Papers by Daniel Luecking
arXiv (Cornell University), Dec 1, 2014
A sequence which is a finite union of interpolating sequences for H ∞ have turned out to be espec... more A sequence which is a finite union of interpolating sequences for H ∞ have turned out to be especially important in the study of Bergman spaces. The Blaschke products B(z) with such zero sequences have been shown to be exactly those such that the multiplication f → f B defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem. We will use D(z, r) for the pseudohyperbolic disk of radius r centered at z, that is, the ball of radius r < 1 in the pseudohyperbolic metric. Let Z = {z k : k = 1, 2, 3,. .. } be a sequence in D without limit points in D. Define the space of sequences l p Z , 0 < p ≤ ∞, to be all those w = (w k) such that
College Mathematics Journal, Mar 1, 2004
The global treaty-based nuclear order is running out of steam. The problems facing it are progres... more The global treaty-based nuclear order is running out of steam. The problems facing it are progressively building up, while problem-solving is losing momentum. The search for a "golden key" to address disarmament and non-proliferation in a way fit for the 21st century prompts decision-makers to look for novel approaches. NATO needs to actively shape this newly emerging space. Acting today from within a tight policy and institutional "corset", the Alliance should strengthen its non-proliferation and disarmament portfolio, and harness its consultative and coordination strengths for agenda-setting, norm-shaping and awareness-raising within the international community.
Pacific Journal of Mathematics, 1984
We extend the results in [5] to several variables and to larger classes of domains. In particular... more We extend the results in [5] to several variables and to larger classes of domains. In particular it is shown that if B is the unit ball in C" and G C B is measurable then for any/? < 0 (\ffdV* const. [\ffdV, J B J G for all analytic functions/in L P (B) if and only if there exist δ > 0 and 0 < r < 1 such that I G Π Q(B(0, r)) |> δ | Q(B(0, r)) | for all automorphisms Q: B-* B. This is actually done for weighted integrals in more general domains. This is easily seen to be a criterion for the operation /~*/IG *° nave dosed range. In addition, some partial results on the closed range of /-» {f(z n)} in some weighted l p spaces are obtained for sequences {z n } in certain domains.
Quotients of L ∞ by Douglas Algebras and Best Approximation
Transactions of the American Mathematical Society, Apr 1, 1983
Page 1. transactions of the american mathematical society Volume 276, Number 2, April 1983 QUOTIE... more Page 1. transactions of the american mathematical society Volume 276, Number 2, April 1983 QUOTIENTS OF L°° BY DOUGLAS ALGEBRAS AND BEST APPROXIMATION BY DANIEL H. LUECKING AND RAHMAN M. YOUNIS Abstract. ...
First-Order Conformal Invariants
Springer eBooks, 1984
The Theme of this section is the following: Suppose you find yourself on a plane domain, with onl... more The Theme of this section is the following: Suppose you find yourself on a plane domain, with only a restricted logic at your disposal; how closely can you determine which domain you are on—up to conformal equivalence? This leads to a study of a system of conformal invariants, the first-order conformal invariants (FOCI), which are obtained from the elementary properties of the algebra (or ring) of analytic functions on plane domains. Although the formal definition of FOCI is given in the terminology of mathematical logic, these invariants are nonetheless all included within the framework of classical function theory. Each of the FOCI corresponds to an elementary assertion about analytic functions that can be understood without any knowledge of mathematical logic.
Mathematische Annalen, Apr 1, 2000
We provide a characterization of the sampling measures for the Bergman spaces. These are the posi... more 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.
Proceedings of the Edinburgh Mathematical Society, Feb 1, 1986
Illinois Journal of Mathematics, Mar 1, 1981
Forward and Reverse Carleson Inequalities for Functions in Bergman Spaces and Their Derivatives
American Journal of Mathematics, Feb 1, 1985
... 85 Page 2. 86 DANIEL LUECKING THEOREM A (Oleinik-Pavlov-Hastings-Stegenga). Let q 2 p &gt... more ... 85 Page 2. 86 DANIEL LUECKING THEOREM A (Oleinik-Pavlov-Hastings-Stegenga). Let q 2 p &gt; 0, c &gt; -1 and let A be a positive measure on U. In order that there exist a constant C such that ... &lt; I f qdg U Page 8. 92 DANIEL LUECKING Cq If IP(l - Iz1)Ydm) = Cq(const.)qp = C ...
American Mathematical Monthly, Mar 1, 1986
The Hahn-Banach Theorem, and Applications
Springer eBooks, 1984
A major tool in the application of duality results (in any locally convex topological vector spac... more A major tool in the application of duality results (in any locally convex topological vector space) is the Hahn-Banach Theorem. We state here one standard version (there are many equivalent versions) and two important corollaries.
Properties of C(G) and H(G)
Springer eBooks, 1984
With little change we can study functions of several variables. To keep the notation simple, we w... more With little change we can study functions of several variables. To keep the notation simple, we will restrict ourselves to two variables. In that case, G is an open set in ℂ × ℂ = ℂ2. Then C(G) is defined just as in the one-variable case. (The distance in ℂ2 between two points (z,w) and (z’,w’) will be denoted d((z,w), (z’,w’)) = (|z - z’|2 + |w - w’|2)½. We also define H(G) as before, but must first define holomorphic.
Transactions of the American Mathematical Society, Feb 1, 1983
We show that L°°/A is not the dual space of any Banach space when A is a Douglas algebra of a cer... more We show that L°°/A is not the dual space of any Banach space when A is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in L°° has a nonunique best approximation in A. We therefore also show that the Douglas algebra Hx + Lf, when F is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.
Proceedings of the American Mathematical Society, Apr 1, 1983
A method is presented for characterizing Carleson-type measures relative to Bergman spaces. This ... more A method is presented for characterizing Carleson-type measures relative to Bergman spaces. This method applies to the standard weighted and unweighted Bergman spaces on the unit ball in C" to yield simple proofs of the known results. It also extends these results to domains more general than balls Let D be a domain in C and A a space of analytic functions with a norm || /1| defined by some integral or integrals of \f\f being bounded. For example if D is the unit disk in C', A could be an Hp space or a Bergman space. If p is a positive measure on D, a common terminology is to call p an A-Carleson measure if there is a constant C > 0 such that (/j/l'^j '<CH/II. This note will deal only with Bergman spaces Ap(w, D) defined for p > 0 and nonnegative w by Ap(w, D) = {/: / is analytic in D and jD \f\pwdm < +00}. The integral is with respect to Lebesgue 2 «-dimensional volume measure (or area if n = 1). Associate with each z E D an open set E(z) containing 2 with the following properties: (1) E(z) E D and XeoÍÜ) 's measurable in D X D. (2) There is a constant C, > 0 such that I U {E(z): E(z) n E(a) ¥> 0)\<Cx\E(a)\. (3) There is a constant C2 > 0 such that for all a E D, w(z,) < C2w(z2) when z,, z2 £ E(a). (| • I denotes the Lebesgue measure of a set.) Example. Let D be the unit disk in C and let r > 0. Let E(z) be the hyperbolic disk about z with hyperbolic radius r. If w(z) = (1-| z |)a for a >-1 then (l)-(3) are satisfied. Let E2(y) = U [E(z): E(y) D E(z) # 0 }. Lemma 1. Suppose there is a constant C > 0 such that \K2W<7rhr\S i^w< fEAp(w,D), IM*) I JE(z) and suppose ft(E2(z)) < CjEi,z)wdm, z ED. Then fi is an Ap(w)-Carleson measure.
Transactions of the American Mathematical Society, Feb 1, 1994
We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman ... more We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman space of a strongly pseudoconvex domain and show that it is equivalent to BMO plus analytic. (Here we mean the Bergman metric BMO of Berger, Coburn and Zhu.) In the course of the proof we obtain new d-estimates that may be of independent interest. Some applications include a decomposition of BMO similar to the classical L°° + L°° , and two characterizations of the dual of VMO (which is also a predual of BMO). In addition, we obtain some partial results on the boundedness of Hankel operators in L1 norm.
Proceedings of the American Mathematical Society, Mar 1, 1986
A Green potential on the unit disk [\z\ < 1} is a function u(z) of the form 1-wz .(,)-/ log ^d a(... more A Green potential on the unit disk [\z\ < 1} is a function u(z) of the form 1-wz .(,)-/ log ^d a(w). where a is a positive measure such that ((1-\w\) da(w) is finite. In this note I give a necessary and sufficient condition on a relatively closed subset F of the unit disk in order that, for all such u(z), liminf(l-\z\)u(z) = 0. F3:^l The condition is that the hyperbolic capacity of the portion of F in arbitrarily small neighborhoods of 1 is bounded away from zero.
Composition Operators Belonging to the Schatten Ideals
American Journal of Mathematics, Oct 1, 1992
Page 1. COMPOSITION OPERATORS BELONGING TO THE SCHATTEN IDEALS By Daniel H. Luecking1 and Kehe Zh... more Page 1. COMPOSITION OPERATORS BELONGING TO THE SCHATTEN IDEALS By Daniel H. Luecking1 and Kehe Zhu2 1. Introduction. Let D denote the unit disk in the complex plane C. Let H2 denote the usual Hardy space ...
arXiv (Cornell University), Nov 20, 2003
The author showed that a sequence Z in the unit disk is a zero sequence for the Bergman space A p... more The author showed that a sequence Z in the unit disk is a zero sequence for the Bergman space A p if and only if the weighted space L p (e pkZ dA) contains a non-zero (equivalently, zero-free) analytic function, where k Z (z) = a∈Z (1 − |a| 2) 2 |1 −āz| 2 |z| 2 2. Here we show that Z is an interpolating sequence for A p if and only if it is separated in the hyperbolic metric and the∂-equation (1 − |z| 2)∂u = f has a solution u satisfying u p,Z ≤ C f p,Z for every f ∈ L p (e pkZ dA), where f p,Z denotes the L p (e pkZ) norm. This holds for all finite p ≥ 1, but also for p < 1 if L p (e pkZ dA) is suitably modified. In addition, we show how this relates to a recent criterion for weighted∂ estimates by J. Ortega-Cerdà, and to K. Seip's criterion for interpolation.
arXiv (Cornell University), Dec 1, 2014
We extend our development of interpolation schemes in [3] to more general weighted Bergman spaces.
arXiv (Cornell University), May 1, 2014
Most characterizations of interpolating sequences for Bergman spaces include the condition that t... more Most characterizations of interpolating sequences for Bergman spaces include the condition that the sequence be uniformly discrete in the hyperbolic metric. We show that if the notion of interpolation is suitably generalized, two of these characterizations remain valid without that condition. The general interpolation we consider here includes the usual simple interpolation and multiple interpolation as special cases.
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Papers by Daniel Luecking