Titlebook: Harmonic Function Theory; Sheldon Axler,Paul Bourdon,Wade Ramey Textbook 19921st edition Springer Science+Business Media New York 1992 Har

AMOR

玷污

Bounded Harmonic Functions,Liouville’s Theorem in complex analysis states that a bounded holo-morphic function on . is constant. A similar result holds for harmonic functions on ... 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.

brachial-plexus

Positive Harmonic Functions,In Chapter 2 we proved that a bounded harmonic function on .. is constant. We now improve that result.

bizarre

迷住

Harmonic Hardy Spaces,In Chapter 1 we defined the Poisson integral of a function . ∈ . to be the function . defined on . by ..

NEG

悲观

Annular Regions,An . is a set of the form {. ∈ .. : .. < ∈. ∈ < ..}; here .. ∈ [0, ∞) and .. ∈ (0, ∞]. Thus an annular region is the region between two concentric spheres, or is a punctured ball, or is the complement of a closed ball, or is .. {0}.

BAN

Harmonic Functions on Half-Spaces,at on . One advantage of . over . is the dilation-invariance of . We have already put this to good use in the section . in Chapter 2. Some disadvantages: . is not compact and Lebesgue measure on . is not finite.

附录

平

The Dirichlet Problem and Boundary Behavior,techniques we developed for the special domains . and . 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 . that cannot be extended harmonically across any part of ∂Ω.