Extension of semi Lipschitz function on quasi-metric spaces

Costica Mustata
(original), Approximation Theory, asymmetric norm, paper

Abstract


The aim of this note is to prove an extension theorem for semi-Lipschitz real functions defined on quasi-metric spaces, similar to McShane extension theorem for real-valued Lipschitz functions defined on a metric space ([2], [4]).

Authors

Costica Mustata
“Tibeiru Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Paper coordinates

C. Mustăţa, Extension of semi Lipschitz function on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx. 30 (2001) nr. 1, 61-67.

PDF

https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art8/Mustata-pdf

About this paper

Journal

Revue d’Analyse Numer. Theor. Approx.

Publisher Name

Publishing Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art8

Print ISSN

2457-6794

Online ISSN

2501-059X

google scholar link

[1] S. Cobzas and C. Mustata, Norm preserving extension of convex Lipschitz functions,J. Approx. Theory,29(1978), 555–569.

[2] J. Czipserand L. Geher, Extension of functions satisfying a Lipschitz condition, ActaMath. Sci. Hungar.,6(1955), 213–220.
[3] P. Fletcherand W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc.,40(1939),837–842.
[5] C. Mustata, Best approximation and unique extension of Lipschitz functions, J. Ap-prox. Theory,19(1977), 222–230.
[6] S. Romaguera and M. Sanchis,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103(2000), 292–301.
[7] J. H. Wellsand L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975

Paper (preprint) in HTML form

2001-Mustata-EXTENSIONS OF SEMI-LIPSCHITZ FUNCTIONSON QUASI-METRIC SPACES

EXTENSIONS OF SEMI-LIPSCHITZ FUNCTIONS ON QUASI-METRIC SPACES

COSTICĂ MUSTĂŢADedicated to the memory of Acad. Tiberiu Popoviciu

Abstract

The aim of this note is to prove an extension theorem for semiLipschitz real functions defined on quasi-metric spaces, similar to McShane extension theorem for real-valued Lipschitz functions defined on a metric space ([2], [4]).

MSC 2000. 46A22, 26A16, 26A48.

1. INTRODUCTION

Let X X XXX be a nonvoid set. A quasi-metric on X X XXX is a function d : X × X → [ 0 , ∞ ) d : X × X → [ 0 , ∞ ) d:X xx X rarr[0,oo)d: X \times X \rightarrow [0, \infty)d:X×X→[0,∞) satisfying the conditions
(i) d ( x , y ) = d ( y , x ) = 0 ⟺ x = y ; x , y ∈ X , (i) d ( x , y ) = d ( y , x ) = 0 ⟺ x = y ; x , y ∈ X , {:(i)d(x","y)=d(y","x)=0Longleftrightarrow x=y;quad x","y in X",":}\begin{equation*} d(x, y)=d(y, x)=0 \Longleftrightarrow x=y ; \quad x, y \in X, \tag{i} \end{equation*}(i)d(x,y)=d(y,x)=0⟺x=y;x,y∈X,
(ii)
d ( x , y ) ≤ d ( x , z ) + d ( z , y ) , x , y , z ∈ X . d ( x , y ) ≤ d ( x , z ) + d ( z , y ) , x , y , z ∈ X . d(x,y) <= d(x,z)+d(z,y),quad x,y,z in X.d(x, y) \leq d(x, z)+d(z, y), \quad x, y, z \in X .d(x,y)≤d(x,z)+d(z,y),x,y,z∈X.
If d d ddd is a quasi-metric on X X XXX, then the pair ( X , d X , d X,dX, dX,d ) is called a quasi-metric space.
The conjugate of quasi-metric d d ddd, denoted by d − 1 d − 1 d^(-1)d^{-1}d−1 is defined by d − 1 ( x , y ) = d ( y , x ) , x , y ∈ X d − 1 ( x , y ) = d ( y , x ) , x , y ∈ X d^(-1)(x,y)=d(y,x),x,y in Xd^{-1}(x, y)= d(y, x), x, y \in Xd−1(x,y)=d(y,x),x,y∈X.
Obviously the function d s : X × X → [ 0 , ∞ ) d s : X × X → [ 0 , ∞ ) d^(s):X xx X rarr[0,oo)d^{s}: X \times X \rightarrow[0, \infty)ds:X×X→[0,∞) defined by
d s ( x , y ) = max { d ( x , y ) , d − 1 ( x , y ) } ; x , y ∈ X d s ( x , y ) = max d ( x , y ) , d − 1 ( x , y ) ; x , y ∈ X d^(s)(x,y)=max{d(x,y),d^(-1)(x,y)};quad x,y in Xd^{s}(x, y)=\max \left\{d(x, y), d^{-1}(x, y)\right\} ; \quad x, y \in Xds(x,y)=max{d(x,y),d−1(x,y)};x,y∈X
is a metric on X X XXX.
If the quasi-metric d d ddd can take the value + ∞ + ∞ +oo+\infty+∞, then it is called an extended quasi-metric.
Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space. A function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is called semiLipschitz if there exists a constant K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 so that
(1) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) (1) f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) {:(1)f(x)-f(y) <= K*d(x","y):}\begin{equation*} f(x)-f(y) \leq K \cdot d(x, y) \tag{1} \end{equation*}(1)f(x)−f(y)≤K⋅d(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X. The number K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 in (1) is called a semi-Lipschitz constant for f f fff.
For a quasi-metric space ( X , d X , d X,dX, dX,d ) the real-valued function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is said to be ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing if
(2) d ( x , y ) = 0 implies f ( x ) − f ( y ) ≤ 0 , x , y ∈ X (2) d ( x , y ) = 0  implies  f ( x ) − f ( y ) ≤ 0 , x , y ∈ X {:(2)d(x","y)=0quad" implies "quad f(x)-f(y) <= 0","quad x","y in X:}\begin{equation*} d(x, y)=0 \quad \text { implies } \quad f(x)-f(y) \leq 0, \quad x, y \in X \tag{2} \end{equation*}(2)d(x,y)=0 implies f(x)−f(y)≤0,x,y∈X
or equivalently,
(3) f ( x ) − f ( y ) > 0 implies d ( x , y ) > 0 , x , y ∈ X . (3) f ( x ) − f ( y ) > 0  implies  d ( x , y ) > 0 , x , y ∈ X . {:(3)f(x)-f(y) > 0quad" implies "quad d(x","y) > 0","quad x","y in X.:}\begin{equation*} f(x)-f(y)>0 \quad \text { implies } \quad d(x, y)>0, \quad x, y \in X . \tag{3} \end{equation*}(3)f(x)−f(y)>0 implies d(x,y)>0,x,y∈X.
Note that every semi-Lipschitz function on quasi-metric space ( X , d ) ( X , d ) (X,d)(X, d)(X,d) is ≤ d − ≤ d − <= _(d^(-))\leq_{d^{-}}≤d− increasing (see (1)).
For a semi-Lipschitz function f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R, where ( X , d X , d X,dX, dX,d ) is a quasi-metric space, denote by ‖ f ‖ d ‖ f ‖ d ||f||_(d)\|f\|_{d}‖f‖d the constant:
(4) ‖ f ‖ d = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 , x , y ∈ X } (4) ‖ f ‖ d = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : d ( x , y ) > 0 , x , y ∈ X {:(4)||f||_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):d(x,y) > 0,quad x,y in X}:}\begin{equation*} \|f\|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: d(x, y)>0, \quad x, y \in X\right\} \tag{4} \end{equation*}(4)‖f‖d=sup{(f(x)−f(y))∨0d(x,y):d(x,y)>0,x,y∈X}
Theorem 1. Let ( X , d X , d X,dX, dX,d ) a quasi-metric space and f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R a semiLipschitz function. Then ‖ f ‖ d ‖ f ‖ d ||f||_(d)\|f\|_{d}‖f‖d defined by (4) is the smallest semi-Lipschitz constant for f f fff.
Proof. If f : X → R f : X → R f:X rarrRf: X \rightarrow \mathbb{R}f:X→R is semi-Lipschitz, then f f fff is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing, and then f ( x ) − f ( y ) > 0 f ( x ) − f ( y ) > 0 f(x)-f(y) > 0f(x)-f(y)>0f(x)−f(y)>0 implies d ( x , y ) > 0 d ( x , y ) > 0 d(x,y) > 0d(x, y)>0d(x,y)>0. It follows that
( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = f ( x ) − f ( y ) d ( x , y ) > 0 ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = f ( x ) − f ( y ) d ( x , y ) > 0 ((f(x)-f(y))vv0)/(d(x,y))=(f(x)-f(y))/(d(x,y)) > 0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=\frac{f(x)-f(y)}{d(x, y)}>0(f(x)−f(y))∨0d(x,y)=f(x)−f(y)d(x,y)>0
The inequalities f ( x ) − f ( y ) ≤ 0 f ( x ) − f ( y ) ≤ 0 f(x)-f(y) <= 0f(x)-f(y) \leq 0f(x)−f(y)≤0 and d ( x , y ) > 0 d ( x , y ) > 0 d(x,y) > 0d(x, y)>0d(x,y)>0 imply
( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = 0 ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = 0 ((f(x)-f(y))vv0)/(d(x,y))=0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=0(f(x)−f(y))∨0d(x,y)=0
Consequently ‖ f ‖ d ≥ 0 ‖ f ‖ d ≥ 0 ||f||_(d) >= 0\|f\|_{d} \geq 0‖f‖d≥0.
For f ( x ) − f ( y ) < 0 f ( x ) − f ( y ) < 0 f(x)-f(y) < 0f(x)-f(y)<0f(x)−f(y)<0 it follows ( f ( x ) − f ( y ) ) / d ( x , y ) ≤ ‖ f ‖ d ( f ( x ) − f ( y ) ) / d ( x , y ) ≤ ‖ f ‖ d (f(x)-f(y))//d(x,y) <= ||f||_(d)(f(x)-f(y)) / d(x, y) \leq\|f\|_{d}(f(x)−f(y))/d(x,y)≤‖f‖d and obviously for f ( x ) − f ( y ) ≤ 0 f ( x ) − f ( y ) ≤ 0 f(x)-f(y) <= 0f(x)-f(y) \leq 0f(x)−f(y)≤0 we have f ( x ) − f ( y ) ≤ 0 ≤ ‖ f ‖ d ⋅ d ( x , y ) f ( x ) − f ( y ) ≤ 0 ≤ ‖ f ‖ d ⋅ d ( x , y ) f(x)-f(y) <= 0 <= ||f||_(d)*d(x,y)f(x)-f(y) \leq 0 \leq\|f\|_{d} \cdot d(x, y)f(x)−f(y)≤0≤‖f‖d⋅d(x,y).
Consequently
f ( x ) − f ( y ) ≤ ‖ f ‖ d ⋅ d ( x , y ) f ( x ) − f ( y ) ≤ ‖ f ‖ d ⋅ d ( x , y ) f(x)-f(y) <= ||f||_(d)*d(x,y)f(x)-f(y) \leq\|f\|_{d} \cdot d(x, y)f(x)−f(y)≤‖f‖d⋅d(x,y)
for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X.
Now let K ≥ 0 K ≥ 0 K >= 0K \geq 0K≥0 such that
f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) , for all x , y ∈ X . f ( x ) − f ( y ) ≤ K ⋅ d ( x , y ) ,  for all  x , y ∈ X . f(x)-f(y) <= K*d(x,y),quad" for all "x,y in X.f(x)-f(y) \leq K \cdot d(x, y), \quad \text { for all } x, y \in X .f(x)−f(y)≤K⋅d(x,y), for all x,y∈X.
The function f f fff is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing, and then
( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = { f ( x ) − f ( y ) d ( x , y ) ≤ K , if f ( x ) − f ( y ) > 0 , 0 ≤ K , if f ( x ) − f ( y ) ≤ 0 , ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) = f ( x ) − f ( y ) d ( x , y ) ≤ K ,       if  f ( x ) − f ( y ) > 0 , 0 ≤ K ,       if  f ( x ) − f ( y ) ≤ 0 , ((f(x)-f(y))vv0)/(d(x,y))={[(f(x)-f(y))/(d(x,y)) <= K","," if "f(x)-f(y) > 0","],[0 <= K","," if "f(x)-f(y) <= 0","]:}\frac{(f(x)-f(y)) \vee 0}{d(x, y)}= \begin{cases}\frac{f(x)-f(y)}{d(x, y)} \leq K, & \text { if } f(x)-f(y)>0, \\ 0 \leq K, & \text { if } f(x)-f(y) \leq 0,\end{cases}(f(x)−f(y))∨0d(x,y)={f(x)−f(y)d(x,y)≤K, if f(x)−f(y)>0,0≤K, if f(x)−f(y)≤0,
Consequently ‖ f ‖ d ≤ K ‖ f ‖ d ≤ K ||f||_(d) <= K\|f\|_{d} \leq K‖f‖d≤K.
For a quasi-metric ( X , d X , d X,dX, dX,d ) let us consider the set:
(5) S Lip X = { f : X → R ∣ f is ≤ d -increasing, sup d ( x , y ) ≠ 0 ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) < ∞ } . (5) S Lip X = f : X → R ∣ f  is  ≤ d -increasing,  sup d ( x , y ) ≠ 0   ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) < ∞ . {:(5)S Lip X={f:X rarrR∣f" is " <= _(d)"-increasing, "s u p_(d(x,y)!=0)((f(x)-f(y))vv0)/(d(x,y)) < oo}.:}\begin{equation*} S \operatorname{Lip} X=\left\{f: X \rightarrow \mathbb{R} \mid f \text { is } \leq_{d} \text {-increasing, } \sup _{d(x, y) \neq 0} \frac{(f(x)-f(y)) \vee 0}{d(x, y)}<\infty\right\} . \tag{5} \end{equation*}(5)SLipX={f:X→R∣f is ≤d-increasing, supd(x,y)≠0(f(x)−f(y))∨0d(x,y)<∞}.
It is straightforward to see that S S SSS Lip X X XXX is exactly the set of all semiLipschitz functions on ( X , d ) ( X , d ) (X,d)(X, d)(X,d) (see [6]).

2. EXTENSIONS OF SEMI-LIPSCHITZ FUNCTIONS

Let Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X where ( X , d X , d X,dX, dX,d ) is a quasi-metric space. Then ( Y , d Y , d Y,dY, dY,d ) is a quasimetric space with the quasi-metric induced by d d ddd (denoted by d d ddd too). Let us denote by S S SSS Lip Y Y YYY the set of all semi-Lipschitz functions defined on Y Y YYY and let
(6) ‖ f ‖ d = sup { ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ Y , d ( x , y ) ≠ 0 } (6) ‖ f ‖ d = sup ( f ( x ) − f ( y ) ) ∨ 0 d ( x , y ) : x , y ∈ Y , d ( x , y ) ≠ 0 {:(6)||f||_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y)!=0}:}\begin{equation*} \|f\|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y) \neq 0\right\} \tag{6} \end{equation*}(6)‖f‖d=sup{(f(x)−f(y))∨0d(x,y):x,y∈Y,d(x,y)≠0}
be the smallest semi-Lipschitz constant for f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY.
If f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY, a function F ∈ S F ∈ S F in SF \in SF∈S Lip X X XXX is called an extension (preserving the smallest semi-Lipschitz constant) of f f fff if:
(7) F | Y = f and ‖ F ‖ d = ‖ f ‖ d . (7) F Y = f  and  ‖ F ‖ d = ‖ f ‖ d . {:(7)F|_(Y)=f quad" and "quad||F||_(d)=||f||_(d).:}\begin{equation*} \left.F\right|_{Y}=f \quad \text { and } \quad\|F\|_{d}=\|f\|_{d} . \tag{7} \end{equation*}(7)F|Y=f and ‖F‖d=‖f‖d.
Denote by E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) the set of all extensions of the function f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY, i.e.
(8) E Y ( f ) = { F ∈ S Lip X : F | Y = f and ‖ F ‖ d = ‖ f ‖ d } (8) E Y ( f ) = F ∈ S Lip X : F Y = f  and  ‖ F ‖ d = ‖ f ‖ d {:(8)E_(Y)(f)={F in S Lip X:F|_(Y)=f" and "||F||_(d)=||f||_(d)}:}\begin{equation*} E_{Y}(f)=\left\{F \in S \operatorname{Lip} X:\left.F\right|_{Y}=f \text { and }\|F\|_{d}=\|f\|_{d}\right\} \tag{8} \end{equation*}(8)EY(f)={F∈SLipX:F|Y=f and ‖F‖d=‖f‖d}
Theorem 2. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space and Y Y YYY a nonvoid subset of X X XXX. Then for every f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY the set E Y ( f ) E Y ( f ) E_(Y)(f)E_{Y}(f)EY(f) is nonvoid.
Proof. Let f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY and the constant ‖ f ‖ d ‖ f ‖ d ||f||_(d)\|f\|_{d}‖f‖d defined by (6).
Consider the function
(9) F ( x ) = inf y ∈ Y { f ( y ) + ‖ f ‖ d d ( x , y ) } , x ∈ X (9) F ( x ) = inf y ∈ Y   f ( y ) + ‖ f ‖ d d ( x , y ) , x ∈ X {:(9)F(x)=i n f_(y in Y){f(y)+||f||_(d)d(x,y)}","x in X:}\begin{equation*} F(x)=\inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}, x \in X \tag{9} \end{equation*}(9)F(x)=infy∈Y{f(y)+‖f‖dd(x,y)},x∈X
a) First we show that F F FFF is well defined.
Let z ∈ Y z ∈ Y z in Yz \in Yz∈Y and x ∈ X x ∈ X x in Xx \in Xx∈X. For any y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
f ( y ) + ‖ f ‖ d d ( x , y ) = f ( z ) + ‖ f ‖ d d ( x , y ) − ( f ( z ) − f ( y ) ) ≥ f ( z ) + ‖ f ‖ d d ( x , y ) − ‖ f ‖ d d ( z , y ) = f ( z ) − ‖ f ‖ d ( d ( z , y ) − d ( x , y ) ) f ( y ) + ‖ f ‖ d d ( x , y ) = f ( z ) + ‖ f ‖ d d ( x , y ) − ( f ( z ) − f ( y ) ) ≥ f ( z ) + ‖ f ‖ d d ( x , y ) − ‖ f ‖ d d ( z , y ) = f ( z ) − ‖ f ‖ d ( d ( z , y ) − d ( x , y ) ) {:[f(y)+||f||_(d)d(x","y)=f(z)+||f||_(d)d(x","y)-(f(z)-f(y))],[ >= f(z)+||f||_(d)d(x","y)-||f||_(d)d(z","y)],[=f(z)-||f||_(d)(d(z","y)-d(x","y))]:}\begin{aligned} f(y)+\|f\|_{d} d(x, y) & =f(z)+\|f\|_{d} d(x, y)-(f(z)-f(y)) \\ & \geq f(z)+\|f\|_{d} d(x, y)-\|f\|_{d} d(z, y) \\ & =f(z)-\|f\|_{d}(d(z, y)-d(x, y)) \end{aligned}f(y)+‖f‖dd(x,y)=f(z)+‖f‖dd(x,y)−(f(z)−f(y))≥f(z)+‖f‖dd(x,y)−‖f‖dd(z,y)=f(z)−‖f‖d(d(z,y)−d(x,y))
The inequality d ( z , y ) − d ( x , y ) ≤ d ( z , x ) = d − 1 ( x , z ) d ( z , y ) − d ( x , y ) ≤ d ( z , x ) = d − 1 ( x , z ) d(z,y)-d(x,y) <= d(z,x)=d^(-1)(x,z)d(z, y)-d(x, y) \leq d(z, x)=d^{-1}(x, z)d(z,y)−d(x,y)≤d(z,x)=d−1(x,z) implies
(10) f ( y ) + ‖ f ‖ d d ( x , y ) ≥ f ( z ) − ‖ f ‖ d ⋅ d − 1 ( x , z ) (10) f ( y ) + ‖ f ‖ d d ( x , y ) ≥ f ( z ) − ‖ f ‖ d ⋅ d − 1 ( x , z ) {:(10)f(y)+||f||_(d)d(x","y) >= f(z)-||f||_(d)*d^(-1)(x","z):}\begin{equation*} f(y)+\|f\|_{d} d(x, y) \geq f(z)-\|f\|_{d} \cdot d^{-1}(x, z) \tag{10} \end{equation*}(10)f(y)+‖f‖dd(x,y)≥f(z)−‖f‖d⋅d−1(x,z)
showing that for every x ∈ X x ∈ X x in Xx \in Xx∈X the set { f ( y ) + ‖ f ‖ d d ( x , y ) : y ∈ Y } f ( y ) + ‖ f ‖ d d ( x , y ) : y ∈ Y {f(y)+||f||_(d)d(x,y):y in Y}\left\{f(y)+\|f\|_{d} d(x, y): y \in Y\right\}{f(y)+‖f‖dd(x,y):y∈Y} is bounded from above by f ( z ) − ‖ f ‖ d d − 1 ( x , z ) f ( z ) − ‖ f ‖ d d − 1 ( x , z ) f(z)-||f||_(d)d^(-1)(x,z)f(z)-\|f\|_{d} d^{-1}(x, z)f(z)−‖f‖dd−1(x,z), and the infimum (9) is finite.
b) We show now that F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y) for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y.
Let y ∈ Y y ∈ Y y in Yy \in Yy∈Y. Then
F ( y ) ≤ f ( y ) + ‖ f ‖ d d ( y , y ) = f ( y ) . F ( y ) ≤ f ( y ) + ‖ f ‖ d d ( y , y ) = f ( y ) . F(y) <= f(y)+||f||_(d)d(y,y)=f(y).F(y) \leq f(y)+\|f\|_{d} d(y, y)=f(y) .F(y)≤f(y)+‖f‖dd(y,y)=f(y).
For any v ∈ Y v ∈ Y v in Yv \in Yv∈Y we have
f ( y ) − f ( v ) ≤ ‖ f ‖ d ⋅ d ( y , v ) f ( y ) − f ( v ) ≤ ‖ f ‖ d ⋅ d ( y , v ) f(y)-f(v) <= ||f||_(d)*d(y,v)f(y)-f(v) \leq\|f\|_{d} \cdot d(y, v)f(y)−f(v)≤‖f‖d⋅d(y,v)
so that
f ( v ) + ‖ f ‖ d ⋅ d ( y , v ) ≥ f ( y ) f ( v ) + ‖ f ‖ d ⋅ d ( y , v ) ≥ f ( y ) f(v)+||f||_(d)*d(y,v) >= f(y)f(v)+\|f\|_{d} \cdot d(y, v) \geq f(y)f(v)+‖f‖d⋅d(y,v)≥f(y)
and
F ( y ) = inf { f ( v ) + ‖ f ‖ d d ( y , v ) : v ∈ Y } ≥ f ( y ) F ( y ) = inf f ( v ) + ‖ f ‖ d d ( y , v ) : v ∈ Y ≥ f ( y ) F(y)=i n f{f(v)+||f||_(d)d(y,v):v in Y} >= f(y)F(y)=\inf \left\{f(v)+\|f\|_{d} d(y, v): v \in Y\right\} \geq f(y)F(y)=inf{f(v)+‖f‖dd(y,v):v∈Y}≥f(y)
It follows F ( y ) = f ( y ) F ( y ) = f ( y ) F(y)=f(y)F(y)=f(y)F(y)=f(y).
c) We prove that ‖ F ‖ d = ‖ f ‖ d ‖ F ‖ d = ‖ f ‖ d ||F||_(d)=||f||_(d)\|F\|_{d}=\|f\|_{d}‖F‖d=‖f‖d.
Since F | Y = f F Y = f F|_(Y)=f\left.F\right|_{Y}=fF|Y=f, the definitions of ‖ F ‖ d ‖ F ‖ d ||F||_(d)\|F\|_{d}‖F‖d and ‖ f ‖ d ‖ f ‖ d ||f||_(d)\|f\|_{d}‖f‖d yield ‖ F ‖ d ≥ ‖ f ‖ d ‖ F ‖ d ≥ ‖ f ‖ d ||F||_(d) >= ||f||_(d)\|F\|_{d} \geq\|f\|_{d}‖F‖d≥‖f‖d.
Let x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Choosing y ∈ Y y ∈ Y y in Yy \in Yy∈Y such that
F ( x 1 ) ≥ f ( y ) + ‖ f ‖ d d ( x 1 , y ) − ε F x 1 ≥ f ( y ) + ‖ f ‖ d d x 1 , y − ε F(x_(1)) >= f(y)+||f||_(d)d(x_(1),y)-epsiF\left(x_{1}\right) \geq f(y)+\|f\|_{d} d\left(x_{1}, y\right)-\varepsilonF(x1)≥f(y)+‖f‖dd(x1,y)−ε
we obtain
F ( x 2 ) − F ( x 1 ) ≤ f ( y ) + ‖ f ‖ d d ( x 2 , y ) − ( f ( y ) + ‖ f ‖ d ⋅ d ( x 1 , y ) − ε ) = ‖ f ‖ d [ d ( x 2 , y ) − d ( x 1 , y ) ] + ε ≤ ‖ f ‖ d ⋅ d ( x 2 , x 1 ) + ε . F x 2 − F x 1 ≤ f ( y ) + ‖ f ‖ d d x 2 , y − f ( y ) + ‖ f ‖ d ⋅ d x 1 , y − ε = ‖ f ‖ d d x 2 , y − d x 1 , y + ε ≤ ‖ f ‖ d ⋅ d x 2 , x 1 + ε . {:[F(x_(2))-F(x_(1)) <= f(y)+||f||_(d)d(x_(2),y)-(f(y)+||f||_(d)*d(x_(1),y)-epsi)],[=||f||_(d)[d(x_(2),y)-d(x_(1),y)]+epsi],[ <= ||f||_(d)*d(x_(2),x_(1))+epsi.]:}\begin{aligned} F\left(x_{2}\right)-F\left(x_{1}\right) & \leq f(y)+\|f\|_{d} d\left(x_{2}, y\right)-\left(f(y)+\|f\|_{d} \cdot d\left(x_{1}, y\right)-\varepsilon\right) \\ & =\|f\|_{d}\left[d\left(x_{2}, y\right)-d\left(x_{1}, y\right)\right]+\varepsilon \\ & \leq\|f\|_{d} \cdot d\left(x_{2}, x_{1}\right)+\varepsilon . \end{aligned}F(x2)−F(x1)≤f(y)+‖f‖dd(x2,y)−(f(y)+‖f‖d⋅d(x1,y)−ε)=‖f‖d[d(x2,y)−d(x1,y)]+ε≤‖f‖d⋅d(x2,x1)+ε.
Since ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 is arbitrary, it follows
F ( x 2 ) − F ( x 1 ) ≤ ‖ f ‖ d ⋅ d ( x 2 , x 1 ) F x 2 − F x 1 ≤ ‖ f ‖ d ⋅ d x 2 , x 1 F(x_(2))-F(x_(1)) <= ||f||_(d)*d(x_(2),x_(1))F\left(x_{2}\right)-F\left(x_{1}\right) \leq\|f\|_{d} \cdot d\left(x_{2}, x_{1}\right)F(x2)−F(x1)≤‖f‖d⋅d(x2,x1)
for any x 1 , x 2 ∈ X x 1 , x 2 ∈ X x_(1),x_(2)in Xx_{1}, x_{2} \in Xx1,x2∈X and ‖ F ‖ d ≤ ‖ f ‖ d ‖ F ‖ d ≤ ‖ f ‖ d ||F||_(d) <= ||f||_(d)\|F\|_{d} \leq\|f\|_{d}‖F‖d≤‖f‖d.
d) The function F F FFF is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing.
Indeed, let be u , v ∈ X u , v ∈ X u,v in Xu, v \in Xu,v∈X and d ( u , v ) = 0 d ( u , v ) = 0 d(u,v)=0d(u, v)=0d(u,v)=0. We have d ( u , y ) ≤ d ( u , v ) + d ( v , y ) d ( u , y ) ≤ d ( u , v ) + d ( v , y ) d(u,y) <= d(u,v)+d(v,y)d(u, y) \leq d(u, v)+d(v, y)d(u,y)≤d(u,v)+d(v,y). Consequently
d ( u , y ) ≤ d ( v , y ) d ( u , y ) ≤ d ( v , y ) d(u,y) <= d(v,y)d(u, y) \leq d(v, y)d(u,y)≤d(v,y)
Then
f ( y ) + ‖ f ‖ d d ( u , y ) ≤ f ( y ) + ‖ f ‖ d d ( v , y ) . f ( y ) + ‖ f ‖ d d ( u , y ) ≤ f ( y ) + ‖ f ‖ d d ( v , y ) . f(y)+||f||_(d)d(u,y) <= f(y)+||f||_(d)d(v,y).f(y)+\|f\|_{d} d(u, y) \leq f(y)+\|f\|_{d} d(v, y) .f(y)+‖f‖dd(u,y)≤f(y)+‖f‖dd(v,y).
It follows that
F ( u ) ≤ F ( v ) , F ( u ) ≤ F ( v ) , F(u) <= F(v),F(u) \leq F(v),F(u)≤F(v),
and consequently d ( u , v ) = 0 d ( u , v ) = 0 d(u,v)=0d(u, v)=0d(u,v)=0 implies F ( u ) ≤ F ( v ) F ( u ) ≤ F ( v ) F(u) <= F(v)F(u) \leq F(v)F(u)≤F(v).
It follows that F ∈ E Y d ( f ) F ∈ E Y d ( f ) F inE_(Y)^(d)(f)F \in E_{Y}^{d}(f)F∈EYd(f) so that E Y d ( t ) ≠ ∅ E Y d ( t ) ≠ ∅ E_(Y)^(d)(t)!=O/E_{Y}^{d}(t) \neq \emptysetEYd(t)≠∅.
Remarks 1. 1 0 1 0 1^(0)1^{0}10 Similarly, the function
(11) G ( x ) = sup y ∈ Y { f ( y ) − ‖ f ‖ d d − 1 ( x , y ) } (11) G ( x ) = sup y ∈ Y   f ( y ) − ‖ f ‖ d d − 1 ( x , y ) {:(11)G(x)=s u p_(y in Y){f(y)-||f||_(d)d^(-1)(x,y)}:}\begin{equation*} G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{d} d^{-1}(x, y)\right\} \tag{11} \end{equation*}(11)G(x)=supy∈Y{f(y)−‖f‖dd−1(x,y)}
is ≤ d ≤ d <= _(d)\leq_{d}≤d-increasing, and G G GGG belongs to E Y d ( f ) E Y d ( f ) E_(Y)^(d)(f)E_{Y}^{d}(f)EYd(f) too.
2 0 2 0 2^(0)2^{0}20 The inequality
(12) G ( x ) ≤ F ( x ) , (12) G ( x ) ≤ F ( x ) , {:(12)G(x) <= F(x)",":}\begin{equation*} G(x) \leq F(x), \tag{12} \end{equation*}(12)G(x)≤F(x),
holds for every x ∈ X x ∈ X x in Xx \in Xx∈X.
Indeed, taking the infimum with respect to z ∈ Y z ∈ Y z in Yz \in Yz∈Y and then the supremum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y in (10) we find
G ( x ) = sup y ∈ Y { f ( y ) − ‖ f ‖ d d − 1 ( x , y ) } ≤ inf z ∈ Y { f ( z ) + ‖ f ‖ d d ( x , z ) } = F ( x ) G ( x ) = sup y ∈ Y   f ( y ) − ‖ f ‖ d d − 1 ( x , y ) ≤ inf z ∈ Y   f ( z ) + ‖ f ‖ d d ( x , z ) = F ( x ) G(x)=s u p_(y in Y){f(y)-||f||_(d)d^(-1)(x,y)} <= i n f_(z in Y){f(z)+||f||_(d)d(x,z)}=F(x)G(x)=\sup _{y \in Y}\left\{f(y)-\|f\|_{d} d^{-1}(x, y)\right\} \leq \inf _{z \in Y}\left\{f(z)+\|f\|_{d} d(x, z)\right\}=F(x)G(x)=supy∈Y{f(y)−‖f‖dd−1(x,y)}≤infz∈Y{f(z)+‖f‖dd(x,z)}=F(x)
In fact, the following theorem holds:
Theorem 3. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, Y Y YYY a nonvoid subset of X X XXX and f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY.
Then for any H ∈ E Y d ( f ) H ∈ E Y d ( f ) H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f)H∈EYd(f) we have
(13) G ( x ) ≤ H ( x ) ≤ F ( x ) , x ∈ X . (13) G ( x ) ≤ H ( x ) ≤ F ( x ) , x ∈ X . {:(13)G(x) <= H(x) <= F(x)","quad x in X.:}\begin{equation*} G(x) \leq H(x) \leq F(x), \quad x \in X . \tag{13} \end{equation*}(13)G(x)≤H(x)≤F(x),x∈X.
Proof. Let H ∈ E Y d ( f ) H ∈ E Y d ( f ) H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f)H∈EYd(f). For arbitrary x ∈ X x ∈ X x in Xx \in Xx∈X and y ∈ Y y ∈ Y y in Yy \in Yy∈Y we have
H ( x ) − H ( y ) ≤ ‖ f ‖ d d ( x , y ) H ( x ) − H ( y ) ≤ ‖ f ‖ d d ( x , y ) H(x)-H(y) <= ||f||_(d)d(x,y)H(x)-H(y) \leq\|f\|_{d} d(x, y)H(x)−H(y)≤‖f‖dd(x,y)
implying
H ( x ) ≤ H ( y ) + ‖ f ‖ d d ( x , y ) = f ( y ) + ‖ f ‖ d ( x , y ) H ( x ) ≤ H ( y ) + ‖ f ‖ d d ( x , y ) = f ( y ) + ‖ f ‖ d ( x , y ) H(x) <= H(y)+||f||_(d)d(x,y)=f(y)+||f||_(d)(x,y)H(x) \leq H(y)+\|f\|_{d} d(x, y)=f(y)+\|f\|_{d}(x, y)H(x)≤H(y)+‖f‖dd(x,y)=f(y)+‖f‖d(x,y)
Taking the imfimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y we get
H ( x ) ≤ inf y ∈ Y { f ( y ) + ‖ f ‖ d d ( x , y ) } = F ( x ) H ( x ) ≤ inf y ∈ Y   f ( y ) + ‖ f ‖ d d ( x , y ) = F ( x ) H(x) <= i n f_(y in Y){f(y)+||f||_(d)d(x,y)}=F(x)H(x) \leq \inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}=F(x)H(x)≤infy∈Y{f(y)+‖f‖dd(x,y)}=F(x)
The inequality H ( x ) ≥ G ( x ) , x ∈ X H ( x ) ≥ G ( x ) , x ∈ X H(x) >= G(x),x in XH(x) \geq G(x), x \in XH(x)≥G(x),x∈X can be proved similarly.
Corollary 4. A function f ∈ S f ∈ S f in Sf \in Sf∈S Lip Y Y YYY has a unique extension in S S SSS Lip X X XXX if and only if the following relation
(14) inf y ∈ Y { f ( y ) + ‖ f ‖ d d ( x , y ) } = sup y ∈ Y { f ( y ) − ‖ f ‖ d ( y , x ) } (14) inf y ∈ Y   f ( y ) + ‖ f ‖ d d ( x , y ) = sup y ∈ Y   { f ( y ) − ‖ f ‖ d ( y , x ) } {:(14)i n f_(y in Y){f(y)+||f||_(d)d(x,y)}=s u p_(y in Y){f(y)-||f||d(y","x)}:}\begin{equation*} \inf _{y \in Y}\left\{f(y)+\|f\|_{d} d(x, y)\right\}=\sup _{y \in Y}\{f(y)-\|f\| d(y, x)\} \tag{14} \end{equation*}(14)infy∈Y{f(y)+‖f‖dd(x,y)}=supy∈Y{f(y)−‖f‖d(y,x)}
holds for every x ∈ X x ∈ X x in Xx \in Xx∈X.
Example.
Let R R R\mathbb{R}R be the real axis and d : R × R → [ 0 , ∞ ) d : R × R → [ 0 , ∞ ) d:RxxRrarr[0,oo)d: \mathbb{R} \times \mathbb{R} \rightarrow[0, \infty)d:R×R→[0,∞) the quasi-metric defined by
d ( x , y ) = { x − y , if x ≥ y 1 , if x < y d ( x , y ) = x − y ,  if  x ≥ y 1 ,  if  x < y d(x,y)={[x-y","," if "quad x >= y],[1","," if "quad x < y]:}d(x, y)=\left\{\begin{array}{cc} x-y, & \text { if } \quad x \geq y \\ 1, & \text { if } \quad x<y \end{array}\right.d(x,y)={x−y, if x≥y1, if x<y
Let Y Y YYY be given by Y = [ 0 , 1 ] ⊂ R Y = [ 0 , 1 ] ⊂ R Y=[0,1]subRY=[0,1] \subset \mathbb{R}Y=[0,1]⊂R and f : Y → R , f ( y ) = 2 y f : Y → R , f ( y ) = 2 y f:Y rarrR,f(y)=2yf: Y \rightarrow \mathbb{R}, f(y)=2 yf:Y→R,f(y)=2y. Then f f fff is semi-Lipschitz on Y Y YYY and ‖ f ‖ d = 2 ‖ f ‖ d = 2 ||f||_(d)=2\|f\|_{d}=2‖f‖d=2. The extension F F FFF defined by (9) is
F ( x ) = { 2 , if x < 0 2 x , if x ≥ 0 F ( x ) = 2 ,  if  x < 0 2 x ,  if  x ≥ 0 F(x)={[2","," if ",x < 0],[2x","," if ",x >= 0]:}F(x)=\left\{\begin{array}{cll} 2, & \text { if } & x<0 \\ 2 x, & \text { if } & x \geq 0 \end{array}\right.F(x)={2, if x<02x, if x≥0
and the extension G G GGG defined by (11) is
G ( x ) = { 2 x , x ≤ 1 0 , x > 1 G ( x ) = 2 x , x ≤ 1 0 , x > 1 G(x)={[2x",",x <= 1],[0",",x > 1]:}G(x)=\left\{\begin{array}{cl} 2 x, & x \leq 1 \\ 0, & x>1 \end{array}\right.G(x)={2x,x≤10,x>1
Obviously, G ( x ) ≤ F ( x ) , x ∈ R G ( x ) ≤ F ( x ) , x ∈ R G(x) <= F(x),x inRG(x) \leq F(x), x \in \mathbb{R}G(x)≤F(x),x∈R.

REFERENCES

[1] S. Cobzas and C. Mustăţa, Norm preserving extension of convex Lipschitz functions, J. Approx. Theory, 29 (1978), 555-569.
[2] J. Czipser and L. Gehér, Extension of functions satisfying a Lipschitz condition, Acta Math. Sci. Hungar., 6 (1955), 213-220.
[3] P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982.
[4] J. A. McShane, Extension of range of functions, Bull. Amer. Math. Soc., 40 (1939), 837-842.
[5] C. Mustăţa, Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19 (1977), 222-230.
[6] S. Romaguera and M. Sanchis, Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103 (2000), 292-301.
[7] J. H. WellS and L. R. Williams, Embeddings and Extensions in Analysis, SpringerVerlag, Berlin, 1975.
Received: August 8, 2000.
  1. "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: cmustata@ictp-acad.math.ubbcluj.ro.
2001

Related Posts