## Publications with Abstracts

8. Bounds on Integral Means of Bergman Projections and their Derivatives.

We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted \(L^p\) norms of certain derivatives of the original function in the \(\theta\) direction. These results easily imply the well known result that the Bergman projection is bounded from the Sobolev space \(W^{k,p}\) into itself for \(1 < p < \infty\). We also apply our results to derive certain regularity results involving extremal problems in Bergman spaces. Lastly, we construct a function that approaches \(0\) uniformly at the boundary of the unit disc but whose Bergman projection is not in \(H^2\).

We bound integral means of the Bergman projection of a function in terms of integral means of the original function. As an application of these results, we bound certain weighted Bergman space norms of derivatives of Bergman projections in terms of weighted \(L^p\) norms of certain derivatives of the original function in the \(\theta\) direction. These results easily imply the well known result that the Bergman projection is bounded from the Sobolev space \(W^{k,p}\) into itself for \(1 < p < \infty\). We also apply our results to derive certain regularity results involving extremal problems in Bergman spaces. Lastly, we construct a function that approaches \(0\) uniformly at the boundary of the unit disc but whose Bergman projection is not in \(H^2\).

7. Extremal Problems in Bergman Spaces and an Extension of Ryabykh's Hardy Space Regularity Theorem For \(1 < p < \infty\).

We study linear extremal problems in the Bergman space \(A^p\) of the unit disc, where \(1 < p < \infty\). Given a functional on the dual space of \(A^p\) with representing kernel \(k \in A^q\), where \(1/p + 1/q = 1\), we show that if \(q \le q_1 < \infty\) and \(k \in H^{q_1}\), then \(F \in H^{(p-1)q_1}\). This result was previously known only in the case where \(p\) is an even integer. We also discuss related results.

We study linear extremal problems in the Bergman space \(A^p\) of the unit disc, where \(1 < p < \infty\). Given a functional on the dual space of \(A^p\) with representing kernel \(k \in A^q\), where \(1/p + 1/q = 1\), we show that if \(q \le q_1 < \infty\) and \(k \in H^{q_1}\), then \(F \in H^{(p-1)q_1}\). This result was previously known only in the case where \(p\) is an even integer. We also discuss related results.

6. Regularity of Extremal Functions in Weighted Bergman and Fock Type Spaces. Submitted.

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function \(k\), the corresponding extremal function is the function with unit norm maximizing \(\mathrm{Re} \int_\Omega f(z) \overline{k(z)}\, \nu(z) \, dA(z) \) over all functions \(f\) of unit norm, where \( \nu \) is the weight function and \( \Omega \) is the domain of the functions in the space. We consider the case where \( \nu(z)\) is a decreasing radial function satisfying some additional assumptions, and where \( \Omega \) is either a disc centered at the origin or the entire complex plane. We show that if \( k \) grows slowly in a certain sense, then \(f\) must grow slowly in a related sense. We also discuss a relation between the integrability and growth of certain log-convex functions, and apply the result to obtain information about the growth of integral means of extremal functions in Fock type spaces.

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function \(k\), the corresponding extremal function is the function with unit norm maximizing \(\mathrm{Re} \int_\Omega f(z) \overline{k(z)}\, \nu(z) \, dA(z) \) over all functions \(f\) of unit norm, where \( \nu \) is the weight function and \( \Omega \) is the domain of the functions in the space. We consider the case where \( \nu(z)\) is a decreasing radial function satisfying some additional assumptions, and where \( \Omega \) is either a disc centered at the origin or the entire complex plane. We show that if \( k \) grows slowly in a certain sense, then \(f\) must grow slowly in a related sense. We also discuss a relation between the integrability and growth of certain log-convex functions, and apply the result to obtain information about the growth of integral means of extremal functions in Fock type spaces.

5. Solution of extremal problems in Bergman spaces using the Bergman projection. Comput. Methods Funct. Theory. 14 (2014), no. 1, 35--61. DOI: 10.1007/s40315-013-0046-7 Published version from Springer.

In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain values of \(p\).

In this paper we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain values of \(p\).

4. Self-Commutators of Toeplitz Operators and Isoperimetric Inequalities. S. Bell, T. Ferguson and E. Lundberg, . Mathematical Proceedings of the Royal Irish Academy 114A (2014), 115-132; doi:10.3318/PRIA.2014.114.033.

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.

For a hyponormal operator, C. R. Putnam's inequality gives an upper bound on the norm of its self-commutator. In the special case of a Toeplitz operator with analytic symbol in the Smirnov space of a domain, there is also a geometric lower bound shown by D. Khavinson (1985) that when combined with Putnam's inequality implies the classical isoperimetric inequality. For a nontrivial domain, we compare these estimates to exact results. Then we consider such operators acting on the Bergman space of a domain, and we obtain lower bounds that also reflect the geometry of the domain. When combined with Putnam's inequality they give rise to the Faber-Krahn inequality for the fundamental frequency of a domain and the Saint-Venant inequality for the torsional rigidity (but with non-sharp constants). We conjecture an improved version of Putnam's inequality within this restricted setting.

3. Linear Extremal Problems in Bergman Spaces and an Extension of Ryabykh’s Theorem. Illinois J. Math., 55, no. 2, 555-573.

We study linear extremal problems in the Bergman space \( A^p \) of the unit disc for \(p\) an even integer. Given a functional on the dual space of \(A^p\) with representing kernel \(k \in A^q\), where \(1/p + 1/q = 1\), we show that if the Taylor coefficients of \(k\) are sufficiently small, then the extremal function \(F \in H^{\infty}\). We also show that if \(q \le q_1 < \infty\), then \(F \in H^{(p-1)q_1}\) if and only if \(k \in H^{q_1}\). These results extend and provide a partial converse to a theorem of Ryabykh.

We study linear extremal problems in the Bergman space \( A^p \) of the unit disc for \(p\) an even integer. Given a functional on the dual space of \(A^p\) with representing kernel \(k \in A^q\), where \(1/p + 1/q = 1\), we show that if the Taylor coefficients of \(k\) are sufficiently small, then the extremal function \(F \in H^{\infty}\). We also show that if \(q \le q_1 < \infty\), then \(F \in H^{(p-1)q_1}\) if and only if \(k \in H^{q_1}\). These results extend and provide a partial converse to a theorem of Ryabykh.

2. Gangster Operators and Invincibility of Positive Semidefinite Matrices. with Charles Johnson. Linear Algebra Appl. 433 (2010), 2096–2110.

Decrease in absolute value of a symmetrically placed pair of off diagonal entries need not preserve positive definiteness of an \( n \times n\) matrix, \(n \ge 3 \). A gangster operator is one that replaces some such pairs by \(0\)'s. Circumstances in which gangster operators preserve positive definiteness are investigated. Certain general circumstances are given, and graphs that ensure preservation are characterized.

Decrease in absolute value of a symmetrically placed pair of off diagonal entries need not preserve positive definiteness of an \( n \times n\) matrix, \(n \ge 3 \). A gangster operator is one that replaces some such pairs by \(0\)'s. Circumstances in which gangster operators preserve positive definiteness are investigated. Certain general circumstances are given, and graphs that ensure preservation are characterized.

1. Continuity of Extremal Elements in Uniformly Convex Spaces. Proc. Amer. Math. Soc. 137 (2009), no. 8, 2645–2653.

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal functional belongs to the corresponding Hardy space.

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal functional belongs to the corresponding Hardy space.