Reverse Carleson Embeddings for Model Spaces - VT Math
Reverse Carleson Embeddings for Model Spaces A. Blandigneres, E. Fricain, F. Gaunard, A. Hartmann, W. Ross
March 2013
W. Ross et al ()
Reverse Carleson
March 2013
1 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Hardy space
dθ 2π Z 2 2 H = f ∈ O(D) : sup |f (rζ)| dm(ζ) < ∞ m=
0 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Goal
Continue the study of reverse Carleson embeddings for model spaces (ΘH 2 )⊥ ,
W. Ross et al ()
Θ inner.
Reverse Carleson
March 2013
7 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example If (kλΘn )n≥1 is an unconditional basis for (ΘH 2 )⊥ and µ=
1 δ Θ k 2 λn kk 2 λ n n≥1 X
then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
9 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example If (kλΘn )n≥1 is an unconditional basis for (ΘH 2 )⊥ and µ=
1 δ Θ k 2 λn kk 2 λ n n≥1 X
then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
9 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥ Example Suppose M ⊂ H 2 is nearly invariant (f ∈ M , f (0) = 0 ⇒ f /z ∈ M ) and g ∈ M satisfies 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
W. Ross et al ()
Reverse Carleson
March 2013
15 / 19
Isometric embedding theorem
Theorem (Aleksandrov (1996); BFGHR (2012)) Suppose Θ is inner and µ ∈ M+ (T). Then the embedding (ΘH 2 )⊥ ,→ L2 (µ) is isometric if and only if there is a b ∈ b(H ∞ ) so that µ = σbΘ .
0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
March 2013
W. Ross et al ()
Reverse Carleson
March 2013
1 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Theme
H ⊂ O(D) Hilbert space µ ∈ M+ (D) Carleson embedding if H ,→ L2 (µ), i.e., kf kµ . kf kH ,
f ∈H
Reverse Carleson embedding if H ,→ L2 (µ) is injective with closed range, i.e., k · kH k · kµ on H
W. Ross et al ()
Reverse Carleson
March 2013
2 / 19
Hardy space
dθ 2π Z 2 2 H = f ∈ O(D) : sup |f (rζ)| dm(ζ) < ∞ m=
0 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Reverse Carleson Embeddings for H 2 Theorem (Queffelec et al (2010), Hartmann et al (2013)) For µ ∈ M+ (D− ) with H 2 ,→ L2 (µ) TFAE 1
the embedding is injective with closed range
2
inf I
µ(S(I)) > 0, m(I)
inf over all open arcs I ⊂ T.
Corollary For µ ∈ M+ (D− ) we have k · kµ k · k2 on H 2 if and only if µ(S(I)) m(I),
W. Ross et al ()
Reverse Carleson
I ⊂ T.
March 2013
6 / 19
Goal
Continue the study of reverse Carleson embeddings for model spaces (ΘH 2 )⊥ ,
W. Ross et al ()
Θ inner.
Reverse Carleson
March 2013
7 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Embeddings of model spaces Theorem (Cohn (1982); Treil-Volberg (1986)) Suppose that Θ is inner such that L(Θ, ) := {z ∈ D : |Θ(z)| < } is connected for some > 0. Then, for µ ∈ M+ (D− ), (ΘH 2 )⊥ ,→ L2 (µ) if and only if µ(S(I)) < +∞, sup m(I) I where the supremum is taken over all arcs I ⊂ T for which S(I) ∩ L(Θ, ) 6= ∅.
W. Ross et al ()
Reverse Carleson
March 2013
8 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example If (kλΘn )n≥1 is an unconditional basis for (ΘH 2 )⊥ and µ=
1 δ Θ k 2 λn kk 2 λ n n≥1 X
then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
9 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example If (kλΘn )n≥1 is an unconditional basis for (ΘH 2 )⊥ and µ=
1 δ Θ k 2 λn kk 2 λ n n≥1 X
then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
9 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥
Example Suppose Θ is inner and σΘ ∈ M+ (T) satisfies Z 1 − |z|2 1 + Θ(z) = dσΘ (ζ) < 1 − Θ(z) |z − ζ|2 This measure is called a Clark measure for Θ. By Clark/Poltoratski the embedding (ΘH 2 )⊥ ,→ L2 (σΘ ) is isometric.
W. Ross et al ()
Reverse Carleson
March 2013
10 / 19
Examples of reverse embeddings for (ΘH 2 )⊥ Example Suppose M ⊂ H 2 is nearly invariant (f ∈ M , f (0) = 0 ⇒ f /z ∈ M ) and g ∈ M satisfies 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
Reverse embedding theorem for (ΘH 2 )⊥ Theorem (BFGHR (2012)) Suppose that Θ is inner such that L(Θ, ) is connected for some > 0 and µ ∈ M+ (D− ) such that (ΘH 2 )⊥ ,→ L2 (µ). Then there exists an N = N (Θ, ) > 1 such that if inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T with S(N I) ∩ L(Θ, ) 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
14 / 19
W. Ross et al ()
Reverse Carleson
March 2013
15 / 19
Isometric embedding theorem
Theorem (Aleksandrov (1996); BFGHR (2012)) Suppose Θ is inner and µ ∈ M+ (T). Then the embedding (ΘH 2 )⊥ ,→ L2 (µ) is isometric if and only if there is a b ∈ b(H ∞ ) so that µ = σbΘ .
0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Reverse embeddings via dominating sets
Theorem (BFGHR (2012)) Let Θ be an inner function, Σ be a dominating set for (ΘH 2 )⊥ , and µ ∈ M+ (D− ) be such that (ΘH 2 )⊥ ,→ L2 (µ). Suppose that inf I
µ(S(I)) > 0, m(I)
where the infimum is taken over all arcs I ⊂ T for which I ∩ Σ 6= ∅, then k · kµ k · k2 on (ΘH 2 )⊥ .
W. Ross et al ()
Reverse Carleson
March 2013
18 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
Advertisement for SEAM 2014 talk For inner Θ, (ΘH 2 )⊥ has kernel kλΘ (z) =
1 − Θ(λ)Θ(z) . 1 − λz
Now let b ∈ H ∞ , kbk∞ 6 1 and let H (b) be the RKHS with kλb (z) :=
1 − b(λ)b(z) . 1 − λz
H (b) ⊂ H 2 (contractively). When kbk∞ < 1, then H (b) = H 2 (with equivalent norm). When b is inner H (b) = (bH 2 )⊥ . Carleson and reverse Carleson embeddings for H (b) spaces. W. Ross et al ()
Reverse Carleson
March 2013
19 / 19
