| Abstrakt: |
In this chapter, we begin by proving some of the elementary properties of harmonic functions. A proof of Weyl΄s lemma (Weyl [2]) is inserted in § 2.3 for later reference; the proof is included at that point since it is closely related to the mean value property. Then in § 2.4 the classical notions of Green΄s functions and elementary functions are in?troduced and these notions and Poisson΄s integral formula for the circle and halfplane are carried over to the v-dimensional case. In § 2.5, the study of potential functions is begun and the formulas for and continuity properties of their first derivatives are derived. In §2.6, the formulas for and continuity properties of the first derivatives of certain ˵generalized potential functions″ are studied and the Hölder continuity of the second derivatives of ordinary potentials of Hölder continuous density functions follow from the general results; an example of a continuous density function whose potential is not everywhere of class C2 is given. In § 2.7 we present a proof of the now famous inequalities of Calderon and Zygmund ([1]) and ([2]) for singular integrals; we confine ourselves to reasonably smooth functions in order to retain the essential simplicity of their proofs. The chapter concludes with the proof of the maximum principle for certain second order elliptic equations which was given by E. Hopf ([1]). [ABSTRACT FROM AUTHOR] |
