Abstract
Authors
Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
Keywords
Paper coordinates
C. Mustăţa, Extension of starshaped bounded semi-Lipschitz function on a quasi metric linear spaces. ”135-th Pannonian Applied Mathematical Meeting” 4-7 oct. 2001, 285-291.
About this paper
Journal
Publisher Name
DOI
Print ISSN
Online ISSN
google scholar link
Paper (preprint) in HTML form
Extension of bounded starshaped semi-Lipschitz functions on quasi-metric linear spaces
1 Introduction
Let XX be a nonvoid set. A function d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) satisfying the conditions:
(i) quad d(x,y)=0Longleftrightarrow x=y\quad d(x, y)=0 \Longleftrightarrow x=y,
(ii) quad d(x,y) <= d(x,z)+d(z,y)\quad d(x, y) \leq d(x, z)+d(z, y), for all x,y,z in Xx, y, z \in X is called a quasi-metric on XX and the pair (X,d)(X, d) is called quasi-metric space. The essential difference with respect to a metric on XX is that a quasi-metric does not satisfy the symmetry condition d(x,y)=d(y,x)d(x, y)= d(y, x).
(i) quad d(x,y)=0Longleftrightarrow x=y\quad d(x, y)=0 \Longleftrightarrow x=y,
(ii) quad d(x,y) <= d(x,z)+d(z,y)\quad d(x, y) \leq d(x, z)+d(z, y), for all x,y,z in Xx, y, z \in X is called a quasi-metric on XX and the pair (X,d)(X, d) is called quasi-metric space. The essential difference with respect to a metric on XX is that a quasi-metric does not satisfy the symmetry condition d(x,y)=d(y,x)d(x, y)= d(y, x).
If XX is a linear space and dd a quasi-metric on XX then the pair (X,d)(X, d) is called a quasi-metric linear space.
Let theta in X\theta \in X be the null element of the linear space XX. A subset YY of XX is called starshaped (with respect to theta\theta ) if it satisfies the condition:
{:(1)AA y in Y quad AA alpha in[0","1]:alpha y in Y.:}\begin{equation*}
\forall y \in Y \quad \forall \alpha \in[0,1]: \alpha y \in Y . \tag{1}
\end{equation*}
If YY is a starshaped subset of the linear space XX then a function f:Y rarrRf: Y \rightarrow \mathbb{R} is called a starshaped function provided:
{:(2)AA y in Y quad AA alpha in[0","1]:f(alpha y) <= alpha f(y).:}\begin{equation*}
\forall y \in Y \quad \forall \alpha \in[0,1]: f(\alpha y) \leq \alpha f(y) . \tag{2}
\end{equation*}
Obviously that the condition (1) implies theta in Y\theta \in Y and the condition (2) implies f(theta) <= 0f(\theta) \leq 0. In what follows we shall consider only starshaped functions on YY which vanish at theta\theta, i.e. f(theta)=0f(\theta)=0.
If ( X,dX, d ) is a quasi-metric linear space and Y sub XY \subset X is a starshaped set, the quasi-metric dd is called starshaped on YY if
{:(3)AA x","y in Y","quad AA alpha in[0","1]:quad d(alpha x","alpha y) <= alpha d(x","y).:}\begin{equation*}
\forall x, y \in Y, \quad \forall \alpha \in[0,1]: \quad d(\alpha x, \alpha y) \leq \alpha d(x, y) . \tag{3}
\end{equation*}
Let (X,d)(X, d) be a quasi-metric space and Y sub X,Y!=O/Y \subset X, Y \neq \emptyset. A function f:Y rarrRf: Y \rightarrow \mathbb{R} is called semi-Lipschitz if it satisfies the condition
{:(4)EEK_(Y) >= 0:f(x)-f(y) <= K_(Y)*d(x","y):}\begin{equation*}
\exists K_{Y} \geq 0: f(x)-f(y) \leq K_{Y} \cdot d(x, y) \tag{4}
\end{equation*}
for all x,y in Yx, y \in Y.
A number K_(Y) >= 0K_{Y} \geq 0 for which (4) holds is called a semi-Lipschitz constant for ff (on YY ).
A number K_(Y) >= 0K_{Y} \geq 0 for which (4) holds is called a semi-Lipschitz constant for ff (on YY ).
One sees that
{:(5)||f||_(Y)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y) > 0}:}\begin{equation*}
\|f\|_{Y}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y)>0\right\} \tag{5}
\end{equation*}
is the smallest semi-Lipschitz constant for the function ff on the set YY (see [6][6] and [4, Th.1]).
Let
{:(6)SLipY:={f:Y rarrR","f" is semi-Lipschitz "}:}\begin{equation*}
S L i p Y:=\{f: Y \rightarrow \mathbb{R}, f \text { is semi-Lipschitz }\} \tag{6}
\end{equation*}
the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space (Y,d),Y sube X(Y, d), Y \subseteq X.
If ( X,dX, d ) is a quasi-metric linear space and YY is a subset of XX containing theta\theta then the set
{:(7)SLip_(0)Y:={f∣f in S Lip Y" and "f(theta)=0}:}\begin{equation*}
\operatorname{SLip}_{0} Y:=\{f \mid f \in S \operatorname{Lip} Y \text { and } f(\theta)=0\} \tag{7}
\end{equation*}
is a semilinear space and the functional ||*||_(Y):S\|\cdot\|_{Y}: S Lip _(0)Y rarrR_(+)_{0} Y \rightarrow \mathbb{R}_{+}defined by
{:(8)||f||_(Y):s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in Y,d(x,y)!=0}:}\begin{equation*}
\|f\|_{Y}: \sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in Y, d(x, y) \neq 0\right\} \tag{8}
\end{equation*}
is a quasi-norm on SLip_(0)XS L i p_{0} X (see [4] and [6]).
A semilinear space satisfies axioms similar to those defining a linear space, excepting the existence of the opposite element (the inverse with respect to ^(+){ }^{+}) and that the multiplication is defined only for positive scalars. (see [6]).
A semilinear space satisfies axioms similar to those defining a linear space, excepting the existence of the opposite element (the inverse with respect to ^(+){ }^{+}) and that the multiplication is defined only for positive scalars. (see [6]).
2 Extensions
The following problem is treated in [4]:
Let ( X,dX, d ) be a quasi-metric space, Y sub XY \subset X and f in Sf \in S Lip YY. One asks to find a function F in S Lip XF \in S \operatorname{Lip} X such that
{:(9)F|_(Y)=f" and "||F||_(X)=||f||_(Y).:}\begin{equation*}
\left.F\right|_{Y}=f \text { and }\|F\|_{X}=\|f\|_{Y} . \tag{9}
\end{equation*}
One shows ([4, Th.2]) that for every f in Sf \in S LipY there exists F in SF \in S LipX satisfying (9). This result is similar to a result of Mc Shane from 1934 (see [2]) asserting that every real valued Lipschitz function defined on a subset of a metric space XX admits an extension to the whole space with the same Lipschitz constant.
In the present paper we study the extension problem for bounded starshaped semi-Lipschitz functions defined on sharshaped subsets of quasi-metric linear spaces:
For a quasi-metric linear space (X,d)(X, d) with the quasi-metric dd starshaped, a starshaped subset YY of XX and a bounded semi-Lipschitz starshaped function f inSLip_(0)Yf \in \operatorname{SLip}_{0} Y find a bounded starshaped function F inSLip_(0)XF \in \operatorname{SLip}_{0} X such that
{:(10)F|_(Y)=f","||F||_(X)=||f||_(Y)" and "||F||_(oo)=||f||_(oo):}\begin{equation*}
\left.F\right|_{Y}=f,\|F\|_{X}=\|f\|_{Y} \text { and }\|F\|_{\infty}=\|f\|_{\infty} \tag{10}
\end{equation*}
where ||*||_(oo)\|\cdot\|_{\infty} denotes the sup-norm.
Observe that if YY is a subspace of XX and f inSLip_(0)Yf \in \operatorname{SLip}_{0} Y is starshaped then it is possible that ff be unbounded on YY.
Observe that if YY is a subspace of XX and f inSLip_(0)Yf \in \operatorname{SLip}_{0} Y is starshaped then it is possible that ff be unbounded on YY.
More exactly we have:
Lemma 1 Let (X,d)(X, d) be a quasi-metric linear space, YY a subspace of XX and f inf \in SLip _(0)Y_{0} Y be starshaped. If there exists x_(0)in Yx_{0} \in Y such that f(x_(0)) > 0f\left(x_{0}\right)>0 then ff is unbounded on YY.
Lemma 1 Let (X,d)(X, d) be a quasi-metric linear space, YY a subspace of XX and f inf \in SLip _(0)Y_{0} Y be starshaped. If there exists x_(0)in Yx_{0} \in Y such that f(x_(0)) > 0f\left(x_{0}\right)>0 then ff is unbounded on YY.
Proof. For every x in Y,x!=0x \in Y, x \neq 0, the function h:(0,oo)rarrRh:(0, \infty) \rightarrow \mathbb{R} defined by h(t)=f(tx)//th(t)=f(t x) / t is non-increasing. Indeed, if 0 < t_(1) < t_(2)0<t_{1}<t_{2} then
h(t_(1))=(f(t_(1)x))/(t_(1))=(f(t_(2)^(-1)*t_(1)(t_(2)x)))/(t_(1)) <= (t_(1))/(t_(2))(f(t_(2)x))/(t_(1))=(f(t_(2)x))/(t_(2))=h(t_(2)).h\left(t_{1}\right)=\frac{f\left(t_{1} x\right)}{t_{1}}=\frac{f\left(t_{2}^{-1} \cdot t_{1}\left(t_{2} x\right)\right)}{t_{1}} \leq \frac{t_{1}}{t_{2}} \frac{f\left(t_{2} x\right)}{t_{1}}=\frac{f\left(t_{2} x\right)}{t_{2}}=h\left(t_{2}\right) .
In particular, for t > 1t>1 and x_(0)in Yx_{0} \in Y with f(x_(0)) > 0f\left(x_{0}\right)>0 we have
0 < f(x_(0)) <= (f(tx_(0)))/(t)0<f\left(x_{0}\right) \leq \frac{f\left(t x_{0}\right)}{t}
so that
f(tx_(0)) >= tf(x_(0))f\left(t x_{0}\right) \geq t f\left(x_{0}\right)
for every t > 1t>1, which shows that ff is unbounded on YY.
The following theorem answers positively the question on the extension of bounded starshaped semi-Lipschitz functions.
The following theorem answers positively the question on the extension of bounded starshaped semi-Lipschitz functions.
Theorem 2 Let ( X,dX, d ) be a quasi-metric linear space and YY a starshaped subset of XX. Suppose that the quasi-metric dd is starshaped on YY.
Let f in Sf \in S Lip _(0)Y_{0} Y be bounded and starshaped. In order to exist a bounded starshaped function F in SLip_(0)XF \in S L i p_{0} X such that
F|_(Y)=f,quad||F||_(X)=||f||_(Y)quad" and "quad||F||_(oo)=||f||_(oo)\left.F\right|_{Y}=f, \quad\|F\|_{X}=\|f\|_{Y} \quad \text { and } \quad\|F\|_{\infty}=\|f\|_{\infty}
it is necessary and sufficient that f(y) <= 0f(y) \leq 0 for all y in Yy \in Y.
Proof. Sufficiency suppose that f(y) <= 0f(y) \leq 0 for all y in Yy \in Y. The function
Proof. Sufficiency suppose that f(y) <= 0f(y) \leq 0 for all y in Yy \in Y. The function
{:(11)H(x)=i n f_(y in Y)[f(y)+||f||_(Y)d(x,y)]","quad x in X:}\begin{equation*}
H(x)=\inf _{y \in Y}\left[f(y)+\|f\|_{Y} d(x, y)\right], \quad x \in X \tag{11}
\end{equation*}
is starshaped on XX and satisfies the conditions
H|_(Y)=f quad" and "quad||H||_(X)=||f||_(Y)\left.H\right|_{Y}=f \quad \text { and } \quad\|H\|_{X}=\|f\|_{Y}
(see [4,Th.2] and [5, Th.8).
Indeed, let z in Yz \in Y and x in Xx \in X. For any y in Yy \in Y we have
Indeed, let z in Yz \in Y and x in Xx \in X. For any y in Yy \in Y we have
{:[f(y)+||f||_(Y)d(x","y)=f(z)+||f||_(Y)d(x","y)-(f(z)-f(y)) >= ],[ >= f(z)+||f||_(Y)d(x","y)-||f||_(Y)d(z","y)=],[=f(z)-||f||_(Y)(d(z","y)-d(x","y))]:}\begin{aligned}
f(y)+\|f\|_{Y} d(x, y) & =f(z)+\|f\|_{Y} d(x, y)-(f(z)-f(y)) \geq \\
& \geq f(z)+\|f\|_{Y} d(x, y)-\|f\|_{Y} d(z, y)= \\
& =f(z)-\|f\|_{Y}(d(z, y)-d(x, y))
\end{aligned}
The inequality d(z,y)-d(x,y) <= d(z,x)d(z, y)-d(x, y) \leq d(z, x) implies
f(y)+||f||_(Y)d(x,y) >= f(z)-||f||_(Y)*d(z,x)f(y)+\|f\|_{Y} d(x, y) \geq f(z)-\|f\|_{Y} \cdot d(z, x)
showing that for every x in Xx \in X the set {f(y)+||f||_(Y)d(x,y):y in Y}\left\{f(y)+\|f\|_{Y} d(x, y): y \in Y\right\} is bounded from above by f(z)-||f||_(Y)d(z,x)f(z)-\|f\|_{Y} d(z, x), and the infimum (11) is finite.
We show now that H(y)=f(y)H(y)=f(y) for all y in Yy \in Y.
Let y in Yy \in Y. Then
Let y in Yy \in Y. Then
H(y) <= f(y)+||f||_(Y)d(y,y)=f(y)H(y) \leq f(y)+\|f\|_{Y} d(y, y)=f(y)
For any v in Yv \in Y we have
f(y)-f(v) <= ||f||_(Y)*d(y,v)f(y)-f(v) \leq\|f\|_{Y} \cdot d(y, v)
so that
f(v)+||f||_(Y)*d(y,v) >= f(y)f(v)+\|f\|_{Y} \cdot d(y, v) \geq f(y)
and
H(y)=i n f{f(v)+||f||_(Y)d(y,v):v in Y} >= f(y).H(y)=\inf \left\{f(v)+\|f\|_{Y} d(y, v): v \in Y\right\} \geq f(y) .
It follows H(y)=f(y)H(y)=f(y).
We prove that ||H||_(Y)=||f||_(Y)\|H\|_{Y}=\|f\|_{Y}.
Since H|_(Y)=f\left.H\right|_{Y}=f, the definitions of ||H||_(Y)\|H\|_{Y} and ||f||_(Y)\|f\|_{Y} yield ||H||_(Y) >= ||f||_(Y)\|H\|_{Y} \geq\|f\|_{Y}.
Let x_(1),x_(2)in Xx_{1}, x_{2} \in X and epsi > 0\varepsilon>0. Choosing y in Yy \in Y such that
We prove that ||H||_(Y)=||f||_(Y)\|H\|_{Y}=\|f\|_{Y}.
Since H|_(Y)=f\left.H\right|_{Y}=f, the definitions of ||H||_(Y)\|H\|_{Y} and ||f||_(Y)\|f\|_{Y} yield ||H||_(Y) >= ||f||_(Y)\|H\|_{Y} \geq\|f\|_{Y}.
Let x_(1),x_(2)in Xx_{1}, x_{2} \in X and epsi > 0\varepsilon>0. Choosing y in Yy \in Y such that
H(x_(1)) >= f(y)+||f||_(Y)d(x_(1),y)-epsiH\left(x_{1}\right) \geq f(y)+\|f\|_{Y} d\left(x_{1}, y\right)-\varepsilon
we obtain
{:[H(x_(2))-H(x_(1)) <= f(y)+||f||_(Y)d(x_(2),y)-(f(y)+||f||_(Y)*d(x_(1),y)-epsi)],[=||f||_(Y)[d(x_(2),y)-d(x_(1),y)]+epsi],[ <= ||f||_(Y)*d(x_(2),x_(1))+epsi.]:}\begin{aligned}
H\left(x_{2}\right)-H\left(x_{1}\right) & \leq f(y)+\|f\|_{Y} d\left(x_{2}, y\right)-\left(f(y)+\|f\|_{Y} \cdot d\left(x_{1}, y\right)-\varepsilon\right) \\
& =\|f\|_{Y}\left[d\left(x_{2}, y\right)-d\left(x_{1}, y\right)\right]+\varepsilon \\
& \leq\|f\|_{Y} \cdot d\left(x_{2}, x_{1}\right)+\varepsilon .
\end{aligned}
Since epsi > 0\varepsilon>0 is arbitrary it follows
H(x_(2))-H(x_(1)) <= ||f||_(Y)*d(x_(2),x_(1))H\left(x_{2}\right)-H\left(x_{1}\right) \leq\|f\|_{Y} \cdot d\left(x_{2}, x_{1}\right)
for any x_(1),x_(2)in Xx_{1}, x_{2} \in X and ||H||_(Y) <= ||f||_(Y)\|H\|_{Y} \leq\|f\|_{Y}. Then ||H||_(X)=||f||_(Y)\|H\|_{X}=\|f\|_{Y}.
We shall show that HH is also starshaped on XX. To this end let x in X,z in Yx \in X, z \in Y and alpha in[0,1]\alpha \in[0,1]. We have
We shall show that HH is also starshaped on XX. To this end let x in X,z in Yx \in X, z \in Y and alpha in[0,1]\alpha \in[0,1]. We have
{:[H(alpha x)=f(alpha z)+||f||_(Y)d(alpha x","alpha z) <= ],[ <= alpha f(z)+alpha||f||_(Y)d(x","z)=],[=alpha[f(z)+||f||_(Y)*d(x,z)].]:}\begin{aligned}
H(\alpha x) & =f(\alpha z)+\|f\|_{Y} d(\alpha x, \alpha z) \leq \\
& \leq \alpha f(z)+\alpha\|f\|_{Y} d(x, z)= \\
& =\alpha\left[f(z)+\|f\|_{Y} \cdot d(x, z)\right] .
\end{aligned}
Taking the infimum with respect to z in Yz \in Y we get
H(alpha x) <= alpha H(x)H(\alpha x) \leq \alpha H(x)
for all x in Xx \in X and all alpha in[0,1]\alpha \in[0,1], showing that the function HH defined by (11) is a starshaped extension of ff.
Consider the function
{:(12)F(x)={[H(x)," if ",H(x) <= 0],[0," if ",H(x) > 0].:}:}F(x)=\left\{\begin{array}{lll}
H(x) & \text { if } & H(x) \leq 0 \tag{12}\\
0 & \text { if } & H(x)>0
\end{array} .\right.
Since F|_(Y)=H|_(Y)=f,||H||_(X)=||f||_(Y)\left.F\right|_{Y}=\left.H\right|_{Y}=f,\|H\|_{X}=\|f\|_{Y} and ||H||_(X) >= ||F||_(X)\|H\|_{X} \geq\|F\|_{X}, it follows that ||F||_(X)=||f||_(Y)\|F\|_{X}=\|f\|_{Y}. Therefore FF is an extension of ff with the same semi-Lipschitz constant.
Let x in Xx \in X. If H(x) <= 0H(x) \leq 0 then
H(alpha x) <= alpha H(x) <= 0H(\alpha x) \leq \alpha H(x) \leq 0
for any alpha in[0,1]\alpha \in[0,1] so that
F(alpha x)=H(alpha x) <= alpha H(x)=alpha F(x).F(\alpha x)=H(\alpha x) \leq \alpha H(x)=\alpha F(x) .
If H(x) > 0H(x)>0 then F(x)=0F(x)=0. Let alpha in(0,1)\alpha \in(0,1). If H(alpha x) > 0H(\alpha x)>0 then
F(alpha x)=0=alpha F(x).F(\alpha x)=0=\alpha F(x) .
If H(alpha x) <= 0H(\alpha x) \leq 0 then F(alpha x)=H(alpha x)F(\alpha x)=H(\alpha x) and
F(alpha x)=H(alpha x) <= 0=alpha F(x).F(\alpha x)=H(\alpha x) \leq 0=\alpha F(x) .
It follows that the function FF defined by (12) is an extension of ff which is starshaped and has the same semi-Lipschitz constant as ff.
Since
s u p{|F(x)|:x in X} >= s u p{|F(y)|:y in Y}=||f||_(oo)\sup \{|F(x)|: x \in X\} \geq \sup \{|F(y)|: y \in Y\}=\|f\|_{\infty}
it follows ||F||_(oo) >= ||f||_(oo)\|F\|_{\infty} \geq\|f\|_{\infty}.
Let x in Xx \in X. For any y in Yy \in Y we have
Let x in Xx \in X. For any y in Yy \in Y we have
f(y) <= f(y)+||f||_(Y)*d(x,y)f(y) \leq f(y)+\|f\|_{Y} \cdot d(x, y)
which implies
i n f_(y inY^('))f(y) <= i n f_(y in Y)[f(y)+||f||_(Y)d(x,y)]=H(x).\inf _{y \in Y^{\prime}} f(y) \leq \inf _{y \in Y}\left[f(y)+\|f\|_{Y} d(x, y)\right]=H(x) .
If H(x) <= 0H(x) \leq 0 then
i n f_(y in Y)f(y) <= H(x)=F(x).\inf _{y \in Y} f(y) \leq H(x)=F(x) .
Since f(y) <= 0f(y) \leq 0 for all y in Yy \in Y we have ||f||_(oo)^((1)/(2))=-i n f f(Y)\|f\|_{\infty}^{\frac{1}{2}}=-\inf f(Y) so that
-||f||_(oo) <= F(x)Longleftrightarrow||f||_(oo) >= -F(x).-\|f\|_{\infty} \leq F(x) \Longleftrightarrow\|f\|_{\infty} \geq-F(x) .
If H(x) > 0H(x)>0 then
||f||_(oo) >= 0=F(x).\|f\|_{\infty} \geq 0=F(x) .
Using the fact that F(x) <= 0F(x) \leq 0, for all x in Xx \in X. and the above inequalities we obtain
||f||_(oo) >= s u p_(x in X)(-F(x))=||F||_(oo).\|f\|_{\infty} \geq \sup _{x \in X}(-F(x))=\|F\|_{\infty} .
It follows that ||F||_(oo)=||f||_(oo)\|F\|_{\infty}=\|f\|_{\infty}.
Necessity. Suppose there exists y in Yy \in Y such that f(y) > 0f(y)>0. By Lemma 1 ff has no bounded starshaped extensions to XX.
Necessity. Suppose there exists y in Yy \in Y such that f(y) > 0f(y)>0. By Lemma 1 ff has no bounded starshaped extensions to XX.
References
[1] Cobzaş, S., Mustăta, C., Norm preserving extension of convex Lipschitz functions, J.A.T. 24(1978), 555-564.
[2] McShane, J.A., Extension of range of functions, Bull. Amer. Math. Soc. 40(1934), 837-842.
[3] Mustăta, C., The extension of starshaped bounded Lipschitz functions, Ann. Numer. Theor. Approx., 9(1980) 1, 93-99.
[4] Mustăta, C., Extensions of semi-Lipschitz functions on quasi-metric space (to appear).
[5] Mustăta, C., Extension of convex semi-Lipschitz functions on quasimetric linear space (to appear)
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz functions in quasi-metric spaces, J.A.T 103(2000), 292-301.
[7] Weaver, N., Lipschitz Algebras, World Scientific, Singapore 1999.
[8] Wels, J.H., Williams, L.R., Embeddings and Extension in Analysis, Springer-Verlag, Berlin, 1975.
[2] McShane, J.A., Extension of range of functions, Bull. Amer. Math. Soc. 40(1934), 837-842.
[3] Mustăta, C., The extension of starshaped bounded Lipschitz functions, Ann. Numer. Theor. Approx., 9(1980) 1, 93-99.
[4] Mustăta, C., Extensions of semi-Lipschitz functions on quasi-metric space (to appear).
[5] Mustăta, C., Extension of convex semi-Lipschitz functions on quasimetric linear space (to appear)
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz functions in quasi-metric spaces, J.A.T 103(2000), 292-301.
[7] Weaver, N., Lipschitz Algebras, World Scientific, Singapore 1999.
[8] Wels, J.H., Williams, L.R., Embeddings and Extension in Analysis, Springer-Verlag, Berlin, 1975.