Asymptotic geometric analysis, harmonic analysis and related by McIntosh A., Portal P. (eds.)

By McIntosh A., Portal P. (eds.)

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1) Then the bounded linear operator f (T ) is defined by the Riesz-Dunford formula 1 f (T ) = (λI − T )−1 f (λ) dλ. 2) 2πi C for any function f satisfying the bounds |f (z)| ≤ Kν |z|s , 1 + |z|2s z ∈ Sν◦ (C). The contour C can be taken to be {z ∈ C : | Im(z)| = tan θ| Re(z)| }, with ω < θ < ν. 2) and a positive number Cν such that f (T ) ≤ Cν f ∞ for all f ∈ H ∞ (Sν◦ (C)). 1. Suppose that T is a one-to-one operator of type ω in H.

10) The pullback of ω(dζ) onto M by the embedding of M into Cn+1 is denoted by the same symbol. Let ζ ∈ κ(Ω). 9), because DC f˜C = 0, so that Gz (ζ)ω(dz)f˜C (z) is a closed C (Cn )-valued differential form in κ(Ω) \ N (ζ). The sum A + B of two subsets A, B of a vector space is the set A + B = {a + b : a ∈ A, b ∈ B }. For each r > 0, let Sn (r) = {x ∈ Rn+1 : |x| = r } be the n-dimensional hypersphere of radius r in Rn+1 . The hypersphere Sn (r) is identified with a subset of Cn+1 via the embedding of Rn+1 in Cn+1 .

Re ζn ) with radius | Im ζ| where Im ζ = (Im ζ1 , . . , Im ζn ). The main result of the paper [3] was that the mapping f −→ f˜ from left monogenic functions f uniformly bounded on subsectors of a fixed sector in Rn+1 to holomorphic functions f˜ uniformly bounded on subsectors of a corresponding sector in Cn is actually a bijection. As a consequence, if D Σ is the n-tuple of differentiation operators on a Lipschitz surface Σ in Rn+1 , then the equality f (D Σ ) = f˜(D Σ ) extends to all monogenic functions f uniformly bounded on subsectors of a fixed sector in Rn+1 determined by the tangent hyperplanes of Σ.

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