# Preliminary Materials

• Yoshihiro Tonegawa
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## Abstract

Throughout, 1 ≤ k < n are integers, $$\mathbb {N}$$ is the set of natural numbers, $$\mathbb {R}^n$$ is the n-dimensional Euclidean space and
$$\displaystyle \mathbb {R}^+:=\{x\in \mathbb {R} : x\geq 0\}.$$
Let $$U\subset \mathbb {R}^n$$ be an open set. The symbol
$$\displaystyle C^l_c(U)$$
denotes the set of l times continuously differentiable functions with compact support in U and
$$\displaystyle C_c(U):=C_c^0(U).$$
A function $$f:U\rightarrow \mathbb {R}$$ is said to be Lipschitz if
$$\displaystyle \mathrm {Lip}(f):=\sup _{x,y\in U}\frac {|f(x)-f(y)|}{|x-y|}<\infty .$$
The set of vector fields with each component in $$C^l_c(U)$$ is denoted by
$$\displaystyle C^l_c(U;\mathbb {R}^n)$$
and
$$\displaystyle C_c(U;\mathbb {R}^n):=C_c^0(U;\mathbb {R}^n).$$
For $$g\in C^1_c(U;\mathbb {R}^n)$$, we define ∇g as the n × n matrix-valued function whose first row is composed of partial derivatives of the first component of g and so forth. By ∘, we indicate the usual matrix multiplication.

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