Wade Ramey

Harmonic Polynomials and Dirichlet-Type Problems

We take a new approach to harmonic polynomials via differ- entiation. Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x| 2−n and observing the patterns that emerge. This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollary 1.8) and a new proof of the identity in Corollary 1.10. We then discuss a fast algorithm for computing the Poisson integral of any polynomial. (Note: The algorithm involves differentiation, but no integration.) We show how this algorithm can be used for many other Dirichlet-type problems with polynomial data. Finally, we show how Lemma 1.4 leads to the identity in (3.2), yielding a new and simple proof that the Kelvin transform preserves harmonic functions. 1. Derivatives of |x| 2−n Unless otherwise stated, we work in R n ,n >2; the function |x| 2−n is then har- monic and nonconstant on R n \{ 0}. (When n = 2 we need to replace |x| 2−n with log |x|; the minor modifications needed in this case are discussed in Section 4.) Letting Dj denote the partial derivative with respect to the j th coordinate vari- able, we list here some standard differentiation formulas that will be useful later: Dj|x| t = txj|x| t−2 ∆|x| t = t(t + n − 2)|x| t−2

Bloch-to-BMOA pullbacks on the disk

For a given holomorphic self map (P of the unit disk, we consider the Bloch-to-BMOA composition property (pullback property) of (p. Our results are (1) (p cannot have the pullback property if (p touches the boundary too smoothly, (2) while (p has the pullback property if (p touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.

Bôcher's Theorem

The usual proofs of Bochers Theorem rely either on the theory of superharmonic functions ([4], Theorem 5.4) or series expansions using spherical harmonics ([5], Chapter X, Theorem XII). (The referee has called our attention to the proof given by G. E. Raynor [7]. Raynor points out that the original proof of Maxime Bocher [2] implicitly uses some non-obvious properties of the level surfaces of a harmonic function.) In this, paper we offer a different and simpler approach to this theorem. The only results about harmonic functions needed are the minimum principle, Harnacks Inequality, and the solvability of the Dirichlet problem in Bn. We will investigate a harmonic function by studying its dilates. For u a function defined on Bn \ {0} and r E (0, 1), the dilate ur is the function defined on (l/r)Bn \ {0} by

On the behavior of harmonic functions near a boundary point

Several results on the behavior of harmonic functions at an individual boundary point are obtained. The results apply to positive harmonic functions as well as to Poisson integrals of functions in BMO.

Harmonic Bergman Spaces

Throughout this chapter, p denotes a number satisfying 1 ≤ p < ∞. The Bergman space b p (Ω) is the set of harmonic functions u on Ω such that

Harmonic Function Theory

{\left\| u \right\|_p} = {\left( {\int_\Omega {{{\left| u \right|}^p}dV} } \right)^{1/p}} < \infty

HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES

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The pointwise Fatou theorem and its converse for positive pluriharmonic functions

Harmonic functions, for us, live on open subsets of real Euclidean spaces. Throughout this book, n will denote a fixed positive integer greater than 1 and Ω will denote an open, non-empty subset of R n . A twice continuously differentiate, complex-valued function u defined on Ω is harmonic on Ω if