Multipliers of Dirichlet subspaces of the Bloch space

Ufimskii Matematicheskii Zhurnal

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.

Mathematische Nachrichten, 2003

2016

We define two notions of Logarithmic Bloch space in the polydisc for which we provide equivalent definitions in terms of symbols of bounded Hankel operators. We also provide a full characterization of the pointwise multipliers between two different Bloch spaces of the unit polydisc. n j=1 (1 − w j z j) 2+α j .

2000

Starting from a nondecreasing function K : [0, ∞) → [0, ∞), we introduce a Möbius-invariant Banach space Q K of functions analytic in the unit disk in the plane. We develop a general theory of these spaces, which yields new results and also, for special choices of K, gives most basic properties of Qp-spaces. We have found a general criterion on the kernels K 1 and K 2 , K 1 ≤ K 2 , such that Q K 2 Q K 1 , as well as necessary and sufficient conditions on K so that Q K = B or Q K = D, where the Bloch space B and the Dirichlet space D are the largest, respectively smallest, spaces of Q K -type. We also consider the meromorphic counterpart Q # K of Q K and discuss the differences between Q K -spaces and Q # K -classes.

Bulletin of the Australian Mathematical Society, 1996

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

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 > p > ∞ L_a^p(B),\;0 > p > \infty , if and only if \[ ∫ B | ∇ ~ f ( z ) | 2 | f ( z ) | p − 2 ( 1 − | z | 2 ) n + 1 d λ ( z ) > ∞ , \int _B {|\tilde \nabla } f(z){|^2}|f(z){|^{p - 2}}{(1 - |z{|^2})^{n + 1}}d\lambda (z) > \infty , \] 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.

2012

We answer affirmatively the problem left open in [4, and prove that for a finite Blaschke product φ, the minimal reducing subspaces of the Bergman space multiplier M φ are pairwise orthogonal and their number is equal to the number q of connected components of the Riemann surface of φ −1 • φ. In particular, the double commutant {M φ , M * φ } ′ is abelian of dimension q. An analytic/arithmetic description of the minimal reducing subspaces of M φ is also provided, along with a list of all possible cases in degree of φ equal to eight.

International Journal of Mathematics and Mathematical Sciences, 1990

The radial limits of the weighted derivative of an bounded analytic function is considered.

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.

Proceedings of the Edinburgh Mathematical Society, 1986

We provide in this note a full characterization of the multiplier algebra of the product Bloch space that is the dual of the Bergman space $A^1(\mathbb D^n)$, where $\mathbb D^n$ is the unit. polydisc.

Illinois Journal of Mathematics, 1985

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).

Revista Matemática Iberoamericana, 2000

Motivated by an old paper of Wells [J. London Math. Soc. 2 (1970), 549-556] we define the space X ⊗Y , where X and Y are "homogeneous" Banach spaces of analytic functions on the unit disk D, by the requirement that f can be represented as f = ∞ j=0 gn * hn, with gn ∈ X, hn ∈ Y and ∞ n=1 gn X hn Y < ∞. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula ((X ⊗ Y ), Z) = (X, (Y, Z)), where (U, V ) denotes the space of multipliers from U to V , and as a special case (X ⊗ Y ) * = (X, Y * ), where U * = (U, H ∞ ). We determine H 1 ⊗ X for a class of spaces that contains H p and p (1 ≤ p ≤ 2), and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.