Analytic extensions of vector-valued functions

Vector - Valued Weakly Analytic

Vector-valued Optimal Lipschitz Extensions

Consider a bounded open set U ⊂ Rn and a Lipschitz function g : ∂U → Rm. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variant...

Holomorphic vector-valued functions

exists. The function f is continuously differentiable when it is differentiable and f ′ is continuous. A k-times continuously differentiable function is C, and a continuous function is C. A V -valued function f is weakly C when for every λ ∈ V ∗ the scalar-valued function λ◦ f is C. This sense of weak differentiability of a function f does not refer to distributional derivatives, but to differe...

Differentiation of Vector-Valued Functions

This important result is listed in the theorem on the next page. Note that the derivative of the vector-valued function is itself a vector-valued function. You can see from Figure 12.8 that is a vector tangent to the curve given by and pointing in the direction of increasing values. tr t r t r f t i g t j lim t→0 f t t f t t i lim t→0 g t t g t t j lim t→0 f t t f t t i g t t g t t j lim t→0 f ...

Hardy space of operator-valued analytic functions

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of C. In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operat...