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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.

We study harmonic Bergman functions on the upper half-space of R n \mathbf {R}^n . Among our main results are: The Bergman projection is bounded for the range 1 > p > ∞ 1> p > \infty ; certain nonorthogonal projections are bounded for the range 1 ≤ p > ∞ 1\leq p > \infty ; the dual space of the Bergman L 1 L^1 -space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range 1 ≤ p > ∞ 1\leq p > \infty ; the Bergman norm is equivalent to a “normal derivative norm” as well as to a “tangential derivative norm”.

The usual proofs of Bocher's 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, Harnack's 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

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

If K ⊂ Ω is compact and u is harmonic on Ω \ K, then u might be badly behaved near both ∂K and ∂Ω; see, for example, Theorem 11.18. In this chapter we will see that u is the sum of two harmonic functions, one extending harmonically across ∂K, the other extending harmonically across ∂Ω. More precisely, u has a decomposition of the form $$u = \nu + w$$ on Ω \ K, where v is harmonic on Ω and w is harmonic on R n \ K. Furthermore, there is a canonical choice for w that makes this decomposition unique.

In this chapter we study harmonic functions defined on the upper half-space H. Harmonic function theory on H has a distinctly different flavor from that on B. One advantage of H over B is the dilation-invariance of H. We have already put this to good use in the section Limits Along Rays in Chapter 2. Some disadvantages: ∂H is not compact and Lebesgue measure on ∂H is not finite.

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 $$\Delta u \equiv 0 $$ where Δ = D 1 2 + ... + D n 2 and D j 2 denotes the second partial derivative with respect to the j th coordinate variable. The operator Δ is called the Laplacian, and the equation Δu = 0 is called Laplace’s equation. We say that a function u defined on a (not necessarily open) set E ⊂ R n is harmonic on E if u can be extended to a function harmonic on an open set containing E.

In Chapter 2 we proved that a bounded harmonic function on R n is constant. We now improve that result.

In Chapter 1 we defined the Poisson integral of a function f ∈ C(S) to be the function P[f] defined on B by $$P\left[ f \right](x) = \int_S {P\left( {x,\zeta } \right)f} \left( \zeta \right)d\sigma \left( \zeta \right)$$ (6.1) .

In this chapter we construct harmonic functions on Ω that behave in a prescribed manner near ∂Ω. Here we are interested in general domains Ω ⊂ Rn; the techniques we developed for the special domains B and H will thus not be available. Most of this chapter will concern the Dirichlet problem. In the last section, however, we will study a different kind of boundary behavior problem—the construction of harmonic functions on Ω that cannot be extended harmonically across any part of ∂Ω.

Liouville’s Theorem in complex analysis states that a bounded holo-morphic function on C is constant. A similar result holds for harmonic functions on R n . The simple proof given below is taken from Edward Nelson’s paper [7], which is one of the rare mathematics papers not containing a single mathematical symbol.

The Kelvin transform performs a role in harmonic function theory analogous to that played by the transformation f(z) ↦ f(1/z) in holomorphic function theory. For example, it transforms a function harmonic inside the unit sphere into a function harmonic outside the sphere. In this chapter, we introduce the Kelvin transform and use it to solve the Dirichlet problem for the exterior of the unit sphere and to obtain a reflection principle for harmonic functions. Later, we will use the Kelvin transform in the study of isolated singularities of harmonic functions.

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

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.