CLASSICAL SPACES OF HOLOMORPHIC FUNCTIONS MARCO M. PELOSO Contents 1. Hardy Spaces on the Unit Disc 1 1.1. Review from complex analysis 1 1.2. Hardy spaces 5 1.3. Harmonic Hardy classes 7 1.4. Fatou's theorem 9 1.5. The zero sets of functions in Hp 13 1.6. Boundary behaviour of functions in Hardy spaces 16 1.7. The Cauchy–Szeg¨oprojection 19 2. Bergman spaces on the unit disc 22 2.1. Function spaces with reproducing kernel 22 2.2. The Bergman spaces 25 2.3. Biholomorphic invariance 28 2.4. Lp-boundedness of a family of integral operators 29 2.5. The Bergman kernel and projection on the unit disc 33 3. The Paley{Weiner and Bernstein spaces 35 3.1. The Fourier transform 35 3.2. The Paley{Wiener theorems 40 3.3. The Paley{Wiener spaces 45 3.4. The Bernstein spaces 49 4. Function theory on the upper half plane 51 4.1. Hardy spaces on the upper-half plane 51 4.2. Factorization and boundary behaviour of functions in Hp(U). 54 4.3. H2 and the Paley{Wiener theorem revisited 55 4.4. H1, the atomic decomposition and the space of bounded mean oscillations 55 4.5. Weighted Bergman spaces on the upper-half plane 55 4.6. The Paley{Wiener theorem for Bergman spaces 55 5. Further topics 56 5.1. Hardy spaces on tube domains in Cn 56 5.2. Weighted Bergman spaces on the unit ball in Cn 56 5.3. The Cauchy integral along Lipschitz curves 56 5.4. Real variable Hardy spaces 56 5.5. Hankel and Toeplitz operators 56 References 56 Appunti per il corso Argomenti Avanzati di Analisi Complessa per i Corsi di Laurea in Matematica dell'Universit`adi Milano, a.a. 2011/12. { November 12, 2011. SPACES OF HOLOMORPHIC FUNCTIONS 1 In these notes we consider spaces of holomorphic functions defined on domains of the complex plane or, toward the end of the notes, on domains of the n-dimensional complex space Cn. We begin with the Hardy spaces on the unit disc. 1. Hardy Spaces on the Unit Disc 1.1. Review from complex analysis. Let C denote the field of complex numbers z = x + iy, where x; y 2 R are the real and imaginary part, respectively, of z. We will denote by D the unit disc, that is, D = z 2 C : jzj < 1 : Here and in what follows, we will denote by D(z0; r) the disc centered at z0 2 C and radius r > 0; we will sometimes abbreviate Dr to denote the disc D(0; r). If f is holomorphic on D then it can be written as sum of a converging power series +1 X k f(z) = akz : k=0 From Cauchy's formula we know that, if γ is a simple closed curve contained in D and z lies in the interior of γ (which is a well-defined domain by the Jordan curve theorem), then 1 Z f(ζ) f(z) = dζ : 2πi γ ζ − z If the function f is actually holomorphic in a slightly larger disc DR with R > 1, then we may take the unit circonference @D as curve of integration and obtain 1 Z f(ζ) f(z) = dζ ; 2πi @D ζ − z for all z 2 D. One of the main themes of this course is trying to recover the holomorphic function f (in a given class) from its boundary values, once these values are suitably defined. Setting z = reiη, we rewrite the above identity as Z 2π iθ iη 1 iθ e f(re ) = f(e ) iθ iη dθ 2π 0 e − re 1 Z 2π 1 = f(eiθ) dθ i(η−θ) 2π 0 1 − re Z 2π 1 i(η−θ) 1 = f(e ) iθ dθ ; (1.1) 2π 0 1 − re using the periodicity of the exponential. Notice that +1 1 X = rneinθ 1 − reiθ n=0 iθ =: Cr(e ) ; (1.2) iθ where Cr(e ) is the Cauchy kernel (thought of as a convolution kernel, see (1.4) below). 2 M. M. PELOSO Therefore, using the uniform convergence on compact subsets of D of the power series, we have that (1.1) gives the identity +1 X 1 Z 2π f(reiη) = rn f(eiθ)ein(η−θ) dθ 2π n=0 0 +1 X 1 Z 2π = rneinη f(eiθ)e−inθ dθ : (1.3) 2π n=0 0 We recall that the unit circle @D = fζ 2 C : ζ = eiθg inherits the multiplicative structure from C and it is homeomorphic (as a topological group) to the interval [−π; π] (or any interval of length 2π) once we identify the two end points of the interval. This group T = R=2πZ is called the torus. Of course, the identification between T and @D is given by the mapping t 7! eit. Given f; g 2 L1(T) (where, we recall, T is identified with the unit circle) the convolution on T is defined as 1 Z 2π f ∗ g(eiη) = f(eiθ)g(ei(η−θ)) dθ : (1.4) 2π 0 Therefore, the identity (1.1) can be expressed as convolution on the group T as iη iη f(re ) = (f ∗ Cr)(e ) : (1.5) Moreover, in (1.3) we recognize the n-th Fourier coefficient of the restriction of f on the unit circle @D (function that, at this stage, is assumed to be continuous) 1 Z 2π f^(n) = f(eiθ)e−inθ dθ : 2π 0 It is clear that in order to define the Fourier coefficients is sufficient to assume that f is an L1-function on the unit circle. Another integral formula, with its relative integral kernel, that will play a key role in these notes is the Poisson reproducing formula for the unit disc D 1 Z 2π 1 − r2 u(reiη) = u(eiθ) dθ i(η−θ) 2 2π 0 j1 − re j = (Pu)(reiη) ; (1.6) where u is a function harmonic in a domain containing D and 0 < r < 1. The operator P is called the Poisson operator and Pu the Poisson integral of the function u defined on the unit circle. More generally, the solution of the Dirichlet problem on the unit D (see [CA-notes]) shows that if we start with a continuous function g on the unit circle @D, the function u := Pg is harmonic in D, continuous on the closure of the unit disc and coincides with g on the unit circle. It is worth to observe that Pg is well defined for all g 2 L1(T) and that Pg is a harmonic function in D. One of the main questions that we are going to address is if, and in which sense, the function Pg admits boundary values and, in this case, if such values coincide with the function g. In SPACES OF HOLOMORPHIC FUNCTIONS 3 other words, a classical question is whether the Dirichlet problem (for the Laplacian on the unit disc) ( ∆f = 0 on D (1.7) f = g on @D ; where g 2 Lp(@D) is an assigned function, 1 ≤ p < 1, admits a solution and in what sense the solution f admits g as values on the boundary. As in the case of the reproducing formula (1.5) we may think of the Poisson integral as a convolution integral operator on T by writing 1 − r2 P (eiθ) = ; (1.8) r j1 − reiθj2 so that the Poisson integral of a function g 2 L1(T) is defined by the formula 1 Z 2π 1 − r2 (Pg)(reiη) = g(eiθ) dθ = (P ∗ g)(eiη) : (1.9) i(η−θ) 2 r 2π 0 j1 − re j In complex analysis we were used to think of the Poisson kernel, that we denote temporarely iθ iη iθ as P~z(e ) as a function of the variables z = re 2 D and e 2 @D, where 1 − r2 1 − r2 P~ (eiθ) = = (1.10) z jeiθ − reiηj2 j1 − rei(η−θ)j2 1 − r2 = : (1.11) 1 − 2r cos(θ − η) + r2 In the current setting it is more convenient to think of the Poisson kernel as a family of iη functions fPrg defined on the torus T, hence depending on the variable e , and of the Poisson integral of a function g as the convolution of g with Pr, as in (1.9). According to this point of view, the collection of functions fPrg0<r<1 on T is a summability kernel (or a approximation of the identity), A family of functions fΦtgt>0 is called a summability kernel if they satisfy the following properties: Z 2π 1 iθ (1) Φt(e ) dθ = 1 for t > 0; 2π 0 (2) kΦtkL1 ≤ C with C independent of t; Z iθ (3) for every δ > 0, lim jΦt(e )j dθ = 0. t!+1 δ≤|θ|≤π p We recall that if fΦtgt>0 is a summability kernel on T and g 2 L (T), 1 ≤ p < 1, then p Φt ∗ g ! g in the L -norm, and that if g 2 C(T), then Φt ∗ g ! g in the sup-norm. We collect here the basic properties of the Poisson kernel and integral. iη iη iη Proposition 1.1. The function P (re ) = Pr(e ) is harmonic in D as function of z = re and Z 2π 1 iη Pr(e ) dη = 1 2π 0 for all 0 < r < 1. Moreover, as collection fPrg of functions on T we have (i) for 1 ≤ p < 1, kf ∗ PrkLp ≤ kfkLp ; (ii) for 1 ≤ p < 1, limr!1− kf ∗ Pr − fkLp = 0; 4 M.