Further Results

The boundary value result in Theorem 3 in Section 19.2 for the Bochner-Martineili kernel can be strengthened. If we only assume that f is of class C^α for 0 < α < 1 on M, then B([M]^0,1 ∧ f) is of class C^α on D − ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2343.tif"/> and D + ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/in2344.tif"/> . This can be established by adapting the proof of the corresponding result for the Cauchy kernel in Lemma 2 in Section 15.4. Cauchy's integral formula used in the proof of that lemma must be replaced by the equation 1 ∫ ζ ∈ M { B ( ζ , z ) f ( ζ ) } n , n − 1 = − B ( [ M ] 0 , 1 f ) = χ D − f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/eq24_1_1.tif"/>