Dirichlet spaces with superharmonic weights

Transcription

1 Stamatis Pouliasis Texas Tech University Complex Analysis and Spectral Theory Celebration of Thomas J. Ransford s 60th birthday Université Laval May 21-25, 2018

2 = {z C : z < 1} A=Area measure efinition (Hardy space H 2 ) Classical spaces f H 2 f 2 H 2 = f (0) π efinition (irichlet space ) f f (z) 2 da(z) < +. efinition (BMOA) f BMOA sup f (z) 2 log w f (z) 2 log 1 da(z) < +. z 1 wz da(z) < +. z w

3 (A. Aleman, 1993) ω : (0, + ], positive superharmonic function ω(z) = log 1 wz 1 z 2 dµ(w) + z w ζ z 2 dν(ζ) = U µ (z) + P ν (z), (1 z )dµ(z) < +, and ν( ) < +. efinition (Weighted irichlet space ω ) f ω f (z) 2 ω(z)da(z) < +.

4 irichlet spaces ν with harmonic weights, ω = P ν, (S. Richter, 1991) We will concentrate on irichlet spaces µ, ω(z) = U µ (z) = log 1 wz z w dµ(w). lim r 1 U µ (rζ) = 0 for almost every ζ. f 2 µ = f 2 H π f (z) 2 U µ (z)da(z). Examples ( p spaces with radial superharmonic weights) ω p (z) = (1 z 2 ) p, p (0, 1), dµ p = ((1 z 2 ) p )da(z), µ p () = +.

5 efinition (Carleson measures) For every arc I with length I, µ is Carleson measure if Theorem µ H 2, µ, S(I ) = {rζ : 1 I 2π µ(s(i )) sup <. I I if µ() < +, BMOA µ H 2, < r < 1, ζ I }. if (1 z 2 )dµ(z) is a Carleson measure, µ.

6 efinition (Balayage) If µ() < +, the balayage of µ is the function S µ (ζ) = 1 1 z 2 dµ(z), ζ. 2π ζ z 2 Note that every f H 2 has radial limit f (ζ) at almost every ζ. efinition (Weighted Hardy spaces H 2 µ) Suppose µ() < +. Hµ 2 = {f H 2 : Theorem (with G. Bao and N. G. Göğüş) If µ is a Carleson measure, then µ = H 2 µ. f (ζ) 2 S µ (ζ) dζ < + }.

7 Corrolary (with G. Bao and N. G. Göğüş) Let µ be a Carleson measure and let ν be a measure on. There exists C > 0 such that ( 1/2 f (z) dν(z)) 2 C f µ, f µ, if and only if there exists C > 0 such that O µ 2 dν C I, S(I ) for every arc I, where ( O µ (z) = exp ζ + z ζ z log 1 Sµ (ζ) ) dζ, z, 2π is an outer function with O µ (ζ) = 1/ S µ (ζ), at almost every ζ.

8 Corrolary (with G. Bao and N. G. Göğüş) Suppose that µ = + n=1 a nδ zn is a Carleson measure, where z n and a n > 0, n N. The reproducing kernel of µ for λ with respect to µ is where and K(z, λ) = K 0 (z, λ) + T µ (z) = exp + n=1 a n K 0 (z, z n )K 0 (z n, λ), z, 1 a n K 0 (z n, z n ) K 0 (z, λ) = T µ(λ) 1 λz T µ(z), z, ( 1 ζ + z 2π ζ z log 1 ), 1 + Sµ (ζ) dζ z.

9 efinition φ H 2 is called inner if φ(ζ) = 1 for almost every ζ. Theorem (Alexander-Taylor-Ullman inequality) If f H 2 with f (0) = 0, then f 2 H 2 A(f ()). π Equality holds if and only if f = cφ where c C and φ is an inner function satisfying φ(0) = 0.

10 Theorem (with G. Bao and N. G. Göğüş) Suppose µ() < +. If f µ with f (0) = 0, f 2 µ (1 + µ()) A(f ()). π Equality holds if and only if the measure µ is of the form µ = a 0 δ n=1 a n δ zn, a n > 0, z n, and f is of the form f = cφ, where c C and φ is an inner function with φ(0) = φ(z n ) = 0, for every n N.

11 Proof. Fix w. f (z) 2 1 wz log z w da(z) = f () f (a)=x f () 1 A(f ()) 2 log 1 wa a w da(x) G f () (x, f (w))da(x) and 2 π = 2 π 2 π f (z) 2 U µ (z)da(z) ( f (z) 2 1 wz ) log z w da(z) dµ(w) 1 µ()a(f ()) A(f ())dµ(w) =. 2 π

12 Suppose that equality holds. Then f = cφ where c C and φ is an inner function satisfying φ(0) = 0. φ(z) 2 = h φ (z) 1 log 1 wz φ(w) 2 da(w) 2π z w = 1 2 log 1 wz φ (w) 2 da(w). π z w A(cφ()) = A(c) = c 2 π, µ() c 2 = 2 cφ (z) 2 U µ (z)da(z) π = c 2 2 φ (z) 2 log 1 wz da(z)dµ(w) π z w = c 2 (1 φ(w) 2 )dµ(w) = µ() c 2 c 2 φ(w) 2 dµ(w).

13 φ(w) 2 dµ(w) = 0, which holds if and only if φ = 0 µ-almost everywhere. Since the zeros of φ are isolated, the above equality holds if and only if µ is of the form µ = a 0 δ n=1 a n δ zn, a n > 0, z n, and the inner function φ satisfies φ(0) = φ(z n ) = 0, for every n N.

14 efinition (The Möbius invariant function space M( µ )) The Möbius invariant function space M( µ ) generated by µ is the class of holomorphic functions f on, with Examples f M(µ) = M(H 2 ) = BMOA, M() =, M( p ) = Q p, p (0, 1). sup f φ f (φ(0)) µ <. φ Aut()

15 Theorem (with G. Bao, J. Mashreghi and H. Wulan) If µ() < +, M( µ ) = BMOA. If µ() = +, the following are equivalent: (1) M( µ ) is not trivial, (2) M( µ ), (3) (1 z 2 )dµ(z) is a Carleson measure. Which inner functions are contained in M( µ ) (µ() = + )?

16 efinition (Carleson-Newman Blaschke products) A Blaschke product B(z) = k=1 a k a k a k z 1 a k z is called Carleson-Newman Blaschke product if k=1 (1 a k 2 )δ ak is a Carleson measure.

17 Theorem (with G. Bao, J. Mashreghi and H. Wulan) Suppose that µ() = + and let I be an inner function. 1 If I M( µ ), I is a Blaschke product. 2 Suppose that I is a Carleson-Newman Blaschke product with zeros {a k } k=1. Then I M( µ) if and only if sup φ Aut() k=1 (1 a k φ(w) 2 )dµ(w) <. 1 a k φ(w)

18 Proof. Let σ a (z) = a z 1 az, a. ν = tδ 1, t > 0, ( S ν (z) = exp t 1 + z ) 1 z S ν (z) = exp ( t 1 ) z 2 1 z 2 S ν M( µ )

19 Fix c > 0. Consider the horodisk c = {z : 1 } z 2 1 z 2 > c, note that and let S ν e tc, on c, µ a = µ σ a, a.

20 (S ν σ a ) (z) 2 U µ (z)da(z) = (1 S ν (σ a (z)) 2 )dµ(z) σ a( c) (1 S ν (σ a (z)) 2 )dµ(z) = (1 S ν (z) 2 )dµ a (z) c (1 e 2tc )µ(σ a ( c )). Let φ r (z) = σ r (z) and note that φ r ( c ) as r 1. Then lim S ν φ r 2 r 1 µ lim(1 e 2tc )µ(φ r ( c )) = (1 e 2tc )µ() = +. r 1 S ν M( µ ).

21 Thank you!