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Sharp L log α L inequalities for conjugate functions

Matts Essén, Daniel F. Shea, Charles S. Stanton (2002)

Annales de l’institut Fourier

We give a method for constructing functions φ and ψ for which H ( x , y ) = φ ( x ) - ψ ( y ) has a specified subharmonic minorant h ( x , y ) . By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class L log α L , for - 1 ≤ α < ∞ . In particular, the case α = 1 yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in L log L . We also apply the method to produce a new proof of the...

Smoothness of the Green function for a special domain

Serkan Celik, Alexander Goncharov (2012)

Annales Polonici Mathematici

We consider a compact set K ⊂ ℝ in the form of the union of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions g ℂ ∖ K is estimated. Markov’s constants of the corresponding set are evaluated.

The fall of the doubling condition in Calderón-Zygmund theory.

Joan Verdera (2002)

Publicacions Matemàtiques

The most important results of standard Calderón-Zygmund theory have recently been extended to very general non-homogeneous contexts. In this survey paper we describe these extensions and their striking applications to removability problems for bounded analytic functions. We also discuss some of the techniques that allow us to dispense with the doubling condition in dealing with singular integrals. Special attention is paid to the Cauchy Integral.[Proceedings of the 6th International Conference on...

The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature.

Hany M. Farag (1999)

Publicacions Matemàtiques

Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L2 norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in Rn, even in any Lk norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability...

Voiculescu’s Entropy and Potential Theory

Thomas Bloom (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We give a new proof, relying on polynomial inequalities and some aspects of potential theory, of large deviation results for ensembles of random hermitian matrices.

Wiener's type regularity criteria on the complex plane

Józef Siciak (1997)

Annales Polonici Mathematici

We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that d ( t - 1 E ∩ | z - a | ≤ 1 ) ≥ c o n s t > 0 for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous....

Zero distributions via orthogonality

Laurent Baratchart, Reinhold Küstner, Vilmos Totik (2005)

Annales de l’institut Fourier

We develop a new method to prove asymptotic zero distribution for different kinds of orthogonal polynomials. The method directly uses the orthogonality relations. We illustrate the procedure in four cases: classical orthogonality, non-Hermitian orthogonality, orthogonality in rational approximation of Markov functions and its non- Hermitian variant.

Zeros of Sequences of Partial Sums and Overconvergence

Kovacheva, Ralitza K. (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 30B40, 30B10, 30C15, 31A15.We are concerned with overconvergent power series. The main idea is to relate the distribution of the zeros of subsequences of partial sums and the phenomenon of overconvergence. Sufficient conditions for a power series to be overconvergent in terms of the distribution of the zeros of a subsequence are provided, and results of Jentzsch-Szegö type about the asymptotic distribution of the zeros of overconvergent subsequences are stated....

Currently displaying 81 – 100 of 118