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

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Analytic pseudo-differential calculus via the Bargmann transform Joachim Toft Linnus University, V axj o, Sweden, joachim.toft@lnu.se The talk is dedicated to professor Bert-Wolfgang Schulze on his 75th birthday Potsdam, Germany,

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Analytic pseudo-differential calculus via the Bargmann transform

Joachim Toft

Linnæus University, V¨ axj¨

  • , Sweden,

joachim.toft@lnu.se

The talk is dedicated to professor Bert-Wolfgang Schulze on his 75th birthday

Potsdam, Germany, March 2019

  • J. Toft

Analytic Ψdo Potsdam, March 2019 1 / 16

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Plan of the talk

1

Test functions, distributions and expansions

2

Images under the Bargmann transform

3

Analytic pseudo-differential and integral operators

  • J. Toft

Analytic Ψdo Potsdam, March 2019 2 / 16

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Important contributors to the topic, e.g.:

  • W. Bauer, F. A. Berezin, L. A. Coburn
  • J. Toft

Analytic Ψdo Potsdam, March 2019 3 / 16

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Important contributors to the topic, e.g.:

  • W. Bauer, F. A. Berezin, L. A. Coburn

The talk is based on the following:

  • J. Toft Images of function and distribution spaces under the

Bargmann transform, J. Pseudo-Differ. Oper. Appl. 8 (2017), 83–139.

  • N. Teofanov, J. Toft Pseudo-differential calculus in a Bargmann

setting (preprint) arXiv:1901.02796.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 3 / 16

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Test functions, distributions and expansions

The Schwartz space and Gelfand-Shilov spaces

The Schwartz space: f P S pRdq ô }xβDαf }L8 ă 8, α, β P Nd. The (Fourier invariant) Gelfand-Shilov space of Roumieu type (Beurling type) of order s ě 0: f P SspRdq ô }xβDαf }L8 À h|α`β|pα!β!qs, α, β P Nd, ` f P ΣspRdq ô }xβDαf }L8 À h|α`β|pα!β!qs, α, β P Nd, ˘ for some (for every) h ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 4 / 16

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SLIDE 6

Test functions, distributions and expansions

The Schwartz space and Gelfand-Shilov spaces

The Schwartz space: f P S pRdq ô }xβDαf }L8 ă 8, α, β P Nd. The (Fourier invariant) Gelfand-Shilov space of Roumieu type (Beurling type) of order s ě 0: f P SspRdq ô }xβDαf }L8 À h|α`β|pα!β!qs, α, β P Nd, ` f P ΣspRdq ô }xβDαf }L8 À h|α`β|pα!β!qs, α, β P Nd, ˘ for some (for every) h ą 0. We have: S1{2 Ď Σs Ď Ss Ď Σs`ε Ď S , s ą 1

2, ε ą 0.

(Dense embeddings.)

SspRdq (ΣspRdq) is non-trivial, if and only if s ě 1

2 (s ą 1 2).

  • J. Toft

Analytic Ψdo Potsdam, March 2019 4 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Hermite function hα with respect to α P Nd is given by hαpxq “ π´d{4p´1q|α|p2|α|α!q´1{2e|x|2{2pBαe´|x|2q. Formal Hermite function expansions: f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Hermite function hα with respect to α P Nd is given by hαpxq “ π´d{4p´1q|α|p2|α|α!q´1{2e|x|2{2pBαe´|x|2q. Formal Hermite function expansions: f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*)

Definition

Let s, σ ą 0. The Pilipovi´ c space of Roumieu / Beurling type, HspRdq / H0,spRdq, consists of all f in (*) such that |cα| À e´r|α|

1 2s holds for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Hermite function hα with respect to α P Nd is given by hαpxq “ π´d{4p´1q|α|p2|α|α!q´1{2e|x|2{2pBαe´|x|2q. Formal Hermite function expansions: f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*)

Definition

Let s, σ ą 0. The Pilipovi´ c space of Roumieu / Beurling type, HspRdq / H0,spRdq, consists of all f in (*) such that |cα| À e´r|α|

1 2s holds for some / every r ą 0.

The Pilipovi´ c space H5σpRdq / H0,5σpRdq consists of all f in (*) such that |cα| À r |α|α!´ 1

2σ holds for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

We also let H0pRdq “ All finite Hermite series expansions in (*).

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

By letting R5 “ R` Y t5σu with convention s ă 5σ1 ă 5σ2 ă 1 2, when s ă 1 2, σ1 ă σ2 it follows H0

Dense

ã Ñ H0,s1

Dense

ã Ñ Hs1

Dense

ã Ñ H0,s2, s1, s2 P R5, s1 ă s2.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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SLIDE 15

Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

Pilipovi´ c proved 1986: Hs “ Ss “ t f ; }p|x|2 ´ ∆qNf }L8 À hNN!2s for some h ą 0 u, s ě 1 2, H0,s “ Σs “ t f ; }p|x|2 ´ ∆qNf }L8 À hNN!2s for every h ą 0 u, s ą 1 2. But . . . Σ1{2 “ t0u ‰ t f ; }p|x|2 ´ ∆qNf }L8 À hNN! for every h ą 0 u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

Recent extension:

  • Thm. (T. 2017)

Let 0 ď s P R. Then: Hs “ t f ; }p|x|2 ´ ∆qNf }L8 À hNN!2s for some h ą 0 u, and H0,s “ t f ; }p|x|2 ´ ∆qNf }L8 À hNN!2s for every h ą 0 u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

We let H1

spRdq / H1 0,spRdq be the set of all f in (*) such that

|cα| À e`r|α|

1 2s , s P R`, and |cα| À r|α|α!` 1 2σ , s “ 5σ, for every / some

r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

We let H1

spRdq / H1 0,spRdq be the set of all f in (*) such that

|cα| À e`r|α|

1 2s , s P R`, and |cα| À r|α|α!` 1 2σ , s “ 5σ, for every / some

r ą 0. . . . and H1

0pRdq be the set of all formal expansions (*).

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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SLIDE 22

Test functions, distributions and expansions

Pilipovi´ c spaces

Let f pxq “ ÿ

αPNd

cαhαpxq, x P Rd, cα P C. (*) Let s, σ ą 0. HspRdq / H0,spRdq “ all f in (*) s.t. |cα| À e´r|α|

1 2s for some / every r ą 0.

H5σpRdq / H0,5σpRdq “ all f in (*) s.t. |cα| À r |α|α!´ 1

2σ for some / every r ą 0.

Then H1

s for s ě 0 and H1 0,s for s ą 0 are the duals of Hs and H0,s under

p ¨ , ¨ qL2.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 5 / 16

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Images under the Bargmann transform

Bargmann transform, basic definitions

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy. Here z “ pz1, . . . , zdq P Cd, xz, wy “

d

ÿ

j“1

zjwj, pz, wq “ xz, wy.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy. ApCdq is the set of all entire functions in Cd

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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SLIDE 27

Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy. ApCdq is the set of all entire functions in Cd A2pCdq is the Hilbert space of entire analytic functions such that }F}A2 ” ´ ż

Cd |Fpzq|2 dµpzq

¯1{2 ă 8.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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SLIDE 28

Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy. ApCdq is the set of all entire functions in Cd A2pCdq is the Hilbert space of entire analytic functions such that }F}A2 ” ´ ż

Cd |Fpzq|2 dµpzq

¯1{2 ă 8. Here dµpzq “ π´de´|z|2 dλpzq, where dλpzq is the Lebesgue measure on Cd.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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SLIDE 29

Images under the Bargmann transform

Bargmann transform, basic definitions

The Bargmann transform (1961): pVdf qpzq “ π´d{4 ż

Rd exp

´ ´ 1 2pxz, zy ` |y|2q ` 21{2xz, yy ¯ f pyq dy. ApCdq is the set of all entire functions in Cd A2pCdq is the Hilbert space of entire analytic functions such that }F}A2 ” ´ ż

Cd |Fpzq|2 dµpzq

¯1{2 ă 8. Here dµpzq “ π´de´|z|2 dλpzq, where dλpzq is the Lebesgue measure on Cd. A2-scalar product: pF, GqA2 “ ż

Cd FpzqGpzq dµpzq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 6 / 16

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Images under the Bargmann transform

  • V. Bargmann 1961 - Mapping properties

He proved:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 7 / 16

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Images under the Bargmann transform

  • V. Bargmann 1961 - Mapping properties

He proved: Vd is a bijective isometry from L2pRdq to A2pCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 7 / 16

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Images under the Bargmann transform

  • V. Bargmann 1961 - Mapping properties

He proved: Vd is a bijective isometry from L2pRdq to A2pCdq. Vdhα “ eαpzq ” zα pα!q1{2 .

  • J. Toft

Analytic Ψdo Potsdam, March 2019 7 / 16

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SLIDE 33

Images under the Bargmann transform

  • V. Bargmann 1961 - Mapping properties

He proved: Vd is a bijective isometry from L2pRdq to A2pCdq. Vdhα “ eαpzq ” zα pα!q1{2 . Hence Vd maps ON-basis thαu in L2 into the ON-basis " zα pα!q1{2 * in A2.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 7 / 16

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Images under the Bargmann transform

  • V. Bargmann 1961 - Mapping properties

He proved: Vd is a bijective isometry from L2pRdq to A2pCdq. Vdhα “ eαpzq ” zα pα!q1{2 . Hence Vd maps ON-basis thαu in L2 into the ON-basis " zα pα!q1{2 * in A2. Reproducing kernel: pΠAFqpzq “ ż

Cd epz,wqFpwq dµpwq,

F admissible. Then pΠAFqpzq “ Fpzq, F P A2, dµpzq “ π´de´|z|2 dλpzq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 7 / 16

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Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

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SLIDE 36

Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*) Smaller spaces:

A0pRdq, the set of all analytic polynomials Fpzq in (*)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

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SLIDE 37

Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*) Smaller spaces:

A0pRdq, the set of all analytic polynomials Fpzq in (*)

Larger spaces:

A1

0pCdq, the set of all formal power

series Fpzq in (*)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

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SLIDE 38

Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*) Smaller spaces:

A0pRdq, the set of all analytic polynomials Fpzq in (*)

Larger spaces:

A1

0pCdq, the set of all formal power

series Fpzq in (*)

(usually denoted by Crrz1, . . . , zdss)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

slide-39
SLIDE 39

Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*) Smaller spaces:

A0pRdq, the set of all analytic polynomials Fpzq in (*)

Larger spaces:

A1

0pCdq, the set of all formal power

series Fpzq in (*)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

slide-40
SLIDE 40

Images under the Bargmann transform

Spaces of power series expansions

In the most general situation we consider the power series expansions Fpzq “ ÿ

αPNd

cαeαpzq, z P Cd, cα P C, eαpzq “ zα pα!q1{2 . (*) Smaller spaces:

A0pRdq, the set of all analytic polynomials Fpzq in (*) For s P R5, let AspCdq (A0,spCdq) be the set of all series expansions Fpzq in (*) such that |cα| À $ & % e´r|α|

1 2s ,

s P R` r |α|α!´ 1

2σ ,

s “ 5σ for some (every) r ą 0.

Larger spaces:

A1

0pCdq, the set of all formal power

series Fpzq in (*) For s P R5, let A1

spCdq (A1 0,spCdq)

the set of all series expansions Fpzq in (*) such that |cα| À $ & % er|α|

1 2s ,

s P R` r |α|α!

1 2σ ,

s “ 5σ for every (some) r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 8 / 16

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SLIDE 41

Images under the Bargmann transform

Bargmann transform on Hermite series expansions

  • J. Toft

Analytic Ψdo Potsdam, March 2019 9 / 16

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SLIDE 42

Images under the Bargmann transform

Bargmann transform on Hermite series expansions

Recall that Vdhα “ eα.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 9 / 16

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SLIDE 43

Images under the Bargmann transform

Bargmann transform on Hermite series expansions

Recall that Vdhα “ eα. For any f “ ÿ

α

cαhα, let Vdf “ ÿ

α

cαeα.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 9 / 16

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SLIDE 44

Images under the Bargmann transform

Bargmann transform on Hermite series expansions

Recall that Vdhα “ eα. For any f “ ÿ

α

cαhα, let Vdf “ ÿ

α

cαeα. By the definitions it follows that Vd : H0,spRdq Ñ A0,spCdq, Vd : HspRdq Ñ AspCdq, Vd : H1

spRdq

Ñ A1

spCdq,

Vd : H1

0,spRdq Ñ A1 0,spCdq

are bijective.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 9 / 16

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SLIDE 45

Images under the Bargmann transform

Characterizations of certain spaces of power series

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-46
SLIDE 46

Images under the Bargmann transform

Characterizations of certain spaces of power series

Any entire function F is equal to a power series expansion ÿ

α

cαeα

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-47
SLIDE 47

Images under the Bargmann transform

Characterizations of certain spaces of power series

Any entire function F is equal to a power series expansion ÿ

α

cαeα such that |cα| À r|α|pα!q1{2, for every r ą 0.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-48
SLIDE 48

Images under the Bargmann transform

Characterizations of certain spaces of power series

Any entire function F is equal to a power series expansion ÿ

α

cαeα such that |cα| À r|α|pα!q1{2, for every r ą 0. This implies A1

51pCdq “ ApCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-49
SLIDE 49

Images under the Bargmann transform

Characterizations of certain spaces of power series

Any entire function F is equal to a power series expansion ÿ

α

cαeα such that |cα| À r|α|pα!q1{2, for every r ą 0. This implies A1

51pCdq “ ApCdq.

From the definitions it now follows for s ě 1

2 and s0 ă 1 2:

A0,s0pCdq Ď As0pCdq Ď A0,spCdq Ď AspCdq Ď A1

spCdq Ď A1 0,spCdq Ď ApCdq Ď A1 s0pCdq Ď A1 0,s0pCdq

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-50
SLIDE 50

Images under the Bargmann transform

Characterizations of certain spaces of power series

Any entire function F is equal to a power series expansion ÿ

α

cαeα such that |cα| À r|α|pα!q1{2, for every r ą 0. This implies A1

51pCdq “ ApCdq.

From the definitions it now follows for s ě 1

2 and s0 ă 1 2:

A0,s0pCdq Ď As0pCdq Ď A0,spCdq Ď AspCdq Ď A1

spCdq Ď A1 0,spCdq Ď ApCdq Ď A1 s0pCdq Ď A1 0,s0pCdq

What about those spaces which are contained in ApCdq??

  • J. Toft

Analytic Ψdo Potsdam, March 2019 10 / 16

slide-51
SLIDE 51

Images under the Bargmann transform

For s0 ă 1

2,

s ě 1

2,

xzy “ 1 ` |z| (Recall: s0 ă 5σ ă 1

2):

The tiny planets (smaller than Gelfand-Shilov): , ,

  • J. Toft

Analytic Ψdo Potsdam, March 2019 11 / 16

slide-52
SLIDE 52

Images under the Bargmann transform

For s0 ă 1

2,

s ě 1

2,

xzy “ 1 ` |z| (Recall: s0 ă 5σ ă 1

2):

The tiny planets (smaller than Gelfand-Shilov): A0,s0 { As0 “ t F P A ; |Fpzq| À erplogxzyq

1 1´2s0 , for every / some r ą 0 u,

A0,5σ { A5σ “ t F P A ; |Fpzq| À er|z|

2σ σ`1 , for every / some r ą 0 u,

A0, 1

2 “ t F P A ; |Fpzq| À er|z|2, for every r ą 0 u,

,

  • J. Toft

Analytic Ψdo Potsdam, March 2019 11 / 16

slide-53
SLIDE 53

Images under the Bargmann transform

For s0 ă 1

2,

s ě 1

2,

xzy “ 1 ` |z| (Recall: s0 ă 5σ ă 1

2):

The tiny planets (smaller than Gelfand-Shilov): A0,s0 { As0 “ t F P A ; |Fpzq| À erplogxzyq

1 1´2s0 , for every / some r ą 0 u,

A0,5σ { A5σ “ t F P A ; |Fpzq| À er|z|

2σ σ`1 , for every / some r ą 0 u,

A0, 1

2 “ t F P A ; |Fpzq| À er|z|2, for every r ą 0 u,

The Gelfand-Shilov world: A0,s { As “ t F P A ; |Fpzq| À e

|z|2 2 ´r|z| 1 s , for every / some r ą 0 u, s ‰ 1

2, A1

s { A1 0,s “ t F P A ; |Fpzq| À e

|z|2 2 `r|z| 1 s , for every / some r ą 0 u,

  • J. Toft

Analytic Ψdo Potsdam, March 2019 11 / 16

slide-54
SLIDE 54

Images under the Bargmann transform

For s0 ă 1

2,

s ě 1

2,

xzy “ 1 ` |z| (Recall: s0 ă 5σ ă 1

2):

The tiny planets (smaller than Gelfand-Shilov): A0,s0 { As0 “ t F P A ; |Fpzq| À erplogxzyq

1 1´2s0 , for every / some r ą 0 u,

A0,5σ { A5σ “ t F P A ; |Fpzq| À er|z|

2σ σ`1 , for every / some r ą 0 u,

A0, 1

2 “ t F P A ; |Fpzq| À er|z|2, for every r ą 0 u,

The Gelfand-Shilov world: A0,s { As “ t F P A ; |Fpzq| À e

|z|2 2 ´r|z| 1 s , for every / some r ą 0 u, s ‰ 1

2, A1

s { A1 0,s “ t F P A ; |Fpzq| À e

|z|2 2 `r|z| 1 s , for every / some r ą 0 u,

Beyond Gelfand-Shilov life: A1

0, 1

2 “ t F P A ; |Fpzq| À er|z|2, for some r ą 0 u,

A1

5σ { A1 0,5σ “ t F P A ; |Fpzq| À er|z|

2σ σ´1 , for every / some r ą 0 u, σ ą 1,

A1

51 “ A

p“ ApCdqq, A1

0,51 “

ď

Rą0

ApBRp0qq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 11 / 16

slide-55
SLIDE 55

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-56
SLIDE 56

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-57
SLIDE 57

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-58
SLIDE 58

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-59
SLIDE 59

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: We have TK “ OpVpaq when Kpz, wq “ apz, wqepz,wq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-60
SLIDE 60

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-61
SLIDE 61

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: If apz, wq “ 1, then we regain the reproducing kernel: OpVpaqFpzq “ pΠAFqpzq “ Fpzq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-62
SLIDE 62

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-63
SLIDE 63

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: OpVpaqFpzq “ ÿ

|α|ďN

aαpzqpBα

z Fqpzq,

apz, wq “ ÿ

|α|ďN

aαpzqwα.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-64
SLIDE 64

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-65
SLIDE 65

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: If apz, wq is analytic, then pOpVpaqFqpzq “ apz, zqFpzq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-66
SLIDE 66

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-67
SLIDE 67

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: If χ is the characteristic function of a polydisc and apz, wq “ χpwq, then OpVpaq is bijective between suitable AspCdq spaces.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-68
SLIDE 68

Analytic pseudo-differential and integral operators

Analytic pseudo-differential and integral operators

Suppose apz, wq and Kpz, wq are suitable functions, analytic in z P Cd.

1 The analytic pseudo-differential operator, OpVpaq, is defined by

pOpVpaqFqpzq “ ż

Cd apz, wqFpwqepz,wq dµpwq,

F P A0pCdq.

2 The (analytic) kernel operator, TK, is defined by

pTKFqpzq “ ż

Cd Kpz, wqFpwq dµpwq,

F P A0pCdq. Examples and remarks: If χ is the characteristic function of a polydisc and apz, wq “ χpwq, then OpVpaq is bijective between suitable AspCdq spaces. Some sorts of analytic Paley-Wiener properties Nabizadeh-Pfeuffer-T. (2018).

  • J. Toft

Analytic Ψdo Potsdam, March 2019 12 / 16

slide-69
SLIDE 69

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-70
SLIDE 70

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-71
SLIDE 71

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces. We also let LpV1, V2q be the set of all linear continuous mappings from the topological vector space V1 to the topological vector space V2.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-72
SLIDE 72

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-73
SLIDE 73

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces.

  • Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017

Let s1 P R5 and s2 P R5. The map K ÞÑ TK is bijective

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-74
SLIDE 74

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces.

  • Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017

Let s1 P R5 and s2 P R5. The map K ÞÑ TK is bijective from

  • A0,s1pC2dq

to LpA1

0,s1pCdq, A0,s1pCdqq,

and from

  • A1

0,s1pC2dq

to LpA0,s1pCdq, A1

0,s1pCdqq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-75
SLIDE 75

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

In what follows we let

  • A1

spC2dq “ t Kpz, wq ; pz, wq ÞÑ Kpz, wq P A1 spC2dq u,

and similarly for other spaces.

  • Thm. Teofanov-T. 2019, Chen-Signahl-T. 2017

Let s1 P R5 and s2 P R5. The map K ÞÑ TK is bijective from

  • A0,s1pC2dq

to LpA1

0,s1pCdq, A0,s1pCdqq,

and from

  • A1

0,s1pC2dq

to LpA0,s1pCdq, A1

0,s1pCdqq.

from

  • As2pC2dq

to LpA1

s2pCdq, As2pCdqq,

and from

  • A1

s2pC2dq

to LpAs2pCdq, A1

s2pCdqq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-76
SLIDE 76

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

LpA1

s2, As2q “ t TK ; K P

  • As2 u,

LpAs2, A1

s2q “ t TK ; K P

  • A1

s2 u

  • etc. . .
  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-77
SLIDE 77

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

LpA1

s2, As2q “ t TK ; K P

  • As2 u,

LpAs2, A1

s2q “ t TK ; K P

  • A1

s2 u

  • etc. . .
  • Thm. Teofanov-T. (2019)

Let t P C, s1 P R5, s1 ď 1

2, and s2 P R5, s2 ă 1

  • 2. Then

Kpz, wq ÞÑ Kpz, wqetpz,wq is a continuous bijection on

  • A1

0,s1pC2dq

and on

  • A1

s2pC2dq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-78
SLIDE 78

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

LpA1

s2, As2q “ t TK ; K P

  • As2 u,

LpAs2, A1

s2q “ t TK ; K P

  • A1

s2 u

  • etc. . .
  • Thm. Teofanov-T. (2019)

Let t P C, s1 P R5, s1 ď 1

2, and s2 P R5, s2 ă 1

  • 2. Then

Kpz, wq ÞÑ Kpz, wqetpz,wq is a continuous bijection on

  • A1

0,s1pC2dq

and on

  • A1

s2pC2dq.

By combining this with the earlier kernel theorems:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-79
SLIDE 79

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

LpA1

s2, As2q “ t TK ; K P

  • As2 u,

LpAs2, A1

s2q “ t TK ; K P

  • A1

s2 u

  • etc. . .
  • Thm. Teofanov-T. (2019)

Let t P C, s1 P R5, s1 ď 1

2, and s2 P R5, s2 ă 1

  • 2. Then

Kpz, wq ÞÑ Kpz, wqetpz,wq is a continuous bijection on

  • A1

0,s1pC2dq

and on

  • A1

s2pC2dq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-80
SLIDE 80

Analytic pseudo-differential and integral operators

Kernel theorems and related mapping properties

pOpVpaqFqpzq “ ş

Cd apz, wqFpwqepz,wq dµpwq,

pTKFqpzq “ ş

Cd Kpz, wqFpwq dµpwq.

LpA1

s2, As2q “ t TK ; K P

  • As2 u,

LpAs2, A1

s2q “ t TK ; K P

  • A1

s2 u

  • etc. . .
  • Thm. Teofanov-T. (2019)

Let t P C, s1 P R5, s1 ď 1

2, and s2 P R5, s2 ă 1

  • 2. Then

Kpz, wq ÞÑ Kpz, wqetpz,wq is a continuous bijection on

  • A1

0,s1pC2dq

and on

  • A1

s2pC2dq.

  • Thm. Teofanov-T. (2019)

Let s1 P R5, s1 ď 1

2, and s2 P R5, s2 ă 1

  • 2. Then

LpA0,s1pCdq, A1

0,s1pCdqq “ t OpVpaq ; a P

A1

0,s1pC2dq u.

LpAs2pCdq, A1

s2pCdqq “ t OpVpaq ; a P

A1

s2pC2dq u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 13 / 16

slide-81
SLIDE 81

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d,

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

slide-82
SLIDE 82

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

slide-83
SLIDE 83

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd that is, ω0 ą 0 and ω0, 1{ω0 P L8

locpCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 84

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 85

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 86

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 87

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 88

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

If instead p P r1, 8s4d, let

  • A p

pωqpC2dq “ t All K ; pz, wq ÞÑ Kpz, wq P A p pωqpC2dq u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 89

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

If instead p P r1, 8s4d, let

  • A p

pωqpC2dq “ t All K ; pz, wq ÞÑ Kpz, wq P A p pωqpC2dq u.

Recently, results of the following type appeared

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 90

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

If instead p P r1, 8s4d, let

  • A p

pωqpC2dq “ t All K ; pz, wq ÞÑ Kpz, wq P A p pωqpC2dq u.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 91

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

If instead p P r1, 8s4d, let

  • A p

pωqpC2dq “ t All K ; pz, wq ÞÑ Kpz, wq P A p pωqpC2dq u.

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 92

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

Let LppCdq — LppR2dq be the mixed Lebesgue space with respect to p P r1, 8s2d, ω0 be a weight on Cd and let ω be a weight on C2d. Let Bp

pω0qpCdq “ t F ; Fpzqe´ 1

2 ¨|z|2ω0p

? 2 ¨ zq P LppCdq u. Let Ap

pω0qpCdq “ Bp pω0qpCdq X ApCdq.

If instead p P r1, 8s4d, let

  • A p

pωqpC2dq “ t All K ; pz, wq ÞÑ Kpz, wq P A p pωqpC2dq u.

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

Then TK is continuous from Ap1

pω1qpCdq to Ap2 pω2qpCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 93

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

Then TK is continuous from Ap1

pω1qpCdq to Ap2 pω2qpCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 94

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

Then TK is continuous from Ap1

pω1qpCdq to Ap2 pω2qpCdq.

By putting some restrictions on ω, ωj 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

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SLIDE 95

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

Then TK is continuous from Ap1

pω1qpCdq to Ap2 pω2qpCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 96

Analytic pseudo-differential and integral operators

Analytic Ψdo with Lebesgue conditions on their symbols

  • Thm. Teofanov-T. (2019)

Suppose ω and K P ApC2dq satisfies GK,ωpz `w, zq P Lp,qpC2dq, GK,ωpz, wq “ e´ 1

2 p|z|2`|w|2qKpz, wq¨ωp

? 2 z, ? 2 wq,

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2pzq ω1pwq À ωpz, wq.

Then TK is continuous from Ap1

pω1qpCdq to Ap2 pω2qpCdq.

  • Thm. Gr¨
  • chenig-Heil, T.

Suppose

1 p1 ´ 1 p2 “ 1 ´ 1 p ´ 1 q,

q ď p2 ď p,

ω2px,ξ`ηq ω1px`y,ξq À ωpx, ξ, η, yq,

a P Mp,q

pωqpR2dq.

Then Oppaq is continuous from Mp1

pω1qpRdq to Mp2 pω2qpRdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 14 / 16

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SLIDE 97

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 98

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 99

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

Usually strong restrictions are imposed on the weights (Feichtinger, Gr¨

  • chenig).
  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 100

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 101

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

Mp

pωqpRdq is a Banach space

(Feichtinger for standard weights, T. (2017) for general weights).

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 102

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

slide-103
SLIDE 103

Analytic pseudo-differential and integral operators

Some further properties

A large family of modulation spaces:

Definition modulation spaces

Let p P r1, 8s2d, φpxq “ e´ 1

2 ¨|x|2, x P Rd, ω is a weight on R2d,

f P H1

51pRdq, and Vφf px, ξq “ xf , e´ix ¨ ,ξyφp ¨ ´ xqy (the STFT of f ).

Then the modulation space Mp

pωqpRdq is the set of all f P H1 51pRdq such

that }f }Mp

pωq ” }Vφf ¨ ω}Lp ă 8.

Vd is bijective and isometric from Mp

pωqpRdq to Ap pωqpCdq

(T. (2017) for the most general weights)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 104

Analytic pseudo-differential and integral operators

Some further properties

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 105

Analytic pseudo-differential and integral operators

Some further properties

Some further properties (T. 2012)

ω0 suitable, then ΠA is continuous projection from Bp

pω0qpCdq to Ap pω0qpCdq.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 106

Analytic pseudo-differential and integral operators

Some further properties

Some further properties (T. 2012)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 107

Analytic pseudo-differential and integral operators

Some further properties

Some further properties (T. 2012)

We may have OpVpa1q “ OpVpa2q for different aj P Bp

pωqpC2dq, but . . .

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 108

Analytic pseudo-differential and integral operators

Some further properties

Some further properties (T. 2012)

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 109

Analytic pseudo-differential and integral operators

Some further properties

Some further properties (T. 2012)

a P Bp

pωqpC2dq, ω suitable, then OpVpaq “ OpVpa0q for unique a0 P

A p

pωqpC2dq

  • J. Toft

Analytic Ψdo Potsdam, March 2019 15 / 16

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SLIDE 110

Analytic pseudo-differential and integral operators

Thank you for your attention.

  • J. Toft

Analytic Ψdo Potsdam, March 2019 16 / 16