令 (X, d_X) 和 (Y, d_Y) 為兩個度量空間。設 f 為定義在 X 上的實數值函數. 若存在大於 0
的實數 M，對所有 X 上的 x 和 y 都滿足 |f(x) − f(y)| ≤ Md_X(x, y), 則稱 f 為利普希茨函
數。f 被稱為局部利普希茨函數，表示對所有 X 上的點 p 皆存在ㄧ個 p 的鄰域 U 使得 f 在 U
上為利普希茨函數。我們使用 Lip(X) 和 Lip_{loc}(X) 分別表示所有定義在 X 上的實數值利普
希茨函數和局部利普希茨函數。如果 f 為利普希茨函數或是局部利普希茨函數，則我們定
義 f 的利普希茨常數 L(f) 為
L(f) = sup_{x≠y∈X} (|f(x) − f(y)|)/d_X(x, y) < +∞,
和 f 在點 p 上的局部利普希茨常數 L_p(f) 為
L_p(f) = lim_{ϵ→0} sup_{x≠y∈B(p,ϵ)} (|f(x) − f(y)|)/d_X(x, y) < +∞,
令 T : Lip(X) → Lip(Y )。若 T 對所有 Lip(X) 中的 f，都有 L(Tf) = L(f)，則我們
稱 T 為保持利普希茨常數算子。如果算子 T : Lip(R^n) → Lip(R^n) 定義為 Tf = h · f ◦ φ
且一對一滿射、保持利普希茨常數。則 h 為常數函數，函數值為 h ≡ 1/L(φ) 或 h ≡ −1/L(φ)，而
且 φ(x) = L(φ)Ax − φ(0)，其中 A 為正交矩陣。
我們也研究了保持局部利普希茨常數或是保持點態利普希茨常數的算子，並推廣此結論
到定義在 flat 流形上的利普希茨函數。

Let (X, d_X) and (Y, d_Y) be metric spaces. A function f : X → R is called Lipschitz if
there exists a real number M > 0 such that |f(x) − f(y)| ≤ Md_X(x, y) for all x, y in X,
locally Lipschtiz if for all x in X there exists a neighborhood U of x such that f restricted to
U is Lipschitz. Let Lip(X) denote the space of all Lipschitz functions on X and Lip_{loc}(X)
denote the space of all locally Lipschtiz functions on X. If f is Lipschitz or locally Lipschtiz
on X, then we define the Lipschitz constant L(f) of f by
L(f) = sup_{x≠y∈X} (|f(x) − f(y)|)/d_X(x, y) < +∞,
and locally Lipschtiz constant L_p(f) of f at p by
L_p(f) = lim_{ϵ→0} sup_{x≠y∈B(p,ϵ)} (|f(x) − f(y)|)/d_X(x, y) < +∞,
Let T : Lip(X) → Lip(Y ), we say that T preserves Lipschitz constants if L(Tf) = L(f)
for all f ∈ Lip(X), and show that if T : Lip(R^n) → Lip(R^n) defined by Tf = h · f ◦ φ
is a bijection and preserve Lipschitz constants, then h ≡ 1/L(φ) or h ≡ −1/L(φ) , and φ(x) =L(φ)Ax − φ(0), where A is an orthogonal matrix.
We also study those operators preserving local Lipschitz constants or pointwise Lipschitz
constants and extend our results to the case of Lipschitz functions on flat manifolds.

1 Introduction 1
2 Preliminaries 3
2.1 Lipschitz functions 3
2.2 Linear isometry operators between Lipschitz function spaces 6
2.3 Disjointness preserving operators between Lipschitz function spaces 9
3 Operators preserve suprema of derivatives of differentiable functions 11
3.1 Operators preserving suprema of derivatives 11
3.2 Operators preserving suprema of derivatives of functions in C^1(D) 15
4 Operators preserving Lipschitz constants 18
4.1 Composition operators on Lip(R^n) 18
4.2 Weighted composition operators on Lip(R^n) 33
4.3 Weighted composition operators on Lip(R^n, d_M) 34
4.4 Weighted composition operators on Lip(X, d_X) 37
5 Operators preserving local Lipschitz constants 39
5.1 Weighted composition operators on Liploc(R^n) 39
5.2 Pointwise Lipschitz constants 45
5.3 Riemannian manifold 47
5.4 Flat Manifold 48

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