Interpolating sequences for the Dirichlet space Nicola Arcozzi, with - PowerPoint PPT Presentation

Interpolating sequences for the Dirichlet space Nicola Arcozzi, with R. Rochberg and E. Sawyer Universit` a di Bologna 18 giugno 2013 Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Norm: � + π D = 1 �� | f ′ ( z ) | 2 dxdy + 1 � f � 2 | f ( e it ) | 2 dt . π 2 π ∆ − π Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Norm: � + π D = 1 �� | f ′ ( z ) | 2 dxdy + 1 � f � 2 | f ( e it ) | 2 dt . π 2 π ∆ − π Reproducing kernel: 1 1 f ( z ) = < f , k z > D with k z ( w ) = zw log 1 − zw . � k z � 2 1 1 D = k z ( z ) = | z | 2 log 1 −| z | 2 . | f ( z ) | ≤ � k z � D � f � D . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Norm: � + π D = 1 �� | f ′ ( z ) | 2 dxdy + 1 � f � 2 | f ( e it ) | 2 dt . π 2 π ∆ − π Reproducing kernel: 1 1 f ( z ) = < f , k z > D with k z ( w ) = zw log 1 − zw . � k z � 2 1 1 D = k z ( z ) = | z | 2 log 1 −| z | 2 . | f ( z ) | ≤ � k z � D � f � D . Trivial estimate Z = { z n : n ∈ N } f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D into ℓ ∞ ( Z ). Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Universally interpolating sequences Z is universally interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D onto ℓ 2 boundedly. Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Universally interpolating sequences Z is universally interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D onto ℓ 2 boundedly. Interpolating sequences Z is interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D ⊆ D onto ℓ 2 . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Universally interpolating sequences Z is universally interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D onto ℓ 2 boundedly. Interpolating sequences Z is interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D ⊆ D onto ℓ 2 . Weakly interpolating sequences Z is weakly interpolating if for all z n there is f n such that f n ( z m ) = δ n ( m ) and � f n � 2 D ≤ C � k z n � 2 1 1 D = C | z n | 2 log 1 −| z n | 2 . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Universally interpolating sequences Z is universally interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D onto ℓ 2 boundedly. Interpolating sequences Z is interpolating if f �→ { f ( z n ) / � k z n � D : n ∈ N } maps D ⊆ D onto ℓ 2 . Weakly interpolating sequences Z is weakly interpolating if for all z n there is f n such that f n ( z m ) = δ n ( m ) and � f n � 2 D ≤ C � k z n � 2 1 1 D = C | z n | 2 log 1 −| z n | 2 . Zero sets Z is a zero set for D if there is 0 � = f ∈ D such that f ( z n ) = 0 for n ∈ N . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Results Trivia Universally interpolating = ⇒ Interpolating = ⇒ Weakly interpolating = ⇒ Zero set Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Results Trivia Universally interpolating = ⇒ Interpolating = ⇒ Weakly interpolating = ⇒ Zero set Elementary Weakly interpolating = ⇒ (Sep) | k z ( w ) | ≤ (1 − ǫ ) � k z � D � k w � D . Universally interpolating = ⇒ (Car) � | f ( z n ) | 2 / � k z n � 2 | f | 2 d µ Z ≤ C � f � 2 D =: � D . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Results Trivia Universally interpolating = ⇒ Interpolating = ⇒ Weakly interpolating = ⇒ Zero set Elementary Weakly interpolating = ⇒ (Sep) | k z ( w ) | ≤ (1 − ǫ ) � k z � D � k w � D . Universally interpolating = ⇒ (Car) � | f ( z n ) | 2 / � k z n � 2 | f | 2 d µ Z ≤ C � f � 2 D =: � D . Theorems Universally interpolating ⇐ ⇒ (Sep) and (Car) [Marshall and Sundberg 1994; Chris Bishop 1994] Weakly interpolating ⇐ ⇒ Interpolating [Bishop 1994] Zero set if � 1 / � k z n � 2 D < ∞ [Shapiro and Schields 1962, after Carleson 1958] Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

What’s next? Open problems 1 Characterization of the zero sets. 2 Characterization of the interpolating sequences. Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

What’s next? Open problems 1 Characterization of the zero sets. 2 Characterization of the interpolating sequences. Partial result on interpolating sequences Bishop 1994; B¨ oe 2001: (Sep) and (Simple) = ⇒ interpolating; 1 (Simple) µ Z ( S ( I )) � log(1 / | I | ) . I ⊆ ∆: arc; S ( I ) = { z : z / | z | ∈ I and 1 − | z | ≤ | I |} : the usual Carleson box based on I . Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Question Bishop: interpolating = ⇒ µ Z (∆) < ∞ ? Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Question Bishop: interpolating = ⇒ µ Z (∆) < ∞ ? Answer NO Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Question Bishop: interpolating = ⇒ µ Z (∆) < ∞ ? Answer NO Theorem There is a sequence Z in ∆ s.t. (i) µ Z (∆) = ∞ (ii) Z is interpolating for D . Z = { z n , j : 1 ≤ j ≤ 2 n ∈ N } and 1 − | z n , j | = 2 − A n , Z n = { z n , j : 1 ≤ j ≤ 2 n } have a Cantor-like structure. Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Sketch of the proof Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Sketch of the proof Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Sketch of the proof Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

B¨ oe’s functions ∆ ∋ w �→ ϕ w ∈ D : ϕ w almost minimizes � ϕ � 2 D with ϕ ( w ) = 1, ϕ (0) = 0. Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space

Remarks interpolating sequences on trees can be completely characterized, but the discrete solution can not in general be made holomorphic by means of B¨ oe’s functions; Bishop’s analogs of B¨ oe’s functions are not very well understood, they might provide the right tool; interpolating sequences on trees can be explained in terms of potential theory for networks (Soardi’s monograph). Nicola Arcozzi, with R. Rochberg and E. Sawyer Alba 2013: About interpolating sequences for the Dirichlet space