Analytic pseudo-differential calculus via the Bargmann transform - PowerPoint PPT Presentation

Test functions, distributions and expansions Pilipovi´ c spaces Let ÿ x P R d , c α P C . f p x q “ c α h α p x q , (*) α P N d Let s , σ ą 0. 1 2 s for some / every r ą 0. H s p R d q / H 0 , s p R d q “ all f in (*) s.t. | c α | À e ´ r | α | H 5 σ p R d q / H 0 , 5 σ p R d q “ all f in (*) s.t. | c α | À r | α | α ! ´ 1 2 σ for some / every r ą 0. We let H 1 s p R d q / H 1 0 , s p R d q be the set of all f in (*) such that 1 2 s , s P R ` , and | c α | À r | α | α ! ` 1 | c α | À e ` r | α | 2 σ , s “ 5 σ , for every / some r ą 0. . . . and H 1 0 p R d q be the set of all formal expansions (*). J. Toft Analytic Ψdo Potsdam, March 2019 5 / 16

Test functions, distributions and expansions Pilipovi´ c spaces Let ÿ x P R d , c α P C . f p x q “ c α h α p x q , (*) α P N d Let s , σ ą 0. 1 2 s for some / every r ą 0. H s p R d q / H 0 , s p R d q “ all f in (*) s.t. | c α | À e ´ r | α | H 5 σ p R d q / H 0 , 5 σ p R d q “ all f in (*) s.t. | c α | À r | α | α ! ´ 1 2 σ for some / every r ą 0. J. Toft Analytic Ψdo Potsdam, March 2019 5 / 16

Test functions, distributions and expansions Pilipovi´ c spaces Let ÿ x P R d , c α P C . f p x q “ c α h α p x q , (*) α P N d Let s , σ ą 0. 1 2 s for some / every r ą 0. H s p R d q / H 0 , s p R d q “ all f in (*) s.t. | c α | À e ´ r | α | H 5 σ p R d q / H 0 , 5 σ p R d q “ all f in (*) s.t. | c α | À r | α | α ! ´ 1 2 σ for some / every r ą 0. Then H 1 s for s ě 0 and H 1 0 , s for s ą 0 are the duals of H s and H 0 , s under p ¨ , ¨ q L 2 . J. Toft Analytic Ψdo Potsdam, March 2019 5 / 16

Images under the Bargmann transform Bargmann transform, basic definitions J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . d ÿ Here z “ p z 1 , . . . , z d q P C d , x z , w y “ z j w j , p z , w q “ x z , w y . j “ 1 J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . A p C d q is the set of all entire functions in C d J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . A p C d q is the set of all entire functions in C d A 2 p C d q is the Hilbert space of entire analytic functions such that ´ ż ¯ 1 { 2 C d | F p z q| 2 d µ p z q } F } A 2 ” ă 8 . J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . A p C d q is the set of all entire functions in C d A 2 p C d q is the Hilbert space of entire analytic functions such that ´ ż ¯ 1 { 2 C d | F p z q| 2 d µ p z q } F } A 2 ” ă 8 . d µ p z q “ π ´ d e ´| z | 2 d λ p z q , d λ p z q Here where is the Lebesgue measure on C d . J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform Bargmann transform, basic definitions The Bargmann transform (1961): ż ´ 1 ´ ¯ p V d f qp z q “ π ´ d { 4 2 px z , z y ` | y | 2 q ` 2 1 { 2 x z , y y R d exp f p y q dy . A p C d q is the set of all entire functions in C d A 2 p C d q is the Hilbert space of entire analytic functions such that ´ ż ¯ 1 { 2 C d | F p z q| 2 d µ p z q } F } A 2 ” ă 8 . d µ p z q “ π ´ d e ´| z | 2 d λ p z q , d λ p z q Here where is the Lebesgue measure on C d . ż A 2 -scalar product: p F , G q A 2 “ C d F p z q G p z q d µ p z q . J. Toft Analytic Ψdo Potsdam, March 2019 6 / 16

Images under the Bargmann transform V. Bargmann 1961 - Mapping properties He proved: J. Toft Analytic Ψdo Potsdam, March 2019 7 / 16

Images under the Bargmann transform V. Bargmann 1961 - Mapping properties He proved: V d is a bijective isometry from L 2 p R d q to A 2 p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 7 / 16

Images under the Bargmann transform V. Bargmann 1961 - Mapping properties He proved: V d is a bijective isometry from L 2 p R d q to A 2 p C d q . z α V d h α “ e α p z q ” p α ! q 1 { 2 . J. Toft Analytic Ψdo Potsdam, March 2019 7 / 16

Images under the Bargmann transform V. Bargmann 1961 - Mapping properties He proved: V d is a bijective isometry from L 2 p R d q to A 2 p C d q . z α Hence V d maps ON-basis t h α u in L 2 into V d h α “ e α p z q ” p α ! q 1 { 2 . z α " * in A 2 . the ON-basis p α ! q 1 { 2 J. Toft Analytic Ψdo Potsdam, March 2019 7 / 16

Images under the Bargmann transform V. Bargmann 1961 - Mapping properties He proved: V d is a bijective isometry from L 2 p R d q to A 2 p C d q . z α Hence V d maps ON-basis t h α u in L 2 into V d h α “ e α p z q ” p α ! q 1 { 2 . z α " * in A 2 . the ON-basis p α ! q 1 { 2 Reproducing kernel: ż C d e p z , w q F p w q d µ p w q , p Π A F qp z q “ F admissible. Then d µ p z q “ π ´ d e ´| z | 2 d λ p z q . F P A 2 , p Π A F qp z q “ F p z q , J. Toft Analytic Ψdo Potsdam, March 2019 7 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d Smaller spaces: A 0 p R d q , the set of all analytic polynomials F p z q in (*) J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d Smaller spaces: Larger spaces: A 0 p R d q , the set of all analytic A 1 0 p C d q , the set of all formal power polynomials F p z q in (*) series F p z q in (*) J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d Smaller spaces: Larger spaces: A 0 p R d q , the set of all analytic A 1 0 p C d q , the set of all formal power polynomials F p z q in (*) series F p z q in (*) (usually denoted by C rr z 1 , . . . , z d ss ) J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d Smaller spaces: Larger spaces: A 0 p R d q , the set of all analytic A 1 0 p C d q , the set of all formal power polynomials F p z q in (*) series F p z q in (*) J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Spaces of power series expansions In the most general situation we consider the power series expansions z α ÿ z P C d , c α P C , e α p z q “ F p z q “ c α e α p z q , (*) p α ! q 1 { 2 . α P N d Smaller spaces: Larger spaces: A 0 p R d q , the set of all analytic A 1 0 p C d q , the set of all formal power polynomials F p z q in (*) series F p z q in (*) For s P R 5 , let A s p C d q ( A 0 , s p C d q ) be For s P R 5 , let A 1 s p C d q ( A 1 0 , s p C d q ) the set of all series expansions F p z q the set of all series expansions F p z q in (*) such that in (*) such that $ 1 $ 1 2 s , 2 s , e ´ r | α | e r | α | s P R ` s P R ` & & | c α | À | c α | À r | α | α ! ´ 1 1 r | α | α ! 2 σ , s “ 5 σ 2 σ , s “ 5 σ % % for some (every) r ą 0. for every (some) r ą 0. J. Toft Analytic Ψdo Potsdam, March 2019 8 / 16

Images under the Bargmann transform Bargmann transform on Hermite series expansions J. Toft Analytic Ψdo Potsdam, March 2019 9 / 16

Images under the Bargmann transform Bargmann transform on Hermite series expansions Recall that V d h α “ e α . J. Toft Analytic Ψdo Potsdam, March 2019 9 / 16

Images under the Bargmann transform Bargmann transform on Hermite series expansions Recall that V d h α “ e α . ÿ ÿ For any f “ c α h α , let V d f “ c α e α . α α J. Toft Analytic Ψdo Potsdam, March 2019 9 / 16

Images under the Bargmann transform Bargmann transform on Hermite series expansions Recall that V d h α “ e α . ÿ ÿ For any f “ c α h α , let V d f “ c α e α . α α By the definitions it follows that V d : H 0 , s p R d q Ñ A 0 , s p C d q , V d : H s p R d q Ñ A s p C d q , V d : H 1 s p R d q Ñ A 1 s p C d q , V d : H 1 0 , s p R d q Ñ A 1 0 , s p C d q are bijective. J. Toft Analytic Ψdo Potsdam, March 2019 9 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series ÿ Any entire function F is equal to a power series expansion c α e α α J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series ÿ Any entire function F is equal to a power series expansion c α e α α such that | c α | À r | α | p α ! q 1 { 2 , for every r ą 0. J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series ÿ Any entire function F is equal to a power series expansion c α e α α such that | c α | À r | α | p α ! q 1 { 2 , for every r ą 0. This implies A 1 5 1 p C d q “ A p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series ÿ Any entire function F is equal to a power series expansion c α e α α such that | c α | À r | α | p α ! q 1 { 2 , for every r ą 0. This implies A 1 5 1 p C d q “ A p C d q . From the definitions it now follows for s ě 1 2 and s 0 ă 1 2 : A 0 , s 0 p C d q Ď A s 0 p C d q Ď A 0 , s p C d q Ď A s p C d q Ď A 1 s p C d q Ď A 1 0 , s p C d q Ď A p C d q Ď A 1 s 0 p C d q Ď A 1 0 , s 0 p C d q J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform Characterizations of certain spaces of power series ÿ Any entire function F is equal to a power series expansion c α e α α such that | c α | À r | α | p α ! q 1 { 2 , for every r ą 0. This implies A 1 5 1 p C d q “ A p C d q . From the definitions it now follows for s ě 1 2 and s 0 ă 1 2 : A 0 , s 0 p C d q Ď A s 0 p C d q Ď A 0 , s p C d q Ď A s p C d q Ď A 1 s p C d q Ď A 1 0 , s p C d q Ď A p C d q Ď A 1 s 0 p C d q Ď A 1 0 , s 0 p C d q What about those spaces which are contained in A p C d q ?? J. Toft Analytic Ψdo Potsdam, March 2019 10 / 16

Images under the Bargmann transform s 0 ă 1 s ě 1 (Recall: s 0 ă 5 σ ă 1 For 2 , 2 , x z y “ 1 ` | z | 2 ): The tiny planets (smaller than Gelfand-Shilov): , , J. Toft Analytic Ψdo Potsdam, March 2019 11 / 16

Images under the Bargmann transform s 0 ă 1 s ě 1 (Recall: s 0 ă 5 σ ă 1 For 2 , 2 , x z y “ 1 ` | z | 2 ): The tiny planets (smaller than Gelfand-Shilov): 1 1 ´ 2 s 0 , for every / some r ą 0 u , A 0 , s 0 { A s 0 “ t F P A ; | F p z q| À e r p log x z yq 2 σ σ ` 1 , for every / some r ą 0 u , A 0 , 5 σ { A 5 σ “ t F P A ; | F p z q| À e r | z | 2 “ t F P A ; | F p z q| À e r | z | 2 , for every r ą 0 u , A 0 , 1 , J. Toft Analytic Ψdo Potsdam, March 2019 11 / 16

Images under the Bargmann transform s 0 ă 1 s ě 1 (Recall: s 0 ă 5 σ ă 1 For 2 , 2 , x z y “ 1 ` | z | 2 ): The tiny planets (smaller than Gelfand-Shilov): 1 1 ´ 2 s 0 , for every / some r ą 0 u , A 0 , s 0 { A s 0 “ t F P A ; | F p z q| À e r p log x z yq 2 σ σ ` 1 , for every / some r ą 0 u , A 0 , 5 σ { A 5 σ “ t F P A ; | F p z q| À e r | z | 2 “ t F P A ; | F p z q| À e r | z | 2 , for every r ą 0 u , A 0 , 1 The Gelfand-Shilov world: s , for every / some r ą 0 u , s ‰ 1 | z | 2 1 2 ´ r | z | A 0 , s { A s “ t F P A ; | F p z q| À e 2 , | z | 2 1 s , for every / some r ą 0 u , A 1 s { A 1 2 ` r | z | 0 , s “ t F P A ; | F p z q| À e J. Toft Analytic Ψdo Potsdam, March 2019 11 / 16

Images under the Bargmann transform s 0 ă 1 s ě 1 (Recall: s 0 ă 5 σ ă 1 For 2 , 2 , x z y “ 1 ` | z | 2 ): The tiny planets (smaller than Gelfand-Shilov): 1 1 ´ 2 s 0 , for every / some r ą 0 u , A 0 , s 0 { A s 0 “ t F P A ; | F p z q| À e r p log x z yq 2 σ σ ` 1 , for every / some r ą 0 u , A 0 , 5 σ { A 5 σ “ t F P A ; | F p z q| À e r | z | 2 “ t F P A ; | F p z q| À e r | z | 2 , for every r ą 0 u , A 0 , 1 The Gelfand-Shilov world: s , for every / some r ą 0 u , s ‰ 1 | z | 2 1 2 ´ r | z | A 0 , s { A s “ t F P A ; | F p z q| À e 2 , | z | 2 1 s , for every / some r ą 0 u , A 1 s { A 1 2 ` r | z | 0 , s “ t F P A ; | F p z q| À e Beyond Gelfand-Shilov life: 2 “ t F P A ; | F p z q| À e r | z | 2 , for some r ą 0 u , A 1 0 , 1 2 σ σ ´ 1 , for every / some r ą 0 u , σ ą 1 , A 1 5 σ { A 1 0 , 5 σ “ t F P A ; | F p z q| À e r | z | A 1 A 1 ď p“ A p C d qq , 5 1 “ A 0 , 5 1 “ A p B R p 0 qq . R ą 0 J. Toft Analytic Ψdo Potsdam, March 2019 11 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: We have T K “ Op V p a q when K p z , w q “ a p z , w q e p z , w q . J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: If a p z , w q “ 1, then we regain the reproducing kernel: Op V p a q F p z q “ p Π A F qp z q “ F p z q . J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: ÿ ÿ a α p z qpB α a α p z q w α . Op V p a q F p z q “ z F qp z q , a p z , w q “ | α |ď N | α |ď N J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: If a p z , w q is analytic, then p Op V p a q F qp z q “ a p z , z q F p z q . J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: If χ is the characteristic function of a polydisc and a p z , w q “ χ p w q , then Op V p a q is bijective between suitable A s p C d q spaces. J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Analytic pseudo-differential and integral operators Suppose a p z , w q and K p z , w q are suitable functions, analytic in z P C d . 1 The analytic pseudo-differential operator, Op V p a q , is defined by ż C d a p z , w q F p w q e p z , w q d µ p w q , F P A 0 p C d q . p Op V p a q F qp z q “ 2 The (analytic) kernel operator, T K , is defined by ż F P A 0 p C d q . p T K F qp z q “ C d K p z , w q F p w q d µ p w q , Examples and remarks: If χ is the characteristic function of a polydisc and a p z , w q “ χ p w q , then Op V p a q is bijective between suitable A s p C d q spaces. Some sorts of analytic Paley-Wiener properties Nabizadeh-Pfeuffer-T. (2018). J. Toft Analytic Ψdo Potsdam, March 2019 12 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. We also let L p V 1 , V 2 q be the set of all linear continuous mappings from the topological vector space V 1 to the topological vector space V 2 . J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017 Let s 1 P R 5 and s 2 P R 5 . The map K ÞÑ T K is bijective J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017 Let s 1 P R 5 and s 2 P R 5 . The map K ÞÑ T K is bijective A 0 , s 1 p C 2 d q L p A 1 0 , s 1 p C d q , A 0 , s 1 p C d qq , � from to and A 1 0 , s 1 p C 2 d q L p A 0 , s 1 p C d q , A 1 � 0 , s 1 p C d qq . from to J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ In what follows we let A 1 s p C 2 d q “ t K p z , w q ; p z , w q ÞÑ K p z , w q P A 1 s p C 2 d q u , � and similarly for other spaces. Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017 Let s 1 P R 5 and s 2 P R 5 . The map K ÞÑ T K is bijective A 0 , s 1 p C 2 d q L p A 1 0 , s 1 p C d q , A 0 , s 1 p C d qq , � from to and A 1 0 , s 1 p C 2 d q L p A 0 , s 1 p C d q , A 1 � 0 , s 1 p C d qq . from to A s 2 p C 2 d q L p A 1 � s 2 p C d q , A s 2 p C d qq , from to and A 1 s 2 p C 2 d q L p A s 2 p C d q , A 1 s 2 p C d qq . � from to J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ L p A 1 L p A s 2 , A 1 A 1 s 2 , A s 2 q “ t T K ; K P A s 2 u , � s 2 q “ t T K ; K P � s 2 u etc. . . J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ L p A 1 L p A s 2 , A 1 A 1 s 2 , A s 2 q “ t T K ; K P A s 2 u , � s 2 q “ t T K ; K P � s 2 u etc. . . Thm. Teofanov-T. (2019) Let t P C , s 1 P R 5 , s 1 ď 1 2 , and s 2 P R 5 , s 2 ă 1 2 . Then K p z , w q ÞÑ K p z , w q e t p z , w q is a continuous bijection on A 1 0 , s 1 p C 2 d q A 1 s 2 p C 2 d q . � � and on J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ L p A 1 L p A s 2 , A 1 A 1 s 2 , A s 2 q “ t T K ; K P A s 2 u , � s 2 q “ t T K ; K P � s 2 u etc. . . Thm. Teofanov-T. (2019) Let t P C , s 1 P R 5 , s 1 ď 1 2 , and s 2 P R 5 , s 2 ă 1 2 . Then K p z , w q ÞÑ K p z , w q e t p z , w q is a continuous bijection on A 1 0 , s 1 p C 2 d q A 1 s 2 p C 2 d q . � � and on By combining this with the earlier kernel theorems: J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ L p A 1 L p A s 2 , A 1 A 1 s 2 , A s 2 q “ t T K ; K P A s 2 u , � s 2 q “ t T K ; K P � s 2 u etc. . . Thm. Teofanov-T. (2019) Let t P C , s 1 P R 5 , s 1 ď 1 2 , and s 2 P R 5 , s 2 ă 1 2 . Then K p z , w q ÞÑ K p z , w q e t p z , w q is a continuous bijection on A 1 0 , s 1 p C 2 d q A 1 s 2 p C 2 d q . � � and on J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Kernel theorems and related mapping properties C d a p z , w q F p w q e p z , w q d µ p w q , C d K p z , w q F p w q d µ p w q . ş ş p Op V p a q F qp z q “ p T K F qp z q “ L p A 1 L p A s 2 , A 1 A 1 s 2 , A s 2 q “ t T K ; K P A s 2 u , � s 2 q “ t T K ; K P � s 2 u etc. . . Thm. Teofanov-T. (2019) Let t P C , s 1 P R 5 , s 1 ď 1 2 , and s 2 P R 5 , s 2 ă 1 2 . Then K p z , w q ÞÑ K p z , w q e t p z , w q is a continuous bijection on A 1 0 , s 1 p C 2 d q A 1 s 2 p C 2 d q . � � and on Thm. Teofanov-T. (2019) Let s 1 P R 5 , s 1 ď 1 2 , and s 2 P R 5 , s 2 ă 1 2 . Then L p A 0 , s 1 p C d q , A 1 A 1 0 , s 1 p C 2 d q u . 0 , s 1 p C d qq “ t Op V p a q ; a P � L p A s 2 p C d q , A 1 A 1 s 2 p C 2 d q u . s 2 p C d qq “ t Op V p a q ; a P � J. Toft Analytic Ψdo Potsdam, March 2019 13 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d that is, ω 0 ą 0 and ω 0 , 1 { ω 0 P L 8 loc p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p p ω 0 q p C d q X A p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p If instead p P r 1 , 8s 4 d , let p ω 0 q p C d q X A p C d q . A p p ω q p C 2 d q “ t All K ; p z , w q ÞÑ K p z , w q P A p p ω q p C 2 d q u . � J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p If instead p P r 1 , 8s 4 d , let p ω 0 q p C d q X A p C d q . A p p ω q p C 2 d q “ t All K ; p z , w q ÞÑ K p z , w q P A p p ω q p C 2 d q u . � Recently, results of the following type appeared J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p If instead p P r 1 , 8s 4 d , let p ω 0 q p C d q X A p C d q . A p p ω q p C 2 d q “ t All K ; p z , w q ÞÑ K p z , w q P A p p ω q p C 2 d q u . � J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p If instead p P r 1 , 8s 4 d , let p ω 0 q p C d q X A p C d q . A p p ω q p C 2 d q “ t All K ; p z , w q ÞÑ K p z , w q P A p p ω q p C 2 d q u . � Thm. Teofanov-T. (2019) A p C 2 d q satisfies Suppose ω and K P � ? ? G K ,ω p z , w q “ e ´ 1 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q ď p 2 ď p , ω 1 p w q À ω p z , w q . q , J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Let L p p C d q — L p p R 2 d q be the mixed Lebesgue space with respect to p P r 1 , 8s 2 d , ω 0 be a weight on C d and let ω be a weight on C 2 d . ? 2 ¨| z | 2 ω 0 p Let B p p ω 0 q p C d q “ t F ; F p z q e ´ 1 2 ¨ z q P L p p C d q u . Let A p p ω 0 q p C d q “ B p If instead p P r 1 , 8s 4 d , let p ω 0 q p C d q X A p C d q . A p p ω q p C 2 d q “ t All K ; p z , w q ÞÑ K p z , w q P A p p ω q p C 2 d q u . � Thm. Teofanov-T. (2019) A p C 2 d q satisfies Suppose ω and K P � ? ? G K ,ω p z , w q “ e ´ 1 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q ď p 2 ď p , ω 1 p w q À ω p z , w q . q , Then T K is continuous from A p 1 p ω 1 q p C d q to A p 2 p ω 2 q p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Thm. Teofanov-T. (2019) Suppose ω and K P � A p C 2 d q satisfies ? ? 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z , w q “ e ´ 1 G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q , q ď p 2 ď p , ω 1 p w q À ω p z , w q . Then T K is continuous from A p 1 p ω 1 q p C d q to A p 2 p ω 2 q p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Thm. Teofanov-T. (2019) Suppose ω and K P � A p C 2 d q satisfies ? ? 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z , w q “ e ´ 1 G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q , q ď p 2 ď p , ω 1 p w q À ω p z , w q . Then T K is continuous from A p 1 p ω 1 q p C d q to A p 2 p ω 2 q p C d q . By putting some restrictions on ω , ω j and taking the counter image of the previous result with respect to the Bargmann transform one gets well-known results of continuity results of real Ψdo on modulation spaces, like J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Thm. Teofanov-T. (2019) Suppose ω and K P � A p C 2 d q satisfies ? ? 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z , w q “ e ´ 1 G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q , q ď p 2 ď p , ω 1 p w q À ω p z , w q . Then T K is continuous from A p 1 p ω 1 q p C d q to A p 2 p ω 2 q p C d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Analytic Ψdo with Lebesgue conditions on their symbols Thm. Teofanov-T. (2019) Suppose ω and K P � A p C 2 d q satisfies ? ? 2 p| z | 2 `| w | 2 q K p z , w q¨ ω p G K ,ω p z , w q “ e ´ 1 G K ,ω p z ` w , z q P L p , q p C 2 d q , 2 z , 2 w q , ω 2 p z q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 q , q ď p 2 ď p , ω 1 p w q À ω p z , w q . Then T K is continuous from A p 1 p ω 1 q p C d q to A p 2 p ω 2 q p C d q . Thm. Gr¨ ochenig-Heil, T. Suppose ω 2 p x ,ξ ` η q a P M p , q p 1 ´ 1 1 p 2 “ 1 ´ 1 p ´ 1 p ω q p R 2 d q . q ď p 2 ď p , ω 1 p x ` y ,ξ q À ω p x , ξ, η, y q , q , Then Op p a q is continuous from M p 1 p ω 1 q p R d q to M p 2 p ω 2 q p R d q . J. Toft Analytic Ψdo Potsdam, March 2019 14 / 16

Analytic pseudo-differential and integral operators Some further properties A large family of modulation spaces: J. Toft Analytic Ψdo Potsdam, March 2019 15 / 16

Analytic pseudo-differential and integral operators Some further properties A large family of modulation spaces: Definition modulation spaces Let p P r 1 , 8s 2 d , φ p x q “ e ´ 1 2 ¨| x | 2 , x P R d , ω is a weight on R 2 d , f P H 1 5 1 p R d q , and V φ f p x , ξ q “ x f , e ´ i x ¨ ,ξ y φ p ¨ ´ x qy (the STFT of f ). Then the modulation space M p p ω q p R d q is the set of all f P H 1 5 1 p R d q such that p ω q ” } V φ f ¨ ω } L p ă 8 . } f } M p J. Toft Analytic Ψdo Potsdam, March 2019 15 / 16

Analytic pseudo-differential and integral operators Some further properties A large family of modulation spaces: Definition modulation spaces Let p P r 1 , 8s 2 d , φ p x q “ e ´ 1 2 ¨| x | 2 , x P R d , ω is a weight on R 2 d , f P H 1 5 1 p R d q , and V φ f p x , ξ q “ x f , e ´ i x ¨ ,ξ y φ p ¨ ´ x qy (the STFT of f ). Then the modulation space M p p ω q p R d q is the set of all f P H 1 5 1 p R d q such that p ω q ” } V φ f ¨ ω } L p ă 8 . } f } M p Usually strong restrictions are imposed on the weights (Feichtinger, Gr¨ ochenig). J. Toft Analytic Ψdo Potsdam, March 2019 15 / 16

Analytic pseudo-differential and integral operators Some further properties A large family of modulation spaces: Definition modulation spaces Let p P r 1 , 8s 2 d , φ p x q “ e ´ 1 2 ¨| x | 2 , x P R d , ω is a weight on R 2 d , f P H 1 5 1 p R d q , and V φ f p x , ξ q “ x f , e ´ i x ¨ ,ξ y φ p ¨ ´ x qy (the STFT of f ). Then the modulation space M p p ω q p R d q is the set of all f P H 1 5 1 p R d q such that p ω q ” } V φ f ¨ ω } L p ă 8 . } f } M p J. Toft Analytic Ψdo Potsdam, March 2019 15 / 16