O zonální funkci harmonické
Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian
The relation between the Jacobian and the orders of a linear invariant family of locally univalent harmonic mapping in the plane is studied. The new order (called the strong order) of a linear invariant family is defined and the relations between order and strong order are established.
On a boundary value problem for the heat equation
On a condition of thinness at infinity
On a conjecture of Král concerning the subharmonic extension of continuously differentiable functions
This note verifies a conjecture of Král, that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.
On a construction of Markov processes associated with time dependent Dirichlet spaces.
On a direct decomposition of the space .
On a generalized asymptotic mean value property.
On a generalized heat potential
On a generalized logarithmic kernel and its potentials
On a generalized reflection law for functions satisfying the Helmholtz equation.
On a heat potential
On a heat potential (Preliminary communication)
On a higher-order Hardy inequality
The Hardy inequality with holds for if is an open set with a sufficiently smooth boundary and if . P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for .
On a maximum principle in potential theory
On a potential problem with incident wave as a field source
On a question of T. Sheil-Small regarding valency of harmonic maps
The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form f(eit) = eiϕ(t), 0 ≤ t ≤ 2π, where ϕ is a continuously non-decreasing function that satisfies ϕ(2π)−ϕ(0) = 2Nπ, assume every value finitely many times in the disc?
On a result by Clunie and Sheil-Small
In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F...
On a Singular Value Problem for the Fractional Laplacian on the Exterior of the Unit Ball
2000 Mathematics Subject Classification: Primary 26A33; Secondary 47G20, 31B05We study a singular value problem and the boundary Harnack principle for the fractional Laplacian on the exterior of the unit ball.
On a subclass of harmonic univalent functions.