On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw - PowerPoint PPT Presentation

On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw Chemnitz, August, 2017 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 1 / 17

❘ ❩ ❩ ❘ ❘ ❘ ❩ Trivial bounds For f ∈ L 1 ( ❘ ) define its Fourier transform by � 1 ˆ e − it ξ f ( t ) dt , √ f ( ξ ) := 2 π ❘ and for f ∈ L 1 ( 0 , 2 π ) define its Fourier coefficients (transform) by � 2 π f ( n ) = 1 ˆ e − int f ( t ) dt , n ∈ ❩ . 2 π 0 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 2 / 17

Trivial bounds For f ∈ L 1 ( ❘ ) define its Fourier transform by � 1 ˆ e − it ξ f ( t ) dt , √ f ( ξ ) := 2 π ❘ and for f ∈ L 1 ( 0 , 2 π ) define its Fourier coefficients (transform) by � 2 π f ( n ) = 1 ˆ e − int f ( t ) dt , n ∈ ❩ . 2 π 0 By the Riemann-Lebesgue Lemma: ˆ (ˆ f ∈ C 0 ( ❘ ) , f ( n )) n ∈ ❩ ∈ c 0 ( ❩ ) . From Plancherel’s (Parseval) theorem: ˆ (ˆ f ∈ L 2 ( ❘ ) f ∈ L 1 ( ❘ ) ∩ L 2 ( ❘ ) , f ( n )) n ∈ ❩ ∈ l 2 dla f ∈ L 2 ( 0 , 2 π )) . je´ sli Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 2 / 17

Problem. Q UESTION : How ‘large’ can the Fourier transform be ? Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 3 / 17

Problem. Q UESTION : How ‘large’ can the Fourier transform be ? A NSWER : The Fourier transform can be ‘as large as possible’. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 3 / 17

Problem. Q UESTION : How ‘large’ can the Fourier transform be ? A NSWER : The Fourier transform can be ‘as large as possible’. O UR AIM : Get the ‘answer’ in the framework of weak orbits of C 0 -semigroups. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 3 / 17

❘ ❘ ❘ Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = ( c ( n )) n ∈ ❩ ∈ c 0 ( ❩ ) there exists f ∈ L 1 ( 0 , 2 π ) such that | ˆ f ( n ) | ≥ | c ( n ) | , n ∈ ❩ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 4 / 17

Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = ( c ( n )) n ∈ ❩ ∈ c 0 ( ❩ ) there exists f ∈ L 1 ( 0 , 2 π ) such that | ˆ f ( n ) | ≥ | c ( n ) | , n ∈ ❩ . 2. Given c ∈ C 0 ( ❘ ) there exists f ∈ L 1 ( ❘ ) such that | ˆ f ( ξ ) | ≥ | c ( ξ ) | , ξ ∈ ❘ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 4 / 17

Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = ( c ( n )) n ∈ ❩ ∈ c 0 ( ❩ ) there exists f ∈ L 1 ( 0 , 2 π ) such that | ˆ f ( n ) | ≥ | c ( n ) | , n ∈ ❩ . 2. Given c ∈ C 0 ( ❘ ) there exists f ∈ L 1 ( ❘ ) such that | ˆ f ( ξ ) | ≥ | c ( ξ ) | , ξ ∈ ❘ . Generalization [Curtis, Figa-Talamanca, 1966]: The same result is true in the context of locally compact abelian groups. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 4 / 17

❩ ❩ ❩ ❘ ❘ ❘ Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 ( ❘ ) (or on L 2 ( 0 , 2 π ) ). Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 5 / 17

❘ ❘ ❘ Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 ( ❘ ) (or on L 2 ( 0 , 2 π ) ). Theorem (de-Leeuw-Kahane-Katznelson-Demailly, 1977-1984) 1. Given { c n } n ∈ ❩ ∈ l 2 ( ❩ ) there exists a 2 π -periodic function f ∈ C ([ 0 , 2 π ]) such that | ˆ f ( n ) | ≥ | c n | , n ∈ ❩ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 5 / 17

Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 ( ❘ ) (or on L 2 ( 0 , 2 π ) ). Theorem (de-Leeuw-Kahane-Katznelson-Demailly, 1977-1984) 1. Given { c n } n ∈ ❩ ∈ l 2 ( ❩ ) there exists a 2 π -periodic function f ∈ C ([ 0 , 2 π ]) such that | ˆ f ( n ) | ≥ | c n | , n ∈ ❩ . 2. Given c ∈ L 2 ( ❘ ) there exists a function f ∈ L 2 ( ❘ ) ∩ C 0 ( ❘ ) such that | ˆ f ( ξ ) | ≥ | c ( ξ ) | for almost every ξ. Many other settings !!! Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 5 / 17

❩ ❩ ❩ ❩ Abstract setting Theorem (K. Ball, Inventiones M. 1991, BLMS 1994) 1. If { x ∗ n : n ∈ ❩ } is a sequence of bounded linear functionals of norm 1 on a Banach space X and a = ( a n ) n ∈ ❩ ∈ l 1 ( ❩ ) , � a � l 1 < 1 , then there exists x ∈ X , � x � ≤ 1 , such that |� x ∗ n , x �| ≥ | a n | , n ∈ ❩ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 6 / 17

Abstract setting Theorem (K. Ball, Inventiones M. 1991, BLMS 1994) 1. If { x ∗ n : n ∈ ❩ } is a sequence of bounded linear functionals of norm 1 on a Banach space X and a = ( a n ) n ∈ ❩ ∈ l 1 ( ❩ ) , � a � l 1 < 1 , then there exists x ∈ X , � x � ≤ 1 , such that |� x ∗ n , x �| ≥ | a n | , n ∈ ❩ . 2. If { x n : n ∈ ❩ } is a sequence of elements of norm 1 in a Hilbert space H and ( a n ) n ∈ ❩ ∈ l 2 ( ❩ ) , � a � l 2 < 1 , then there exists x ∈ H , � x � ≤ 1 , such that | ( x n , x ) | ≥ | a n | , n ∈ ❩ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 6 / 17

❩ ❩ ❩ ❘ ❘ ❩ ❩ ❩ Implication for the Fourier transform: Define the bounded linear functionals x ∗ n , n ∈ ❩ , on L 1 ( ❘ ) by � x ∗ e − int x ( t ) dt , x ∈ L 1 ( ❘ ) . n ( x ) := ❘ Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 7 / 17

❩ ❩ Implication for the Fourier transform: Define the bounded linear functionals x ∗ n , n ∈ ❩ , on L 1 ( ❘ ) by � x ∗ e − int x ( t ) dt , x ∈ L 1 ( ❘ ) . n ( x ) := ❘ Then � x ∗ n � = 1 , n ∈ ❩ , and for every a = ( a n ) n ∈ ❩ ∈ l 1 ( ❩ ) , � a � l 1 < 1 , there exists x ∈ L 1 ( ❘ ) , � x � L 1 ( ❘ ) ≤ 1 , such that | x ∗ n ( x ) | ≥ | a n | , n ∈ ❩ . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 7 / 17

Implication for the Fourier transform: Define the bounded linear functionals x ∗ n , n ∈ ❩ , on L 1 ( ❘ ) by � x ∗ e − int x ( t ) dt , x ∈ L 1 ( ❘ ) . n ( x ) := ❘ Then � x ∗ n � = 1 , n ∈ ❩ , and for every a = ( a n ) n ∈ ❩ ∈ l 1 ( ❩ ) , � a � l 1 < 1 , there exists x ∈ L 1 ( ❘ ) , � x � L 1 ( ❘ ) ≤ 1 , such that | x ∗ n ( x ) | ≥ | a n | , n ∈ ❩ . This is still far from the result by Kolmogorov and Titchmarsh where ( a n ) n ∈ ❩ ∈ c 0 ( ❩ )! Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 7 / 17

❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ The Fourier transform via operator (semi-)groups Let ( U ( t )) t ∈ ❘ ⊂ L ( L 2 ( ❘ )) be a family of unitary operators on L 2 ( ❘ ) defined by ( U ( t ) f )( s ) = e − its f ( s ) , s ∈ ❘ Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 8 / 17

❘ ❘ The Fourier transform via operator (semi-)groups Let ( U ( t )) t ∈ ❘ ⊂ L ( L 2 ( ❘ )) be a family of unitary operators on L 2 ( ❘ ) defined by ( U ( t ) f )( s ) = e − its f ( s ) , s ∈ ❘ Observe that for fixed f , g ∈ L 2 ( ❘ ) : � � e − its f ( t ) g ( t ) dt = e − its ϕ ( t ) dt , g ∈ L 1 ( ❘ ) . ϕ := f ¯ ( U ( t ) f , g ) = ❘ ❘ | ( U ( t ) f , g ) | = | ( U ( − t ) f , g ) | , t ≥ 0 . it is enough to study bounds for one-sided weak orbits of ( U ( t )) t ∈ ❘ . NOTE : ( U ( t )) t ∈ ❘ is a strongly continuous operator (semi-)group. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 8 / 17

The Fourier transform via operator (semi-)groups Let ( U ( t )) t ∈ ❘ ⊂ L ( L 2 ( ❘ )) be a family of unitary operators on L 2 ( ❘ ) defined by ( U ( t ) f )( s ) = e − its f ( s ) , s ∈ ❘ Observe that for fixed f , g ∈ L 2 ( ❘ ) : � � e − its f ( t ) g ( t ) dt = e − its ϕ ( t ) dt , g ∈ L 1 ( ❘ ) . ϕ := f ¯ ( U ( t ) f , g ) = ❘ ❘ | ( U ( t ) f , g ) | = | ( U ( − t ) f , g ) | , t ≥ 0 . it is enough to study bounds for one-sided weak orbits of ( U ( t )) t ∈ ❘ . NOTE : ( U ( t )) t ∈ ❘ is a strongly continuous operator (semi-)group. NOTE : The weak orbit ( U ( t ) f , g ) of ( U ( t )) t ∈ ❘ is the Fourier transform of the L 1 ( ❘ ) -function f ¯ g . Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 8 / 17

❘ A bit of theory: Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, 2017 9 / 17