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Get Advanced calculus: Of real-valued functions of a real PDF

By Hans Sagan

ISBN-10: 0395170907

ISBN-13: 9780395170908

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Extra resources for Advanced calculus: Of real-valued functions of a real variable and vector-valued functions of a vector variable

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Note that when 1 ≤ p < ∞ the space Lp (Bn , dvα ) is a Banach space with the norm p,α . If 0 < p < 1, the space Lp (Bn , dvα ) is a complete metric space with the following distance: ρ(f, g) = f − g p p,α . In the special case when p = 2, L2 (Bn , dvα ) is a Hilbert space whose inner product will be denoted by , α . Regardless of what p is, we are going to call p,α the norm on Lp (Bn , dvα ). The following result shows how fast a function in Apα can grow near the boundary of Bn . 1. Suppose 0 < p < ∞ and α > −1.

Similarly, the mean value property for holomorphic functions gives f (a)p = 0≤r< . f (a + rζ)p dσ(ζ), Sn Taking the modulus on both sides, we obtain |f (a)|p ≤ Sn |f (a + rζ)|p dσ(ζ) for all 0 ≤ r < . 28, the functions log |f (z)| and |f (z)|p are subharmonic in Bn . 8 Interpolation of Banach Spaces Interpolation of Banach spaces is a powerful tool in analysis. In this section we introduce the notion of complex interpolation and present a version of the Marcinkiewicz interpolation theorem. Results in this section will be stated without proof, because they are just tools needed later, and because the techniques required for the proof are much different from those in the rest of the book.

Proof. Fix z ∈ Bn and write ϕz (w) = (ϕ1 (w), · · · , ϕn (w)), w ∈ Bn . By the chain rule, n (∆f )(z) = ∆(f ◦ ϕz )(0) = 4 i,j=1 ∂2f (z) ∂zi ∂z j n k=1 ∂ϕi ∂ϕj (0) (0). ∂zk ∂zk The definition of ϕz shows that it admits the expansion sz ϕz (w) = z − sz w + w, z z + · · · , 1 + sz where sz = that 1 − |z|2 and the omitted terms have w-degree 2 or higher. It follows ∂ϕi sz (0) = −sz δik + z k zi , ∂zk 1 + sz and, after some simplification, n k=1 ∂ϕi ∂ϕj (0) (0) = (1 − |z|2 )(δij − zi z j ). ∂zk ∂zk This completes the proof.

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Advanced calculus: Of real-valued functions of a real variable and vector-valued functions of a vector variable by Hans Sagan


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