On Bergman Spaces in Clifford Analysis (original) (raw)

Ann. Acad. Sci. Fenn. Math, 2008

We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Bergman kernels for rectangular domains and multiperiodic functions in Clifford analysis

Mathematical Methods in the Applied Sciences, 2002

In this paper, we consider rectangular domains in real Euclidean spaces of dimension at least 2, where the sides can be ÿnite, semi-inÿnite, or fully inÿnite. The Bergman reproducing kernel for the space of monogenic and square integrable functions on such a domain is obtained in closed form as a ÿnite sum of monogenic multiperiodic functions. The reproducing property leads to an estimate of the ÿrst derivative of the single-periodic cotangent function in terms of the classical real-valued Eisenstein series.

Addendum to the Paper “A note on Weighted bergman Spaces and the cesaro operator”

Nagoya Mathematical Journal, 2005

LetH(Dn)be the space of holomorphic functions on the unit polydisk Dn, and let, wherep, q> 0,α = (α1,…,αn) with αj> -1,j =1,...,n, be the class of all measurable functions f defined on Dnsuch thatwhereMp(f,r)denote thep-integral means of the functionf. Denote the weighted Bergman space on. We provide a characterization for a functionfbeing in. Using the characterization we prove the following result: Letp> 1, then the Cesàro operator is bounded on the space.

Interpolating sequences for weighted Bergman spaces of the ball

The Michigan Mathematical Journal, 1996

Interpolating sequences for weighted Bergman spaces B p α , 0 < p ≤ ∞, α ≥ −1/p are studied. We show that the natural inclusions between B p α for various p and α are also verified by the corresponding spaces of interpolating sequences. We also give conditions (necessary or sufficient) for the B p α -interpolating sequences. These are similar to the known conditions for the spaces H p and A −α , which in our notation correspond respectively to the particular cases α = −1/p and p = ∞.

Sampling measures for Bergman spaces on the unit disk

Mathematische Annalen, 2000

We provide a characterization of the sampling measures for the Bergman spaces. These are the positive measures µ on the unit disk D for which there exists a constant C > 0 such that 1 C |f | p dA ≤ |f | p (1 − |z| 2) 2 dµ ≤ C |f | p dA for all f ∈ A p. These are the continuous analogues of the sets of sampling characterized by K. Seip [13,14] and A. Schuster [12]. Our characterization is in terms of weak* limits of the Moebius transformations of the measure µ, and mimics the notion for sequences that sampling means being uniformly far from zero sets.

Operators on weighted Bergman spaces

Contemporary Mathematics, 2006

Let ρ : (0, 1] → R + be a weight function and let X be a complex Banach space. We denote by A 1,ρ (D) the space of analytic functions in the disc D such that D |f (z)|ρ(1 − |z|)dA(z) < ∞ and by Bloch ρ (X) the space of analytic functions in the disc D with values in X such that sup |z|<1 1−|z| ρ(1−|z|) F (z) < ∞. We prove that, under certain assumptions on the weight, the space of bounded operators L(A 1,ρ (D), X) is isomorphic to Bloch ρ (X) and some applications of this result are presented. Several properties of generalized vector-valued Bloch functions are also considered.

Recent progress and open problems in the Bergman space

The following text is a modified and updated version of the problem collection , which was written in 1993 but became publicly available only in 1995. It was a survey of various open problems; a general survey of the field was provided in [41, 42] in 1998, written in 1995 and 1996, respectively. Since then, a number of new developments have taken place, which in their turn have led to new questions. We feel it is time to update the problem collection.