Operators related to the Jacobi setting, for all admissible - PowerPoint PPT Presentation

Operators related to the Jacobi setting, for all admissible parameter values Peter Sjögren University of Gothenburg Joint work with A. Nowak and T. Szarek Alba, June 2013 () 1 / 18

Let P α,β be the classical Jacobi polynomials, seen via the n transformation x = cos θ as functions of θ ∈ [ 0 , π ] . Here α, β > − 1. () 2 / 18

Let P α,β be the classical Jacobi polynomials, seen via the n transformation x = cos θ as functions of θ ∈ [ 0 , π ] . Here α, β > − 1. The P α,β are orthogonal with respect to the measure n sin θ � 2 α + 1 � cos θ � 2 β + 1 � in d µ α,β ( θ ) = d θ [ 0 , π ] , 2 2 and we take them to be normalized in L 2 ( d µ α,β ) . () 2 / 18

Let P α,β be the classical Jacobi polynomials, seen via the n transformation x = cos θ as functions of θ ∈ [ 0 , π ] . Here α, β > − 1. The P α,β are orthogonal with respect to the measure n sin θ � 2 α + 1 � cos θ � 2 β + 1 � in d µ α,β ( θ ) = d θ [ 0 , π ] , 2 2 and we take them to be normalized in L 2 ( d µ α,β ) . They are eigenfunctions of the Jacobi operator J α,β = − d 2 d θ 2 − α − β + ( α + β + 1 ) cos θ d � α + β + 1 � 2 d θ + , sin θ 2 � 2 � n + α + β + 1 with eigenvalues . 2 () 2 / 18

√ J α,β ) , t > 0, can be The corresponding Poisson operator exp ( − t defined spectrally and has the integral kernel ∞ e − t | n + α + β + 1 |P α,β H α,β ( θ ) P α,β � ( θ, ϕ ) = ( ϕ ) , 2 n n t n = 0 called the Jacobi-Poisson kernel. () 3 / 18

√ J α,β ) , t > 0, can be The corresponding Poisson operator exp ( − t defined spectrally and has the integral kernel ∞ e − t | n + α + β + 1 |P α,β H α,β ( θ ) P α,β � ( θ, ϕ ) = ( ϕ ) , 2 n n t n = 0 called the Jacobi-Poisson kernel. By means of H α,β , one can express the kernels of many operators t associated with the Jacobi setting. () 3 / 18

√ J α,β ) , t > 0, can be The corresponding Poisson operator exp ( − t defined spectrally and has the integral kernel ∞ e − t | n + α + β + 1 |P α,β H α,β ( θ ) P α,β � ( θ, ϕ ) = ( ϕ ) , 2 n n t n = 0 called the Jacobi-Poisson kernel. By means of H α,β , one can express the kernels of many operators t associated with the Jacobi setting. We list some of these operators and their kernels. () 3 / 18

Riesz transforms D N ( J α,β ) − N / 2 of order N ≥ 1, with kernels � ∞ ( θ, ϕ ) t N − 1 dt . θ H α,β ∂ N c N t 0 () 4 / 18

Riesz transforms D N ( J α,β ) − N / 2 of order N ≥ 1, with kernels � ∞ ( θ, ϕ ) t N − 1 dt . θ H α,β ∂ N c N t 0 � √ � Laplace transform type multipliers m J α,β with m given by m ( z ) = z L φ ( z ) , where L denotes the Laplace transform and φ ∈ L ∞ ( R + ) . () 4 / 18

Riesz transforms D N ( J α,β ) − N / 2 of order N ≥ 1, with kernels � ∞ ( θ, ϕ ) t N − 1 dt . θ H α,β ∂ N c N t 0 � √ � Laplace transform type multipliers m J α,β with m given by m ( z ) = z L φ ( z ) , where L denotes the Laplace transform and φ ∈ L ∞ ( R + ) . The kernel is � ∞ φ ( t ) ∂ t H α,β − ( θ, ϕ ) dt . t 0 () 4 / 18

Riesz transforms D N ( J α,β ) − N / 2 of order N ≥ 1, with kernels � ∞ ( θ, ϕ ) t N − 1 dt . θ H α,β ∂ N c N t 0 � √ � Laplace transform type multipliers m J α,β with m given by m ( z ) = z L φ ( z ) , where L denotes the Laplace transform and φ ∈ L ∞ ( R + ) . The kernel is � ∞ φ ( t ) ∂ t H α,β − ( θ, ϕ ) dt . t 0 With φ ( t ) = const . t − 2 i γ , γ � = 0, this gives the imaginary power ( J α,β ) i γ of the Jacobi operator. () 4 / 18

Riesz transforms D N ( J α,β ) − N / 2 of order N ≥ 1, with kernels � ∞ ( θ, ϕ ) t N − 1 dt . θ H α,β ∂ N c N t 0 � √ � Laplace transform type multipliers m J α,β with m given by m ( z ) = z L φ ( z ) , where L denotes the Laplace transform and φ ∈ L ∞ ( R + ) . The kernel is � ∞ φ ( t ) ∂ t H α,β − ( θ, ϕ ) dt . t 0 With φ ( t ) = const . t − 2 i γ , γ � = 0, this gives the imaginary power ( J α,β ) i γ of the Jacobi operator. � √ � Laplace-Stieltjes transform type multipliers m J α,β , where m = L ν and ν is a signed or complex Borel measure on e − t | α + β + 1 | / 2 d | ν | ( t ) < ∞ . The kernel is now � ( 0 , ∞ ) satisfying � H α,β ( θ, ϕ ) d ν ( t ) . t ( 0 , ∞ ) () 4 / 18

These kernels are scalar-valued. But allowing kernels taking values in a Banach space B , one can write more operator kernels. () 5 / 18

These kernels are scalar-valued. But allowing kernels taking values in a Banach space B , one can write more operator kernels. Examples: The Poisson maximal operator , with kernel H α,β � � ( θ, ϕ ) t t > 0 and B = L ∞ ( R + ; dt ) . () 5 / 18

These kernels are scalar-valued. But allowing kernels taking values in a Banach space B , one can write more operator kernels. Examples: The Poisson maximal operator , with kernel H α,β � � ( θ, ϕ ) t t > 0 and B = L ∞ ( R + ; dt ) . The mixed square functions , with kernels t H α,β ∂ N θ ∂ M � � ( θ, ϕ ) t t > 0 and B = L 2 ( R + ; t 2 M + 2 N − 1 dt ) . Here M + N > 0 . () 5 / 18

Old result: Theorem (Nowak and Sjögren, J.F.A.A. 2012) For α, β ≥ − 1 / 2 , the following among the operators listed above are bounded on L p ( d µ α,β ) , 1 < p < ∞ , and of weak type (1,1) for µ α,β : () 6 / 18

Old result: Theorem (Nowak and Sjögren, J.F.A.A. 2012) For α, β ≥ − 1 / 2 , the following among the operators listed above are bounded on L p ( d µ α,β ) , 1 < p < ∞ , and of weak type (1,1) for µ α,β : The Riesz transforms, the imaginary powers of the Jacobi operator, the maximal operator and the mixed square functions. () 6 / 18

Old result: Theorem (Nowak and Sjögren, J.F.A.A. 2012) For α, β ≥ − 1 / 2 , the following among the operators listed above are bounded on L p ( d µ α,β ) , 1 < p < ∞ , and of weak type (1,1) for µ α,β : The Riesz transforms, the imaginary powers of the Jacobi operator, the maximal operator and the mixed square functions. These operators are also bounded between the corresponding weighted spaces with a weight in the Muckenhoupt class A α,β , p defined with respect to the measure µ α,β . () 6 / 18

Old result: Theorem (Nowak and Sjögren, J.F.A.A. 2012) For α, β ≥ − 1 / 2 , the following among the operators listed above are bounded on L p ( d µ α,β ) , 1 < p < ∞ , and of weak type (1,1) for µ α,β : The Riesz transforms, the imaginary powers of the Jacobi operator, the maximal operator and the mixed square functions. These operators are also bounded between the corresponding weighted spaces with a weight in the Muckenhoupt class A α,β , p defined with respect to the measure µ α,β . New result: Theorem (Nowak, Sjögren and Szarek, 2013) The preceding theorem holds for all α, β > − 1 and all the operators listed above. () 6 / 18

To prove both theorems, one shows that the operators are well-behaved Calderón-Zygmund operators () 7 / 18

To prove both theorems, one shows that the operators are well-behaved Calderón-Zygmund operators on the space of homogeneous type defined by the interval [ 0 , π ] with the measure µ α,β and the ordinary distance. () 7 / 18

To prove both theorems, one shows that the operators are well-behaved Calderón-Zygmund operators on the space of homogeneous type defined by the interval [ 0 , π ] with the measure µ α,β and the ordinary distance. The operators can be seen to be bounded on L 2 ( d µ α,β ) (or L ∞ ( d µ α,β ) in the case of the maximal operator). () 7 / 18

To prove both theorems, one shows that the operators are well-behaved Calderón-Zygmund operators on the space of homogeneous type defined by the interval [ 0 , π ] with the measure µ α,β and the ordinary distance. The operators can be seen to be bounded on L 2 ( d µ α,β ) (or L ∞ ( d µ α,β ) in the case of the maximal operator). So the main crux is to verify that their off-diagonal kernels satisfy the standard estimates in this space. () 7 / 18

To prove both theorems, one shows that the operators are well-behaved Calderón-Zygmund operators on the space of homogeneous type defined by the interval [ 0 , π ] with the measure µ α,β and the ordinary distance. The operators can be seen to be bounded on L 2 ( d µ α,β ) (or L ∞ ( d µ α,β ) in the case of the maximal operator). So the main crux is to verify that their off-diagonal kernels satisfy the standard estimates in this space. It also requires some effort to prove that the kernels are really as described above. () 7 / 18

The argument for the old theorem is based on the following integral formula for the Poisson kernel �� H α,β Ψ α,β ( t , θ, ϕ, u , v ) d Π α ( u ) d Π β ( v ) , ( θ, ϕ ) = t valid for α, β > − 1 / 2, () 8 / 18

The argument for the old theorem is based on the following integral formula for the Poisson kernel �� H α,β Ψ α,β ( t , θ, ϕ, u , v ) d Π α ( u ) d Π β ( v ) , ( θ, ϕ ) = t valid for α, β > − 1 / 2, where c α,β sinh t Ψ α,β ( t , θ, ϕ, u , v ) = 2 2 − 1 + q ( θ, ϕ, u , v )) α + β + 2 , ( cosh t () 8 / 18