Conjugate harmonic functions and Clifford algebras

We generalize a Hardy-Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions.     

On meromorphically harmonic starlike functions with respect to symmetric conjugate points

In a previous paper M.P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu, Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25-34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. Then, the same method is employed to functions meromorphic by Z.Z. Zou and Z.-R. Wu [Zhong Zhu Zou, Zhuo-Ren Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 261 (2001) 17-27]. Now, the method can be employed to functions meromorphic harmonic in the punctured disc 0<|z|<1. Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.     

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain we prove that multipole pluricomplex Green functions are dense in the cone in of negative plurisubharmonic functions with zero boundary values.

     

Derivatives of the restrictions of harmonic functions on the Sierpinski gasket to segments

We give an explicit derivative computation for the restriction of a harmonic function on SG to segments at specific points of the segments: The derivative is zero at points dividing the segment in ratio 1:3. This shows that the restriction of a harmonic function to a segment of SG has the following curious property: The restriction has infinite derivatives on a dense subset of the segment (at junction points) and vanishing derivatives on another dense subset.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

Generalized harmonic number summation formulae via hypergeometric series and digamma functions

By computing the derivatives of five classical hypergeometric summation theorems, and applying the related properties of the digamma function, we derive a large number of closed summation formulae for generalized harmonic numbers.     

Approximation by harmonic functions in theC m -Norm and harmonicC m -capacity of compact sets in ℝ n

We study the function Λ ^m (X), 0<m<1, of compact setsX in ℝ ⁿ , n≥2, defined as the distance in the spaceC ^m (X)≡lip ^m(X) from the function |x|² to the subspaceH ^m (X) which is the closure inC ^m (X) of the class of functions harmonic in the neighborhood ofX (each function in its own neighborhood). We prove the equivalence of the conditions Λ ^m (X)=0 andC ^m (X)=H ^m (X). We derive an estimate from above that depends only on the geometrical properties of the setX (on its volume). Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 372–382, September, 1997. Translated by I. P. Zvyagin     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

Some characterizations of

We show that a function on the unit disk extends continuously to , the maximal ideal space of iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.

     

A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.

     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Approximation of Sobolev classes by polynomials and ridge functions

Let be the usual Sobolev class of functions on the unit ball in , and be the subclass of all radial functions in . We show that for the classes and , the orders of best approximation by polynomials in coincide. We also obtain exact orders of best approximation in of the classes by ridge functions and, as an immediate consequence, we obtain the same orders in for the usual Sobolev classes .     

Multiplier family of harmonic univalent functions

The aim of this paper is to study a multiplier family of harmonic univalent functions using the sequences {c_n} and {d_n} of positive real numbers. By specializing {c_n} and {d_n}, the generalized Bernardi–Libera–Livingston integral operator is modified for such functions and the closure of the multiplier family under the modified integral operator is determined. Also, convolution products, closure properties, distortion theorems, convex combinations and neighborhoods for such functions are given.     

Vitali's and Harnack's type results for vector-valued functions

In this brief note, we extend Vitali's theorem for holomorphic functions obtained by Arendt and Nikolski to nets of functions of sheaves of smooth vector-valued functions. As a consequence we also extend a Harnack's theorem for compact operator-valued harmonic functions recently obtained by Enflo and Smithies to bounded operator-valued harmonic functions, avoiding the assumption that the Hilbert space H where the operators are defined is separable.