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By Choquet-Bruhat Y., Dewitt-Morette C., M.Dillard-Bleick

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159], and [76, p. 197]. They do use the Cayley transform and the Hp(][J) spaces but with a twist: Hoo (U) = {f : f o r E Hoo (][J)) } , Hp(U) = {(z + i) -2Ip fez) : f o r E Hp (][J))} , 0 < p < 00. The presence of the extra factor (z + i) 2lp in the definition of Hp(U) can be explained as follows. Let us temporarily use an extra argument for a measure to indicate where it is supported. For example, A(e, P) = (27·0 -l e is the normalized Lebesgue measure for the interval P = (-:IT, :IT]. We have identified this previously with the measure A (t, 1I') on the unit circle where t = eiO by ill setting A(e , 1I') = A(e, P).

1 (Schwarz 's lemma). Suppose f E S([J) and f(O) = O. Then I f'(O) 1 :::: 1 and If(z) l :::: Izl , z Equality holds if and only if E [J). (1. 14) f ez) = c z with I cl = 1. Proof. This is a classical result and we are not going to prove it here. See, for example, Ref. D [46, p. 191] or [92, p. 1]. Note that for f E S ([J) \ B ([J) , it can never be true that f(O) = 0; thus the lemma actually gives a statement about functions in B([J) . A Mobius transform is a linear fractional transform that conformally maps the unit circle/disk onto itself.

J eilJIJ + z dJL«(}) , '- e1 - Z C E JR, z E j[J). 27) This function is analytic in j[J) and belongs to Hp for all p < 1 [76, p. e. The constant c is the imaginary part of nfL (O) = 1 + ic. This integral representation of C functions is called the Riesz­ Herglotz representation for functions of class C. Conversely, every C function can be represented in this form. The relation between JL and nfL is one to one except for the constant c. Since JL is uniquely defined by nI" we shall refer to it as the Riesz-Herglotz measure for nIL" Note that the real part of the kernel D (t , z), t E T, z E j[J) is given by P (t, z) = Re D (t, Z) = l - lzl 2 t E T.

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Analysis, manifolds and physics by Choquet-Bruhat Y., Dewitt-Morette C., M.Dillard-Bleick

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