Abstract
We consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.
Highlights
Let D be the open unit disc in the complex plane C and denote by ∂z = ∂/∂z and ∂ ̄z = ∂/∂zthe usual complex partial derivatives
This work is concerned with second order partial differential operators of the form
We say that a function u is ( p, q)-harmonic if u is twice continuously differentiable in D (in symbols u ∈ C2(D)) and L p,q u = 0 in D, where L p,q is as in (0.1)
Summary
As a consequence we have that u ∈ C∞(D) if u is ( p, q)-harmonic (see Corollary 4.9) This characterization of ( p, q)-harmonic functions improves on a result by Ahern et al [1, Theorem 2.1] when specialized to our setting. |z |2 )zm , z∈D (see Theorem 6.3) This latter series expansion generalizes a well-known partial fraction decomposition formula for the classical Poisson kernel for D which is obtained for p = q = 0. A main contribution of this paper concerns series expansion of ( p, q)-harmonic functions. Apart from its intrinsic interest, this limit theorem provides an efficient tool for the study of limit properties of homogeneous parts of ( p, q)-harmonic functions.
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