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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

SLIDE 1

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

SLIDE 2

Norm: f 2

D = 1

π

|f ′(z)|2dxdy + 1 2π +π

−π

|f (eit)|2dt.

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

SLIDE 3

Norm: f 2

D = 1

π

|f ′(z)|2dxdy + 1 2π +π

−π

|f (eit)|2dt. Reproducing kernel: f (z) =< f , kz >D with kz(w) =

1 zw log 1 1−zw .

kz2

D = kz(z) = 1 |z|2 log 1 1−|z|2 .

|f (z)| ≤ kzDf D.

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

SLIDE 4

Norm: f 2

D = 1

π

|f ′(z)|2dxdy + 1 2π +π

−π

|f (eit)|2dt. Reproducing kernel: f (z) =< f , kz >D with kz(w) =

1 zw log 1 1−zw .

kz2

D = kz(z) = 1 |z|2 log 1 1−|z|2 .

|f (z)| ≤ kzDf D. Trivial estimate Z = {zn : n ∈ N} f → {f (zn)/kznD : n ∈ N} maps D into ℓ∞(Z).

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

SLIDE 5

Universally interpolating sequences Z is universally interpolating if f → {f (zn)/kznD : n ∈ N} maps D onto ℓ2 boundedly.

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

SLIDE 6

Universally interpolating sequences Z is universally interpolating if f → {f (zn)/kznD : n ∈ N} maps D onto ℓ2 boundedly. Interpolating sequences Z is interpolating if f → {f (zn)/kznD : n ∈ N} maps D ⊆ D

nto ℓ2.

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

SLIDE 7

Universally interpolating sequences Z is universally interpolating if f → {f (zn)/kznD : n ∈ N} maps D onto ℓ2 boundedly. Interpolating sequences Z is interpolating if f → {f (zn)/kznD : n ∈ N} maps D ⊆ D

nto ℓ2.

Weakly interpolating sequences Z is weakly interpolating if for all zn there is fn such that fn(zm) = δn(m) and fn2

D ≤ Ckzn2 D = C 1 |zn|2 log 1 1−|zn|2 .

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

SLIDE 8

Universally interpolating sequences Z is universally interpolating if f → {f (zn)/kznD : n ∈ N} maps D onto ℓ2 boundedly. Interpolating sequences Z is interpolating if f → {f (zn)/kznD : n ∈ N} maps D ⊆ D

nto ℓ2.

Weakly interpolating sequences Z is weakly interpolating if for all zn there is fn such that fn(zm) = δn(m) and fn2

D ≤ Ckzn2 D = C 1 |zn|2 log 1 1−|zn|2 .

Zero sets Z is a zero set for D if there is 0 = f ∈ D such that f (zn) = 0 for n ∈ N.

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

SLIDE 9

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

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Results

Trivia Universally interpolating = ⇒ Interpolating = ⇒ Weakly interpolating = ⇒ Zero set Elementary Weakly interpolating = ⇒ (Sep) |kz(w)| ≤ (1 − ǫ)kzDkwD. Universally interpolating = ⇒ (Car) |f (zn)|2/kzn2

D =:

|f |2dµZ ≤ Cf 2

D.

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

SLIDE 11

Results

Trivia Universally interpolating = ⇒ Interpolating = ⇒ Weakly interpolating = ⇒ Zero set Elementary Weakly interpolating = ⇒ (Sep) |kz(w)| ≤ (1 − ǫ)kzDkwD. Universally interpolating = ⇒ (Car) |f (zn)|2/kzn2

D =:

|f |2dµZ ≤ Cf 2

D.

Theorems Universally interpolating ⇐ ⇒ (Sep) and (Car) [Marshall and Sundberg 1994; Chris Bishop 1994] Weakly interpolating ⇐ ⇒ Interpolating [Bishop 1994] Zero set if 1/kzn2

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

SLIDE 12

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

SLIDE 13

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¨

e 2001: (Sep) and (Simple) =

⇒ interpolating; (Simple) µZ(S(I)) 1 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

SLIDE 14

Question Bishop: interpolating = ⇒ µZ(∆) < ∞?

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

SLIDE 15

Question Bishop: interpolating = ⇒ µZ(∆) < ∞? Answer NO

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

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Question Bishop: interpolating = ⇒ µZ(∆) < ∞? Answer NO Theorem There is a sequence Z in ∆ s.t. (i) µZ(∆) = ∞ (ii) Z is interpolating for D. Z = {zn,j : 1 ≤ j ≤ 2n ∈ N} and 1 − |zn,j| = 2−An, Zn = {zn,j : 1 ≤ j ≤ 2n} have a Cantor-like structure.

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

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Sketch of the proof

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

SLIDE 18

Sketch of the proof

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

SLIDE 19

Sketch of the proof

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

SLIDE 20

B¨

e’s functions

∆ ∋ w → ϕw ∈ D: ϕw almost minimizes ϕ2

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

SLIDE 21

Remarks

interpolating sequences on trees can be completely characterized, but the discrete solution can not in general be made holomorphic by means of B¨

e’s functions;

Bishop’s analogs of B¨

e’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