The Bloch Space of Analytic functions

Integral Equations and Operator Theory, 1993

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch space ~ is the image P(Ll) of space of all continuous functions on the maximal ideal space of H ~176 under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).

Annales Academiae Scientiarum Fennicae Mathematica, 2019

If 0 < p < ∞ and α > −1, the space of Dirichlet type D p α consists of those functions f which are analytic in the unit disc D and have the property that f ′ belongs to the weighted Bergman space A p α. Of special interest are the spaces D p p−1 (0 < p < ∞) and the analytic Besov spaces B p = D p p−2 (1 < p < ∞). Let B denote the Bloch space. It is known that the closure of B p (1 < p < ∞) in the Bloch norm is the little Bloch space B 0. A description of the closure in the Bloch norm of the spaces H p ∩ B has been given recently. Such closures depend on p. In this paper we obtain a characterization of the closure in the Bloch norm of the spaces D p α ∩ B (1 ≤ p < ∞, α > −1). In particular, we prove that for all p ≥ 1 the closure of the space D p p−1 ∩ B coincides with that of H 2 ∩ B. Hence, contrary with what happens with Hardy spaces, these closures are independent of p. We apply these results to study the membership of Blaschke products in the closure in the Bloch norm of the spaces D p α ∩ B.

Mathematische Nachrichten, 2003

Transactions of the American Mathematical Society, 1995

In this paper we prove that, in the unit ball B B of C n {{\mathbf {C}}^n} , a holomorphic function f f is in the Bergman space L a p ( B ) , 0 &gt; p &gt; ∞ L_a^p(B),\;0 &gt; p &gt; \infty , if and only if \[ ∫ B | ∇ ~ f ( z ) | 2 | f ( z ) | p − 2 ( 1 − | z | 2 ) n + 1 d λ ( z ) &gt; ∞ , \int _B {|\tilde \nabla } f(z){|^2}|f(z){|^{p - 2}}{(1 - |z{|^2})^{n + 1}}d\lambda (z) &gt; \infty , \] where ∇ ~ \tilde \nabla and λ \lambda denote the invariant gradient and invariant measure on B B , respectively. Further, we give some characterizations of Bloch functions in the unit ball B B , including an exponential decay characterization of Bloch functions. We also give the analogous results for BMOA ⁡ ( ∂ B ) \operatorname {BMOA} (\partial B) functions in the unit ball.

Studia Mathematica, 2004

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc verifying that sup |z|<1 (1−|z| 2)||f (z)|| < ∞. A sequence (T n) n of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and 1 (Y) if the map ∞ n=0 x n z n → (T n (x n)) n defines a bounded linear operator from Bloch(X) into 1 (Y). It is shown that if X is a Hilbert space then (T n) n is a multiplier from Bloch(X) into 1 (Y) if and only sup k 2 k+1 n=2 k ||T n || 2 < ∞. Several results about Taylor coefficient of vector-valued Bloch functions depending on properties on X, such as Rademacher and Fourier type p, are presented.

Complex Variables and Elliptic Equations, 2017

Let f be a complex-valued harmonic mapping defined in the unit disk D. We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies sup z∈D (1 − |z| 2) |J f (z)| < ∞. This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic ϕ is Bloch if and only if there exists c > 0 and a univalent ψ such that ϕ = c log ψ ′ .

Bulletin of the Australian Mathematical Society, 1999

Bloch space and the little Bloch space are characterised. (2) Ph{z) = f-^-2 dA(w). JD (1-wz) Note that, using (2), we can extend P to a linear operator from V-{D) into A(D). Recall that the Bloch space B consists of the analytic functions / satisfying | 2 | 2) < 00, and the little Bloch space Bo consists of the analytic functions / satisfying The following result is well known:

Journal of Operator Theory, 2014

For 0 < p < ∞ we let D p p−1 denote the space of those functions f which are analytic in the unit disc D and satisfy D (1 − |z|) p−1 |f ′ (z)| p dA(z) < ∞.

Journal of Function Spaces

We study the Banach space BHα (α&gt;0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When…

Pacific Journal of Mathematics, 1987

It is known that for 0 < p < oo the Hardy space H p contains a residual set of functions, each of which has range equal to the whole plane at every boundary point of the unit disk. With quite new general techniques, we are able to show that this result holds for numerous other spaces. The space BMOA of analytic functions of bounded mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet spaces D a f or 0 < a < 1/2 are examples. Our methods involve hyperbolic geometry, cluster set analysis and the "depth" function which we have used previously for determining geometric properties of the image surfaces of functions. Denote by D(a, r) the open disc in C centered at a and of radius r. Denote by D the unit disc D(0,l) and let Δ(a,r) = D Π D{a,r) for a e 3D. Brown and Hansen [4] proved that each Hardy space H p 9

Journal für die reine und angewandte Mathematik (Crelles Journal), 1986

ON BLOCH FUNCTIONS AND GAP SERIES DANIEL GIRELA Kennedy obtained sharp estimates of the growth of the Nevanlinna characteristic of the derivative of a function f analytic and with bounded characteristic in the unit disc. Actually, Kennedy's results are sharp even for VMOA functions. It is well known that any BMOA function is a Bloch function and any VIVIOA function belongs to the little Bloch space. In this paper we study the possibility of extending Kennedy's results to certain classes of Bloch functions. Also, we prove some more general results obtaining sharp comparison results between the integral means Mp (r, f) with T(r, f) for certain classes of functions f analytic in the unit disc .

Collectanea Mathematica, 1997

We introduce the convolution of functions in the vector valued spaces H 1 (L p ) and H 1 (L q ) by means of Young's Theorem, and we use this to show that Bloch functions taking values in certain space of operators define bilinear bounded maps in the product of those spaces for 1 ≤ p, q ≤ 2. As a corollary, we get a Marcinkiewicz-Zygmund type result.

Proceedings of The American Mathematical Society, 2005

A subspace X of the Hardy space H 1 is said to have the f -property if h/I ∈ X whenever h ∈ X and I is an inner function with h/I ∈ H 1 . We let B denote the space of Bloch functions and B 0 the little Bloch space. Anderson proved in 1979 that the space B 0 ∩H ∞ does not have the f -property. However, the question of whether or not B ∩ H p (1 ≤ p < ∞) has the f -property was open. We prove that for every p ∈ [1, ∞) the space B ∩ H p does not have the f -property.

Bulletin of the Australian Mathematical Society, 1989

We will prove local and global Besov-type characterisations for the Bloch space and the little Bloch space. As a special case we obtain that the Bloch space consists of those analytic functions on the unit disc whose restrictions to pseudo-hyperbolic discs (of fixed pseudo-hyperbolic radius) uniformly belong to the Besov space. We also generalise the results to Bloch functions and little Bloch functions on the unit ball in . Finally we discuss the related spaces BMOA and VMOA.

Mathematical Proceedings of the Cambridge Philosophical Society, 2000

A function f , analytic in the unit disc ∆, is said to be a Bloch function if sup z∈∆ (1 − |z| 2 )|f (z)| < ∞. In this paper we study the zero sequences of non-trivial Bloch functions. Among other results we prove that if f is a Bloch function with f (0) 0 and {z k } is the sequence of ordered zeros of f, then N k=1 † D.G. has been supported in part by a grant from 'El Ministerio de Educación y Cultura, Spain' (PB97-1081) and by a grant from 'La Junta de Andalucía'.