Contemporary Mathematics The classical Dirichlet space William T. Ross Dedicated to J. A. Cima on the occasion of his 70th birthday and to over 45 years of dedicated service to mathematics. 1. Introduction In this survey paper, we will present a selection of results concerning the class of analytic functions f on the open unit disk D := fz 2 C : jzj < 1g which have ¯nite Dirichlet integral Z 1 D(f) := jf 0j2 dx dy: ¼ D In particular, we will cover the basic structure of these functions - their boundary values and their zeros - along with two important operators that act on this space of functions - the forward and backward shifts. This survey is by no means complete. For example, we will not cover the Toeplitz or Hankel operators on these functions, nor will we cover the important topic of interpolation. These topics are surveyed in a nice paper of Wu [64]. In order to make this survey more manageable, we will also restrict ourselves to this space of functions with ¯nite Dirichlet integral and will not try to cover the many related Dirichlet-type spaces. We refer the reader to the papers [11, 41, 47, 54] for more on this. 2. Basic de¯nitions An analytic function f on the open unit disk D belongs to the classical Dirichlet space D if it has ¯nite Dirichlet integral Z 1 D(f) := jf 0j2 dA; ¼ D where dA is two dimensional Lebesgue area measure. Thinking of f as a mapping 0 2 from D to some region f(D), one computes the Jacobian determinant Jf to be jf j and so the two dimensional area of f(D), counting multiplicities, is Z Z Z 0 2 1 dA = jJf j dA = jf j dA: f(D) D D 2000 Mathematics Subject Classi¯cation. Primary 31C25; Secondary 30C15. °c 0000 (copyright holder) 1 2 WILLIAM T. ROSS Thus a function has ¯nite Dirichlet integral exactly when its image has ¯nite area (counting multiplicities). Another interesting geometric observation to make is that for each ³ 2 T, the quantity Z 1 (2.1) L(f; ³) := jf 0(r³)j dr 0 is the length of the curve ff(r³) : 0 < r < 1g, which is the image of the ray fr³ : 0 < r < 1g under the mapping f. If D(f) < 1, we can apply the Cauchy- Schwartz inequality to show that Z 2¼ L(f; eiθ) dθ 6 cD(f) < 1 0 and so L(f; eiθ) < 1 for almost every θ. A theorem of Beurling (see Theorem 5.5 below) says that L(f; eiθ) < 1 for quasi-every θ in the sense of logarithmic capacity. Writing X1 n f(z) = anz n=0 as a power series and setting z = reiθ, one can integrate in polar coordinates to see that X1 2 D(f) = njanj : n=0 Closely related to the Dirichlet space is the classical Hardy space H2 of analytic functions f on D for which Z 2 2 kfkH2 := sup jf(r³)j dm(³) < 1: 0<r<1 T Here dm denotes Lebesgue measure on the unit circle T = @D normalized so that m(T) = 1. Observe that Z X1 2 2n 2 jf(r³)j dm(³) = r janj T n=0 and so X1 2 2 kfkH2 = janj : n=0 Thus D ½ H2 and so, via Fatou's theorem on radial limits [22], functions in D have radial boundary values almost everywhere on T, that is to say, (2.2) f(³) := lim f(r³) r!1¡ exists and is ¯nite for almost every ³ 2 T. It will turn out (see Theorem 5.10 and Theorem 5.12) that functions in D have much stronger regularity near T than H2 functions. The quantity D(f) is not a norm, since D(c) = 0 for any constant function c. However, one can endow D with the norm kfk, where X1 2 2 2 (2.3) kfk := kfkH2 + D(f) = (1 + n)janj : n=0 THE CLASSICAL DIRICHLET SPACE 3 An easy estimate yields ¡ ¢ jf(0)j2 + D(f) 6 kfk2 6 2 jf(0)j2 + D(f) and several authors use the quantity p jf(0)j2 + D(f) to de¯ne an equivalent norm on D which is sometimes more convenient to use. With the norm k ¢ k in eq.(2.3) above, one de¯nes an inner product Z Z 1 hf; gi := fg dm + f 0g0 dA: T ¼ D Simple computations show that the quantities Z 1 (zf)0(zg)0 dA ¼ D and X1 (n + 1)anbn; n=0 where the an are the Taylor coe±cients of f and the bn are those for g, are both equal to hf; gi. Another power series computation shows that if we de¯ne kernels kz(w) by 1 1 k (w) := log ; z; w 2 D; z wz 1 ¡ wz then f(z) = hf; kzi: Moreover, by the Cauchy-Schwartz inequality, 1=2 1=2 jf(z)j 6 kfkkkzk = kfkhkz; kzi = kfkkz(z) and so f satis¯es the pointwise estimate µ ¶ 1 1=2 jf(z)j 6 ckfk log : 1 ¡ jzj2 In particular, this pointwise estimate shows that if fn ! f in norm of D, then fn ! f uniformly on compact subsets of D. Hence D is a reproducing kernel Hilbert space of analytic functions on D. Finally, it is clear from the de¯nition of the norm that if X1 n f(z) = anz ; n=0 belongs to D, then ° ° ° N °2 1 °X ° X ° a zn ¡ f° = (n + 1)ja j2 ° n ° n n=0 n=N+1 which goes to zero as N ! 1: Thus the polynomials form a dense subset of D. 4 WILLIAM T. ROSS 3. The Douglas and Carleson formulas In his investigations of the Plateau problem, Jesse Douglas [20] proved the following formula for the Dirichlet integral Z Z ¯ ¯ ¯ ¯2 ¯f(³) ¡ f(»)¯ D(f) = ¯ ¯ dm(³) dm(»); T T ³ ¡ » where we understand that the function ³ 7! f(³) on T is the almost everywhere de¯ned boundary function via radial boundary values eq.(2.2). The inner integral Z ¯ ¯ ¯ ¯2 ¯f(³) ¡ f(»)¯ D³ (f) := ¯ ¯ dm(») T ³ ¡ » is called the local Dirichlet integral and one can use it to make some interesting observations [47]. First notice how Z D³ (f) dm(³) = D(f) T and so for a Dirichlet function, the local Dirichlet integral is ¯nite almost every- where. 2 Proposition 3.1. Suppose ³ 2 T and f 2 H with D³ (f) < 1. Then the oricyclic limit of f exists at ³. Thus any f 2 D has an oricyclic limit almost everywhere. Here the oricyclic approach regions with contact point ³ are 1=2 A2;c(³) := fz 2 D : j³ ¡ zj 6 c(1 ¡ jzj) g; c > 0: For example, A2;2(³) contains the disk with center ³=2 and radius 1=2. We say that f has oricyclic limit L at ³ if lim f(z) = L z!³;z2A2;c(³) for every c > 0. Compare this to the non-tangential approach regions with contact point ³ A1;c(³) := fz 2 D : j³ ¡ zj 6 c(1 ¡ jzj)g; c > 0; which are triangle shaped regions with vertex at ³. We will discuss much stronger results in Section 5. We also mention that whenever D³ (f) < 1, the Fourier series of f converges to f(³). A function f 2 H2 has the standard Nevanlinna factorization [22] (3.2) f = bs¹F; where Y1 a a ¡ z b(z) = zm n n ja j 1 ¡ a z n=1 n n is the Blaschke factor with zeros at z = 0 and (an)n>1 ½ Dnf0g (repeated according to multiplicity), ½ Z ¾ ³ + z s¹(z) = exp ¡ d¹(³) ; T ³ ¡ z is the singular inner factor with positive singular measure ¹ on T (i.e., ¹ ? m), and ½Z ¾ ³ + z F (z) = exp u(³) dm(³) ; u(³) = log jf(³)j; T ³ ¡ z THE CLASSICAL DIRICHLET SPACE 5 is the outer factor. The two inner factors b and s¹ are bounded analytic functions on D with unimodular boundary values almost everywhere. The outer factor F belongs to H2. 2 If f 2 D, then f 2 H and as such has a factorization f = bs¹F as in eq.(3.2). Here, the inner factors b and s¹ do not belong to D unless b is a ¯nite Blaschke product and s¹ ´ 1 (i.e., ¹ ´ 0). However, the outer factor does belong to D (see Corollary 3.5 below). This following theorem of Carleson [14], which has proven to be a quite a workhorse in the subject, computes D(f) in terms of the Nevanlinna factorization. Theorem 3.3 (Carleson). For f = bs¹F 2 D as in eq.(3.2), Z Ã ! Z Z X1 2 ¼D(f) = m + P (³) jf(³)j2 dm(³) + d¹(³)jf(»)j2 dm(») an j³ ¡ »j2 T n=1 T T Z Z (e2u(³) ¡ e2u(»))(u(³) ¡ u(»)) + 2 dm(³) dm(»): T T j³ ¡ »j In the above formula, 1 ¡ jaj2 P (³) = ; a 2 D; ³ 2 T; a j³ ¡ aj2 is the usual Poisson kernel. We also agree to the understanding that if any of the factors in eq.(3.2) are missing, then those corresponding components of the Carleson formula are zero. Carleson's formula has some nice consequences. For example, two applications of Fubini's theorem with Carleson's formula yields: Corollary 3.4. An inner function bs¹ belongs to D if and only if ¹ = 0 and b is a ¯nite Blaschke product. One can also use standard facts about inner functions and the Carleson formula to prove the division property for D (often called the F -property for D).