Weighted Dirichlet spaces and Q p

Transcription

1 Weighted Dirichlet spaces and Q p Nihat Gökhan Göğüş (partly joint with G. Bao and S. Pouliasis) Sabanci University CAFT 2018, Heraklion

2 Dirichlet type spaces SETUP D = {z : z < 1} open unit disk in C. η M(D) positive measure on D, write η = µ + ν, µ = η D, ν = η D. ω(z) = log 1 wz dµ(w) + D z w := U µ (z) + P ν (z). T 1 z 2 ζ z 2 dν(ζ) Weighted Dirichlet space D ω is the space of f Hol(D) f (z) 2 ω(z)da(z) < +. D

3 Dirichlet type spaces H 2 = H 2 (D) the Hardy space, µ = δ 0, ν 0 f 2 H = f (0) f (z) 2 log 1 2 π D z da(z) f (0) f (z) 2 (1 z )da(z) π D Dirichlet space, µ 0, ν = arclength measure on T f 2 D = f 2 H + f (z) 2 da(z) 2 D D

4 Dirichlet type spaces D p = {f H(D) : D f (z) 2 (1 z ) p da(z) < } D ν when µ 0 introduced by Stefan Richter (1991) - two isometries. D ω general studied in Habilitation thesis of Alexandru Aleman (1993). D ω H 2 We are interested in Möbius invariant spaces Q p in connection with Dirichlet type spaces

5 Möbius invariant spaces ϕ Aut(D) = ϕ(z) = e iθ σ a (z), σ a (z) = a z 1 az, θ R, a D X a Banach space of analytic functions on D is called Möbius invariant if f X, ϕ Aut(D) = f ϕ X, f ϕ X = f X Bloch space B : f B = sup z D (1 z 2 ) f (z) < BMOA: analytic functions on D with boundary values of bdd mean oscillation f BMOA = f (0) + sup f σ a f (a) H 2 a D

6 Möbius invariant spaces Q p, 0 p < : 1995, R. Aulaskari, J. Xiao and R. Zhao, f 2 Q p = sup a D D f (z) 2 ( 1 σ a (z) 2) p da(z) < p = 0 = Q 0 = D Dirichlet space p = 1 = Q 1 = BMOA, 1 < p < = Q p = B

7 Möbius invariant spaces (X,. X ) a Banach space of analytic functions in D containing all constants. A. Aleman and A. Simbotin: M(X ) the Möbius invariant function space generated by X f M(X ) = sup f ϕ f (ϕ(0)) X < ϕ Aut(D) X = H p, 0 < p < = M(X ) = BMOA X = A p, 0 < p < = M(X ) = B X = D p, 0 < p < 1 = M(X ) = Q p

8 Space D µ,p Weighted Green function Let µ be a positive Borel measure on D, p > 0, U µ,p (z) = (1 σ z (w) 2 ) p dµ(w) D D µ,p : f 2 D µ,p = D f (z) 2 U µ,p (z)da(z) < (BGP, 2017) 0 < p 1 = U µ,p is superharmonic, otherwise not µ = δ 0 = D µ,p = D p

9 Möbius invariant spaces Denote by F the set of all finite positive Borel measures and by P the set of all probability measures on D. Theorem (BGP, 2017) Let µ F and 0 < p <. Then the following are true. (i) Q p D µ,p. (ii) Q p = M(D µ,p ). (iii) Q p = µ P D µ,p. Moreover, f Qp = sup f Dµ,p µ P

10 Proof of (ii) and (iii) For (ii), Q p D µ,p D p implies Q p = M(Q p ) M(D µ,p ) M(D p ) = Q p Q p µ P D µ,p clear! Suppose f Q p, that is, sup w f Dδw,p =. Choose w k D so that β k = f Dδwk,p 2k. Set ν = 2 k δ wk. k=1 Then ν P, and f 2 D ν,p = k=1 2 k β k =. Hence, f µ P D µ,p.

11 Composition operators ϕ : D D analytic self-map of unit disk D induces a composition operator C ϕ f (z) = f (ϕ(z)), z D, f H(D) Studied like crazy on most known spaces Nevanlinna counting function of ϕ with respect to µ 0, p > 0 N ϕ,µ,p (z) = U µ,p (a), z D, ϕ(a)=z multiplicities are taken into account

12 Change of variables (f ϕ) (z) 2 U µ,p (z)da(z) = D D f (z) 2 N ϕ,µ,p (z)da(z) Subaveraging for 0 < p 1 N ϕ,µ,p (z) 1 Area( z ) z N ϕ,µ,p (w)da(w) for any open disk z D with center at z.

13 Main theorem Theorem (G, 201?) Let p > 1, p > 0, and let ϕ be an analytic self-map of D. Then the following conditions are equivalent. 1 C ϕ : B = Q p Q p is bounded. 2 0 < p 1 N ϕ,µ,p (z) sup µ P D (1 z 2 da(z) <. ) 2 3 For every µ P, there exists a ν P such that C ϕ : D ν,p D µ,p.

14 Main theorem, part 2 Theorem (G, 201?) Let p > 1, 0 < p 1, and let ϕ be an analytic self-map of D. Then the following conditions are equivalent. 1 C ϕ : B = Q p Q p is compact. 2 For every µ P, lim inf sup N ϕ,µ,p (z) = 0. z 1 ν P z U ν,p (z) 3 For every µ P, there exists a ν P such that is compact. C ϕ : D ν,p D µ,p

15 Composition operators Previous work When p = 1, equivalence of (i) and (ii) in Main Theorem generalizes a characterization of bounded/compact C ϕ : B BMOA by S. Makhmutov and M. Tjani. C ϕ : B B studied by many others: K. Madigan and A. Matheson; M. Tjani; H. Wulan, D. Zheng and K. Zhu,...

16 Composition operators Theorem (BGP, 2018 when p=1) Let µ be a positive Borel measure on D, 0 < p 1, and let ϕ be an analytic self-map of D. Then the following conditions are equivalent. 1 C ϕ is bounded on D µ,p. 2 3 N ϕ,µ,p (w) = O(U µ,p (w)), as w 1. 1 N ϕ,µ,p (z)da(z) = O(U µ,p (w)), as w 1, A( w ) w where w = {z D : z w < 1 (1 w )}. 2

17 Composition operators Theorem (BGP, 2018 when p=1) Let µ be a positive Borel measure on D, 0 < p 1, and let ϕ : D D be analytic. Then the following conditions are equivalent. 1 C ϕ is compact on D µ,p. 2 3 N ϕ,µ,p (w) = o(u µ,p (w)), as w 1. 1 N ϕ,µ,p (z)da(z) = o(u µ,p (w)), as w 1, A( w ) w where w = {z D : z w < 1 (1 w )}. 2

18 Composition operators Theorem (G, 201?) Let µ be a positive Borel measure on D, p > 0, 0 < p 1, and let ϕ be an analytic self-map of D. Then the following conditions are equivalent. 1 C ϕ : D µ,p Q p is bounded. 2 sup N ϕ,ν,p (w) = O(U µ,p (w)), as w 1. ν P In this case, C ϕ sup 1 ϕ(0) < w <1 sup N ϕ,ν,p (w) ν P U, and 2 µ,p(w) N ϕ,ν,p (w) C ϕ e lim sup sup w 1 ν P U µ,p (w)

19 Composition operators Theorem (G, 201?) Let µ be a positive Borel measure on D, p > 0, 0 < p 1, and let ϕ be an analytic self-map of D. Then the following conditions are equivalent. 1 C ϕ : D µ,p Q p is compact. 2 sup N ϕ,ν,p (w) = o(u µ,p (w)), as w 1. ν P

20 Open problem It is currently an open problem to characterize (in terms of function-theoretic properties of ϕ) bounded or compact composition operators on Q p for 0 < p < 1.

21 References A. Aleman, The Multiplication Operator on Hilbert Spaces of Analytic Functions, Habilitation, FernUniversität Hagen, A. Aleman, Hilbert spaces of analytic functions between the Hardy and the Dirichlet space, Proc. Amer. Math. Soc. 115 (1992), A. Aleman and A. Simbotin, Estimates in Möbius invariant spaces of analytic functions, Complex Var. Theory Appl., 49 (2004), J. Arazy, S. D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), R. Aulaskari, J. Xiao and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis, 15 (1995),

22 References G. Bao, N. G. Göğüş and S. Pouliasis,Q p spaces and Dirichlet type spaces, Canad. Math. Bull. 60 (2017), no. 4, G. Bao, N. G. Göğüş and S. Pouliasis, On Dirichlet spaces with a class of superharmonic weights, Canad. J. Math. 70 (2018), no. 4, S. Makhmutov and M. Tjani, Composition operators on some Möbius invariant Banach spaces. Bull. Austral. Math. Soc. 62 (2000), no. 1, S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), J. Xiao, Holomorphic Q Classes, Springer, LNM 1767, Berlin, J. Xiao, Geometric Q p Functions, Birkhäuser Verlag, Basel-Boston-Berlin, 2006.