Antonios D. Melas

A lower bound for sums of eigenvalues of the Laplacian

Let λ k (Ω) be the kth Dirichlet eigenvalue of a bounded domain Ω in R n . According to Weyls asymptotic formula we have λ k (Ω) ∼ C n (k/V(Ω)) 2/n . The optimal in view of this asymptotic relation lower estimate for the sums Σ λ j (Ω) has been proven by P.Li and S.T.Yau (Comm. Math. Phys. 88 (1983), 309-318). Here we will improve this estimate by adding to its righthand side a term of the order of k that depends on the ratio of the volume to the moment of inertia of Ω.

On various types of universal taylor series

We denote by SN (f, ζ)(z) the Nth partial sum of the Taylor development of a holomorphic function f on a domain Ω with center ζ ε Ω. Let Ω be a Jordan domain with rectifiable boundary, for example the unit disc. We prove the existence of a holomorphic function f εH(Ω) with the following property: For every compact set with and K c connected and every function h: , continuous on K and holomorphic in K 0, there exists a sequence λ n ε {0,1,2,…}n =1,2,…, such that . We also show that the set of functions fεH(Ω) with the previous property is contained in A ∞(Ω) and is a Gρ-dense subset under the natural topology of A ∞(Ω).

Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov's inequality

For each q < 1 we precisely evaluate the main Bellman functions associated with the behavior of dyadic maximal operators on ℝ n on integrable functions. Actually we do that in the more general setting of tree-like maximal operators. These are related to and refine the corresponding Kolmogorov inequality, which we show is actually sharp. For this we use the effective linearization introduced by the first author in 2005 for such maximal operators on an adequate set of functions.

On the centered Hardy-Littlewood maximal operator

We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant C in the L 1 → weak L1 inequality for this operator. In fact we will show that 11+√61/12 = 1.5675208... ≤ C ≤ 5/3 = 1.66....

On the growth of universal functions

AbstractIn this paper, we study the growth of universal functions (Taylor series) on the unit discD. We describe a class of growth types prohibited for such a function. By investigating the relation between growth and value distribution, we prove that every universal function assumes every complex value, with at most one exception, on sequences inD that approach ϖD rather slowly. We use this to get a large class of equations, including polynomials with Nevanlinna coefficients and equations of iterates, that no universal function satisfies. Finally, we produce a universal function whose growth is bounded by

L^{\lowercase{p}}

\exp (\exp (\frac{M}{{1 - \left| z \right|}}\log \log \frac{4}{{1 - \left| z \right|}})),

A_{1}

, which is close to the rates of growth prohibited.

WEIGHTS IN

The exact best possible range of p is determined such that any dyadic