) dw(ξ).
  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 18 / 43

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SLIDE 30

Dunkl translations

Dunkl translations =generalization of translations

The Dunkl translation τxf of f ∈ S(RN) by x ∈ RN is defined by τxf (y) = c−1

k

  • RN E(iξ, x) E(iξ, y) Ff (ξ) dw(ξ).

Dunkl convolution =generalization of convolution

The Dunkl convolution of two reasonable functions is defined by (f ∗ g)(x) = ck F−1[(Ff )(Fg)](x) =

  • RN τxf (−y)g(y) dw(y).
  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 18 / 43

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SLIDE 31

Essential difference

Dunkl translations don’t form a group

τxτy = τx+y

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 19 / 43

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SLIDE 32

Positivity of the Dunkl translations

Positivity of the Dunkl translations (radial function)

Suppose that f ∈ L2(dw) is radial and f 0 a.e. Then τxf 0 a. e. for all x ∈ RN.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 20 / 43

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SLIDE 33

Positivity of the Dunkl translations

Positivity of the Dunkl translations (radial function)

Suppose that f ∈ L2(dw) is radial and f 0 a.e. Then τxf 0 a. e. for all x ∈ RN.

Nonpositivity of the Dunkl translations

There are: root system R, multiplicity function k 0, x ∈ RN, and L2(dw) ∋ f 0 a.e. such that τxf < 0 on the set of positive Lebesgue measure.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 20 / 43

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SLIDE 34

Boundedness of the Dunkl translations

L2(dw)-case

By the Plancherel’s theorem for the Dunkl transform sup

x∈RN τxL2(dw)→L2(dw) = 1.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 21 / 43

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SLIDE 35

Boundedness of the Dunkl translations

L2(dw)-case

By the Plancherel’s theorem for the Dunkl transform sup

x∈RN τxL2(dw)→L2(dw) = 1.

Open problem

Let 1 < p < ∞. Then sup

x∈RN τxLp(dw)→Lp(dw) < ∞.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 21 / 43

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SLIDE 36

Radial case

Exception

If f ∈ Lp(dw) is radial, then for all x ∈ RN we have τxf Lp(dw) f Lp(dw).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 22 / 43

slide-37
SLIDE 37

Radial case

Exception

If f ∈ Lp(dw) is radial, then for all x ∈ RN we have τxf Lp(dw) f Lp(dw). Conclusion: Dunkl translation of radial function is easier to treat.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 22 / 43

slide-38
SLIDE 38
  • 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 τxf (−·)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 23 / 43

slide-39
SLIDE 39

Distance of orbits

Let us define the distance of the orbits O(x) and O(y) to be d(x, y) = min

σ∈G σ(x) − y.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 24 / 43

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SLIDE 40

Distance of orbits

Let us define the distance of the orbits O(x) and O(y) to be d(x, y) = min

σ∈G σ(x) − y.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 24 / 43

slide-41
SLIDE 41

Distance of orbits

Let us define the distance of the orbits O(x) and O(y) to be d(x, y) = min

σ∈G σ(x) − y x − y.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 24 / 43

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SLIDE 42

Dunkl Laplacian and Dunkl semigroup

Dunkl Laplacian

The Dunkl Laplacian associated with G and k is the differential-difference

  • perator

∆ =

N

  • j=1

T 2

j .

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 25 / 43

slide-43
SLIDE 43

Dunkl Laplacian and Dunkl semigroup

Dunkl Laplacian

The Dunkl Laplacian associated with G and k is the differential-difference

  • perator

∆ =

N

  • j=1

T 2

j .

Heat semigroup

The operator ∆ generates the semigroup Ht = et∆ of linear self-adjoint contractions on L2(dw). The semigroup has the form et∆f (x) =

  • RN τxht(−y)f (y) dw(y),

where ht is classical heat kernel, which is radial.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 25 / 43

slide-44
SLIDE 44

Heat kernel estimates

Theorem (J.-P.. Anker, J. Dziubański, A.H.)

There are C, c > 0 such that for all x, y ∈ RN and t > 0 we have ht(x, y) C w(B(x, √ t ))−1 e−c d(x,y)2/t,

  • ht(x, y) − ht(x, y′)
  • C

y−y′

√t

  • w(B(x,

√ t ))−1 e−c d(x,y)2/t.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 26 / 43

slide-45
SLIDE 45
  • 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 τxf (−·)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 27 / 43

slide-46
SLIDE 46

Derivatives of higher order

Semigroup operator associated with Dunkl derivatives of higher order

We define L = (−1)ℓ+1

N

  • j=1

T 2ℓ

j .

It is generator of semigroup of operators on L2(dw) with kernels of the form qt(x, y) = τx(Fq)t(−y), where q is the associated nonradial multiplier.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 28 / 43

slide-47
SLIDE 47

Derivatives of higher order

Semigroup operator associated with Dunkl derivatives of higher order

We define L = (−1)ℓ+1

N

  • j=1

T 2ℓ

j .

It is generator of semigroup of operators on L2(dw) with kernels of the form qt(x, y) = τx(Fq)t(−y), where q is the associated nonradial multiplier.

Theorem (J. Dziubański, A.H.)

There are constants C, c > 0 such that for all x, y ∈ RN we have |q1(x, y)| C(w(B(x, 1)))−1 exp(−cd(x, y)2ℓ/(2ℓ−1)).

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 28 / 43

slide-48
SLIDE 48

H¨

  • rmander’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

t>0

ψ(·)m(t·)W s

2 < ∞

for some s > N, then Tmf = F−1(mFf ), is (A) of weak type (1, 1), (B) of strong type (p, p) for 1 < p < ∞, (C) bounded on the Hardy space H1

atom.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 29 / 43

slide-49
SLIDE 49
  • 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 τxf (−·)

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 30 / 43

slide-50
SLIDE 50

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-51
SLIDE 51

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-52
SLIDE 52

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-53
SLIDE 53

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

3 Pass to kernel pointwise estimate

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-54
SLIDE 54

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

3 Pass to kernel pointwise estimate no

group property of translations!

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-55
SLIDE 55

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

3 Pass to kernel pointwise estimate no

group property of translations!

4 Perturbation of operator by

Laplacian in order to use convolution with radial function technique.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-56
SLIDE 56

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

3 Pass to kernel pointwise estimate no

group property of translations!

4 Perturbation of operator by

Laplacian in order to use convolution with radial function technique. H¨

  • rmander’s multiplier

theorem

1 Imitate the classical

proof of H¨

  • rmander.
  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-57
SLIDE 57

Idea of the proofs

Kernel estimate for L

1 Follow attept of Dziubański &

Hulanicki, 1989 and Dziubański & Hebisch & Zienkiewicz, 1994: introduce (exponential) weighted L2 bilinear forms.

2 Prove G¨

arding inequality and use theorem of Lions no Leibniz rule!

3 Pass to kernel pointwise estimate no

group property of translations!

4 Perturbation of operator by

Laplacian in order to use convolution with radial function technique. H¨

  • rmander’s multiplier

theorem

1 Imitate the classical

proof of H¨

  • rmander.

2 Convolution with

radial function technique.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 31 / 43

slide-58
SLIDE 58

Convolution with radial function technique

L = (−1)ℓ+1 N

j=1 T 2ℓ j , associated multiplier q(ξ) = exp(− N j=1 |ξj|2ℓ).

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SLIDE 59

Convolution with radial function technique

L = (−1)ℓ+1 N

j=1 T 2ℓ j , associated multiplier q(ξ) = exp(− N j=1 |ξj|2ℓ).

Introduce L(ε) = L − ε∆, associated multiplier is q(ε)(ξ) = exp(−

N

  • j=1

|ξj|2ℓ + ε|ξ|2).

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SLIDE 60

Convolution with radial function technique

L = (−1)ℓ+1 N

j=1 T 2ℓ j , associated multiplier q(ξ) = exp(− N j=1 |ξj|2ℓ).

Introduce L(ε) = L − ε∆, associated multiplier is q(ε)(ξ) = exp(−

N

  • j=1

|ξj|2ℓ + ε|ξ|2). Then q(ξ) = q(ε)(ξ)e−ε|ξ|2,

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SLIDE 61

Convolution with radial function technique

L = (−1)ℓ+1 N

j=1 T 2ℓ j , associated multiplier q(ξ) = exp(− N j=1 |ξj|2ℓ).

Introduce L(ε) = L − ε∆, associated multiplier is q(ε)(ξ) = exp(−

N

  • j=1

|ξj|2ℓ + ε|ξ|2). Then q(ξ) = q(ε)(ξ)e−ε|ξ|2, so τx(Fq)(−y) = τx(Fq(ε) ∗ hε)(−y) = (Fq(ε)) ∗ τx(hε)(−y). Translation is on radial function!

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SLIDE 62

Further problem

g-radial, f - not necessary radial

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SLIDE 63

Further problem

g-radial, f - not necessary radial

Classical case (trivial)

τx(f ∗ g)L1(dw) f L1(dw)gL∞

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SLIDE 64

Further problem

g-radial, f - not necessary radial

Classical case (trivial)

τx(f ∗ g)L1(dw) f L1(dw)gL∞

Dunkl case (no L1-boundedness!)

τx(f ∗ g)L1(dw) f (·)(1 + | · |)N/2L1(dw)g(·)(1 + | · |)NL∞

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Dunkl semigroups Będlewo, 21.05.2019 33 / 43

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SLIDE 65

Further problem

g-radial, f - not necessary radial

Classical case (trivial)

τx(f ∗ g)L1(dw) f L1(dw)gL∞

Dunkl case (no L1-boundedness!)

τx(f ∗ g)L1(dw) f (·)(1 + | · |)N/2L1(dw)g(·)(1 + | · |)NL∞ Idea: Pass to L2 by Cauchy–Schwarz. τx(f ∗ g)L1(dw) w(supp τx(f ∗ g))1/2τx(f ∗ g)L2(dw)

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Dunkl semigroups Będlewo, 21.05.2019 33 / 43

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SLIDE 66
  • 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 τxf (−·)

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SLIDE 67

Question

Suppose that f ∈ L2(dw) is such that supp f ⊆ B(0, 1).

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SLIDE 68

Question

Suppose that f ∈ L2(dw) is such that supp f ⊆ B(0, 1). If we consider fx = f (x − ·), then supp fx ⊆ B(x, 1).

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SLIDE 69

Question

Suppose that f ∈ L2(dw) is such that supp f ⊆ B(0, 1). If we consider fx = f (x − ·), then supp fx ⊆ B(x, 1). Question: What about supp τxf (−·)?

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SLIDE 70

What was known? 1/2

Results of Amri, Anker and Sifi (Paley-Wiener approach) imply supp τxf (−·) ⊆ {y : x − 1 y x + 1}.

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SLIDE 71

What was known? 2/2

Results of R¨

  • sler imply that if f is radial, then

supp τxf (− ·) ⊆ O(B(x, 1)) =

  • g∈G

B(g(x), 1).

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SLIDE 72

Theorem

Theorem (J. Dziubański, A.H.)

Let f ∈ L2(dw), supp f ⊆ B(0, 1), and x ∈ RN. Then supp τxf (− ·) ⊆ O(B(x, 1)).

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SLIDE 73

What is the point?

The measure of O(B(x, 1)) is much smaller than the measure of {y : x − 1 y x + 1}.

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SLIDE 74

Sketch of the proof 1/2

Let us denote gL(x) = max{0, (1 − x2)}L.

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SLIDE 75

Sketch of the proof 1/2

Let us denote gL(x) = max{0, (1 − x2)}L. For α = (α1, α2, . . . , αN) ∈ NN

0 = (N ∪ {0})N we define

T 0

j = I,

T α := T α1

1

  • T α2

2

  • . . . ◦ T αN

N .

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Dunkl semigroups Będlewo, 21.05.2019 40 / 43

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SLIDE 76

Sketch of the proof 1/2

Let us denote gL(x) = max{0, (1 − x2)}L. For α = (α1, α2, . . . , αN) ∈ NN

0 = (N ∪ {0})N we define

T 0

j = I,

T α := T α1

1

  • T α2

2

  • . . . ◦ T αN

N . 1 By induction we show that if p is a polynomial of degree d, then

p(x)gL(x) can be written as p(x)gL(x) =

d

  • ℓ=0
  • αℓ

cℓ,αT α(gL+ℓ)(x) for some cℓ,α ∈ C.

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Dunkl semigroups Będlewo, 21.05.2019 40 / 43

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SLIDE 77

Sketch of the proof 1/2

Let us denote gL(x) = max{0, (1 − x2)}L. For α = (α1, α2, . . . , αN) ∈ NN

0 = (N ∪ {0})N we define

T 0

j = I,

T α := T α1

1

  • T α2

2

  • . . . ◦ T αN

N . 1 By induction we show that if p is a polynomial of degree d, then

p(x)gL(x) can be written as p(x)gL(x) =

d

  • ℓ=0
  • αℓ

cℓ,αT α(gL+ℓ)(x) for some cℓ,α ∈ C. The key point is the fact that the Leibniz rule can be applied Tj(pgL) = gL(Tjp) + p(TjgL).

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Dunkl semigroups Będlewo, 21.05.2019 40 / 43

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SLIDE 78

Sketch of the proof 2/2

2 The set {p(·)g1(·) : p is a polynomial} is dense in L2(B(0, 1), dw).

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SLIDE 79

Sketch of the proof 2/2

2 The set {p(·)g1(·) : p is a polynomial} is dense in L2(B(0, 1), dw). 3 τx is a contraction on L2(dw), so for any ε > 0 there is a polynomial

p such that τxf − τx(pg1)L2(dw) f − pg1L2(dw) < ε. (⋆)

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SLIDE 80

Sketch of the proof 2/2

2 The set {p(·)g1(·) : p is a polynomial} is dense in L2(B(0, 1), dw). 3 τx is a contraction on L2(dw), so for any ε > 0 there is a polynomial

p such that τxf − τx(pg1)L2(dw) f − pg1L2(dw) < ε. (⋆)

4 The Dunkl translations commute with the Dunkl operators, so

τx(pg1)(−y) =

d

  • ℓ=0
  • αℓ

cℓ,αT ατx(g1+ℓ)(−y).

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Dunkl semigroups Będlewo, 21.05.2019 41 / 43

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SLIDE 81

Sketch of the proof 2/2

2 The set {p(·)g1(·) : p is a polynomial} is dense in L2(B(0, 1), dw). 3 τx is a contraction on L2(dw), so for any ε > 0 there is a polynomial

p such that τxf − τx(pg1)L2(dw) f − pg1L2(dw) < ε. (⋆)

4 The Dunkl translations commute with the Dunkl operators, so

τx(pg1)(−y) =

d

  • ℓ=0
  • αℓ

cℓ,αT ατx(g1+ℓ)(−y).

5 By the results of R¨

  • sler supp T ατx(g1+ℓ) ⊆ O(B(x, 1)), so (⋆)

implies the claim.

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SLIDE 82

Thank you for your attention.

Thank you for your attention.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 42 / 43

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SLIDE 83

Bibliography

J.-Ph. Anker, J. Dziubański, A. Hejna, Harmonic functions, conjugate harmonic functions and the Hardy space H1 in the rational Dunkl setting, [doi:10.1007/s00041-019-09666-0], to appear in J. Fourier

  • Anal. Appl.
  • J. Dziubański, A. Hulanicki, On semigroups generated by left-invariant

positive differential operators on nilpotent Lie groups, Studia Math. 94 (1989), 81–95.

  • J. Dziubański, W. Hebisch, and J. Zienkiewicz, Note on semigroups

generated by positive Rockland operators on graded homogeneous groups, Studia.Math. 110 (1994), 115–126.

  • J. Dziubański and A. Hejna, H¨
  • rmander’s multiplier theorem for the

Dunkl transform, [doi:10.1016/j.jfa.2019.03.002], to appear in JFA.

  • J. Dziubański and A.Hejna, On semigroups generated by sums of even

powers of Dunkl operators, arxiv.

  • A. Hejna (IM UWr.)

Dunkl semigroups Będlewo, 21.05.2019 43 / 43