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Costica Mustata
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania
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C. Mustăţa, Uniqueness of extension of semi-Lipschitz functions on quasi-metric spaces, Bul. Ştiinţ. Univ. Baia Mare, Seria B, Fascicola matematică-informatică, 16 (2000) no. 2, pp. 207-212.
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[1] Cobzas, S., Mustata, C., Norm Preserving Extension of Convex Lipschitz Functions , J. Approx. Theory, 24 (1978), 555-564.
[2] Fletcher, P., Lindgren, W.F., Quasi-Uniform Spaces, Dekker, York, 1982.
[3] McShane, J. A,, Extension of range of functions, Bull. Amer. Math. Soc. 40(1934) 837-842.
[4] Mustata, C., Best Approximation and Unique Extension of Lipschitz Functions, J. Approx. Theory 19(1977), 222-230.
[5] Mustata, C., On the Extension of Semi-Lipschitz Functions on Quasi-Metric Space (to appear).
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces, J. Approx. Theory 103 (2000) 292-301.
[7] Wells, J.H., Williams, L.R., Embeddings and Extensions in Analysis, Springer- Verlag, Berlin 1975.
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Dedicated to Maria S. Pop on her 60^("th ")60^{\text {th }} anniversary
UNIQUENESS OF THE EXTENSION OF SEMI-LIPSCHITZ FUNCTIONS ON QUASI - METRIC SPACES
Costică MUSTĂTA
Let XX be a nonvoid set and d:X xx X rarr[0,oo)d: X \times X \rightarrow[0, \infty) a function satisfying the following conditions:
(i) d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=y
(ii) d(x,y) <= d(x,z)+d(z,y)d(x, y) \leq d(x, z)+d(z, y)
for all x,y,z in Xx, y, z \in X. We call dd a quasi-metric on XX and the pair ( X,dX, d ), a quasimetric space. Remark that the main difference with respect to a metric is the symmetry condition, d(x,y)=d(y,x)d(x, y)=d(y, x), which is not satisfied by a quasimetric.
The conjugate of a quasi-metric dd,denoted by d^(-1)d^{-1} is defined by
{:(1)d^(-1)(x","y)=d(y","x):}\begin{equation*}
d^{-1}(x, y)=d(y, x) \tag{1}
\end{equation*}
for all x,y in Xx, y \in X. Obviously, that the mapping d^(s):X xx X rarr[0,oo)d^{s}: X \times X \rightarrow[0, \infty) defined by
{:(2)d^(s)(x","y)=max{d(x,y),d^(-1)(x,y)}","x","y in X:}\begin{equation*}
d^{s}(x, y)=\max \left\{d(x, y), d^{-1}(x, y)\right\}, x, y \in X \tag{2}
\end{equation*}
is a metric on XX, i.e. d^(s)d^{s} satisfies the conditions (i), (ii) and the symmetry condition:
{:(iii)d^(s)(x","y)=d^(s)(y","x)","quad x","y in X:}\begin{equation*}
d^{s}(x, y)=d^{s}(y, x), \quad x, y \in X \tag{iii}
\end{equation*}
A function f:X rarrRf: X \rightarrow \mathbb{R},defined on a quasi-metric space ( X,dX, d ) is called semi-Lipschitz provided there exists a number K >= 0K \geq 0 such that
{:(3)f(x)-f(y) <= Kd(x","y):}\begin{equation*}
f(x)-f(y) \leq K d(x, y) \tag{3}
\end{equation*}
for all x,y in Xx, y \in X. A function f:X rarrRf: X \rightarrow \mathbb{R} is called <= _(d^(-))\leq_{d^{-}}increasing if
{:(3a)d(x","y)=0Longrightarrow f(x)-f(y) <= 0:}\begin{equation*}
d(x, y)=0 \Longrightarrow f(x)-f(y) \leq 0 \tag{3a}
\end{equation*}
for all x,y in Xx, y \in X.
The definition of <= _(d)\leq_{d} - increasing function f:X rarrRf: X \rightarrow \mathbb{R} is consistent for T_(0)T_{0} - separated quasi-metric space (X,d)(X, d) (see [6]). In this note the quasi-metric space ( X.dX . d ) is T_(1)T_{1} - separated (see the condition (i) and (ii)).
The definition of <= _(d)\leq_{d} - increasing function f:X rarrRf: X \rightarrow \mathbb{R} is consistent for T_(0)T_{0} - separated quasi-metric space (X,d)(X, d) (see [6]). In this note the quasi-metric space ( X.dX . d ) is T_(1)T_{1} - separated (see the condition (i) and (ii)).
Since d(x,y)=0Longleftrightarrow x=yd(x, y)=0 \Longleftrightarrow x=y, it follows that f(x) <= f(y)f(x) \leq f(y) for any function f:X rarrRf: X \rightarrow \mathbb{R} i.e. any real - valued function on a quasi - metric space XX is <= _(d)\leq_{d} - increasing.
Theorem 1 Let f:X rarrRf: X \rightarrow \mathbb{R} be such that
{:(4)||f||_(d)=s u p{((f(x)-f(y))vv0)/(d(x,y)):x,y in X,d(x,y) > 0} < oo:}\begin{equation*}
\|f\|_{d}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d(x, y)}: x, y \in X, d(x, y)>0\right\}<\infty \tag{4}
\end{equation*}
Then ff satisfies the inequality
{:(5)f(x)-f(y) <= ||f||_(d)*d(x","y)","AA x","y in X:}\begin{equation*}
f(x)-f(y) \leq\|f\|_{d} \cdot d(x, y), \forall x, y \in X \tag{5}
\end{equation*}
and ||f||_(d)\|f\|_{d} is the smallest constant for which the inequality (3) holds.
P roof. nce ff is <= _(d)\leq_{d} - increasing (see (3a)) it follows that f(x)-f(y) > 0f(x)-f(y)>0 implies d(x,y) > 0d(x, y)>0. But then
P roof. nce ff is <= _(d)\leq_{d} - increasing (see (3a)) it follows that f(x)-f(y) > 0f(x)-f(y)>0 implies d(x,y) > 0d(x, y)>0. But then
(f(x)-f(y))/(d(x,y)) > 0" and "||f||_(d)=s u p_(d(x,y) > 0)((f(x)-f(y))vv0)/(d(x,y)) >= (f(x)-f(y)/(d(x,y))\frac{f(x)-f(y)}{d(x, y)}>0 \text { and }\|f\|_{d}=\sup _{d(x, y)>0} \frac{(f(x)-f(y)) \vee 0}{d(x, y)} \geq \frac{f(x)-f(y}{d(x, y)}
implying
f(x)-f(y) <= ||f||_(d)*d(x,y).f(x)-f(y) \leq\|f\|_{d} \cdot d(x, y) .
If f(x)-f(y) <= 0f(x)-f(y) \leq 0 then
((f(x)-f(y))vv0)/(d(x,y))=0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=0
implying f(x)-f(y) <= ||f||_(d)*d(x,y)f(x)-f(y) \leq\|f\|_{d} \cdot d(x, y).
Let now K >= 0K \geq 0 be such that
Let now K >= 0K \geq 0 be such that
f(x)-f(y) <= K*d(x,y)f(x)-f(y) \leq K \cdot d(x, y)
for all x,y in Xx, y \in X. Then ff is <= _(d)\leq_{d} - increasing and
((f(x)-f(y))vv0)/(d(x,y))=(f(x)-f(y))/(d(x,y)) <= K quad" if "quad f(x)-f(y) > 0\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=\frac{f(x)-f(y)}{d(x, y)} \leq K \quad \text { if } \quad f(x)-f(y)>0
and
((f(x)-f(y))vv0)/(d(x,y))=0 <= K quad" if "quad f(x)-f(y) <= 0.\frac{(f(x)-f(y)) \vee 0}{d(x, y)}=0 \leq K \quad \text { if } \quad f(x)-f(y) \leq 0 .
Consequently, ||f||_(d) <= K\|f\|_{d} \leq K.
Denoting by SLip XX the set of all real - valued semi - Lipschitz functions defined on a quasi - metric space ( X,dX, d ) we have
Denoting by SLip XX the set of all real - valued semi - Lipschitz functions defined on a quasi - metric space ( X,dX, d ) we have
{:(6)" SLip "X={f:X rarrR,s u p_(d(x,y) > 0)((f(x)-f(y))vv0)/(d(x,y)) < oo}:}\begin{equation*}
\text { SLip } X=\left\{f: X \rightarrow \mathbb{R}, \sup _{d(x, y)>0} \frac{(f(x)-f(y)) \vee 0}{d(x, y)}<\infty\right\} \tag{6}
\end{equation*}
Let Y sub X,Y!=O/Y \subset X, Y \neq \emptyset, where (X,d)(X, d) is a quasi - metric space. It follows that ( Y,dY, d ) is a quasi - metric space, too, and let's denote by SLip YY the set of all semi - Lipschitz functions on YY.
The following extension problem arises naturally: for f in SLipYf \in S L i p Y find F inF \in SLip XX such that
{:(7)F|_(Y)=f" and "||F||_(d)=||f||_(d):}\begin{equation*}
\left.F\right|_{Y}=f \text { and }\|F\|_{d}=\|f\|_{d} \tag{7}
\end{equation*}
The answer is affirmative. In [5] it was shown that the functions
{:(8)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} \cdot d(x, y)\right], x \in X \tag{8}
\end{equation*}
{:(9)G(x)=s u p_(y in Y)[f(y)-||f||_(d)*d^(-1)(x,y)]","x in X:}\begin{equation*}
G(x)=\sup _{y \in Y}\left[f(y)-\|f\|_{d} \cdot d^{-1}(x, y)\right], x \in X \tag{9}
\end{equation*}
satisfy the equalities
F|_(Y)=G|_(Y)=f quad" and "quad||F||_(d)=||G||_(d)=||f||_(d).\left.F\right|_{Y}=\left.G\right|_{Y}=f \quad \text { and } \quad\|F\|_{d}=\|G\|_{d}=\|f\|_{d} .
In other words, for any f in SLipYf \in S L i p Y the set
{:(10)E_(Y)^(d)(f):={H in S Lip X:H|_(Y)=f quad" and "quad||H||_(d)=||f||_(d)}:}\begin{equation*}
E_{Y}^{d}(f):=\left\{H \in S \operatorname{Lip} X:\left.H\right|_{Y}=f \quad \text { and } \quad\|H\|_{d}=\|f\|_{d}\right\} \tag{10}
\end{equation*}
of all extensions of ff which preserve the smallest Lipschitz constant is nonvoid.
Concerning the unicity of the extension ( cardE_(Y)^(d)(f)=1\operatorname{card} E_{Y}^{d}(f)=1 ) one can prove:
Theorem 2 Let (X,d)(X, d) be a quasi-metric space, Y sub XY \subset X and f in SLipYf \in S L i p Y. Then
a) For every H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f) the following inequalities hold:
Theorem 2 Let (X,d)(X, d) be a quasi-metric space, Y sub XY \subset X and f in SLipYf \in S L i p Y. Then
a) For every H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f) the following inequalities hold:
{:(11)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{11}
\end{equation*}
where the functions F,GF, G are defined by (8), (9);
b) card E_(Y)^(d)(f)=1E_{Y}^{d}(f)=1 if and only if
b) card E_(Y)^(d)(f)=1E_{Y}^{d}(f)=1 if and only if
{:(12)s u p_(y in Y)[f(y)-||f||_(d)d^(-1)(x,y)]=i n f_(y in Y)[f(y)+||f||_(d)d(x,y)]:}\begin{equation*}
\sup _{y \in Y}\left[f(y)-\|f\|_{d} d^{-1}(x, y)\right]=\inf _{y \in Y}\left[f(y)+\|f\|_{d} d(x, y)\right] \tag{12}
\end{equation*}
for all x in Xx \in X.
P roof. t H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f). Then we have for every x in Xx \in X and y in Yy \in Y :
P roof. t H inE_(Y)^(d)(f)H \in E_{Y}^{d}(f). Then we have for every x in Xx \in X and y in Yy \in Y :
{:[H(x)-H(y) <= ||f||_(d)d(x","y)],[H(y)-H(x) <= ||f||_(d)d(y","x)=||f||_(d)*d^(-1)(x","y).]:}\begin{aligned}
& H(x)-H(y) \leq\|f\|_{d} d(x, y) \\
& H(y)-H(x) \leq\|f\|_{d} d(y, x)=\|f\|_{d} \cdot d^{-1}(x, y) .
\end{aligned}
The first inequality implies
H(x) <= H(y)+||f||_(d)*d(x,y)=f(y)+||f||_(d)d(x,y)H(x) \leq H(y)+\|f\|_{d} \cdot d(x, y)=f(y)+\|f\|_{d} d(x, y)
and, taking the infimum with respect to y in Yy \in Y, we have
H(x) <= F(x),quad x in X.H(x) \leq F(x), \quad x \in X .
Similarly, we get
H(x) >= H(y)-||f||_(d)d^(-1)(x,y)=f(y)-||f||_(d)d^(-1)(x,y).H(x) \geq H(y)-\|f\|_{d} d^{-1}(x, y)=f(y)-\|f\|_{d} d^{-1}(x, y) .
Taking the supremum with respect to y in Yy \in Y one obtains
H(x) >= G(x),quad x in X.H(x) \geq G(x), \quad x \in X .
The assertion b) is a direct consequence of the inequalities (11).
Remark. 1^(0)1^{0}. If the function f:X rarrRf: X \rightarrow \mathbb{R} is constant on XX then ||f||_(d)=0\|f\|_{d}=0, and the equality (12) holds.
Remark. 1^(0)1^{0}. If the function f:X rarrRf: X \rightarrow \mathbb{R} is constant on XX then ||f||_(d)=0\|f\|_{d}=0, and the equality (12) holds.
Consider on R\mathbb{R} de quasi-metric
d(x,y)={[x-y",",x >= y],[0",",x < y]:}d(x, y)=\left\{\begin{array}{cc}
x-y, & x \geq y \\
0, & x<y
\end{array}\right.
and let Y=[0,1]Y=[0,1] and f(y)=2y,y in Yf(y)=2 y, y \in Y.Then ||f||_(d)=2\|f\|_{d}=2 and the extremal extensions F,GF, \mathrm{G} are
F(x)={[2,x < 0],[2x,x >= 0]quad" and "quad G(x)={[2x,x <= 1],[0,x > 1]:}F(x)=\left\{\begin{array}{cc}
2 & x<0 \\
2 x & x \geq 0
\end{array} \quad \text { and } \quad G(x)=\left\{\begin{array}{cc}
2 x & x \leq 1 \\
0 & x>1
\end{array}\right.\right.
which are distinct.
2^(0)2^{0}. By Theorem 2, if f in SLipYf \in S L i p Y has a unique extension then the equality (12) holds and, since
2^(0)2^{0}. By Theorem 2, if f in SLipYf \in S L i p Y has a unique extension then the equality (12) holds and, since
{:[i n f_(y in Y)[f(y)+||f||_(d)*d(x,y)] >= i n f_(y in Y)f(y)+||f||_(d)*d(x","Y)],[s u p_(y in Y)[f(y)-||f||_(d)*d^(-1)(x,y)] <= s u p_(y in Y)f(y)-||f||_(d)*d^(-1)(x","Y)]:}\begin{aligned}
\inf _{y \in Y}\left[f(y)+\|f\|_{d} \cdot d(x, y)\right] & \geq \inf _{y \in Y} f(y)+\|f\|_{d} \cdot d(x, Y) \\
\sup _{y \in Y}\left[f(y)-\|f\|_{d} \cdot d^{-1}(x, y)\right] & \leq \sup _{y \in Y} f(y)-\|f\|_{d} \cdot d^{-1}(x, Y)
\end{aligned}
where
d(x,Y)=i n f{d(x,y):y in Y}d(x, Y)=\inf \{d(x, y): y \in Y\}
and
d^(-1)(x,Y)=i n f{d(y,x):y in Y}d^{-1}(x, Y)=\inf \{d(y, x): y \in Y\}
we obtain the inequality
{:(13)d(x","Y)+d^(-1)(x","Y) <= (1)/(||f||_(d))(s u p_(y in Y)f(y)-i n f_(y in Y)f(y)):}\begin{equation*}
d(x, Y)+d^{-1}(x, Y) \leq \frac{1}{\|f\|_{d}}\left(\sup _{y \in Y} f(y)-\inf _{y \in Y} f(y)\right) \tag{13}
\end{equation*}
Theorem 3 Let (X,d)(X, d) be a quasi-metric space and Y sub X,Y!=XY \subset X, Y \neq X, containing at least one cluster point. If each function f in SLipYf \in S L i p Y has a unique extension then bar(Y)=X\bar{Y}=X.
P\mathbf{P} roof. ty_(0)in Y\mathrm{t} y_{0} \in Y be a cluster point of the set YY and let y_(n)in Y\\{y_(0)}y_{n} \in Y \backslash\left\{y_{0}\right\}, n=1,2dotsn=1,2 \ldots, be such that lim_(n rarr oo)d(y_(n),y_(0))=0\lim _{n \rightarrow \infty} d\left(y_{n}, y_{0}\right)=0.
P\mathbf{P} roof. ty_(0)in Y\mathrm{t} y_{0} \in Y be a cluster point of the set YY and let y_(n)in Y\\{y_(0)}y_{n} \in Y \backslash\left\{y_{0}\right\}, n=1,2dotsn=1,2 \ldots, be such that lim_(n rarr oo)d(y_(n),y_(0))=0\lim _{n \rightarrow \infty} d\left(y_{n}, y_{0}\right)=0.
Claim: There exists x_(0)in Xx_{0} \in X such that d(x_(0),y_(0)) > 0d\left(x_{0}, y_{0}\right)>0 and d(x_(0),y_(n)) > 0d\left(x_{0}, y_{n}\right)>0, n=1,2,dotsn=1,2, \ldots.
Indeed, if contrary, then for every x in X,d(x,y_(0))=0x \in X, d\left(x, y_{0}\right)=0 or d(x,y_(n))=0d\left(x, y_{n}\right)=0 for all n inNn \in \mathbb{N}.In the first case x=y_(0)in Yx=y_{0} \in Y and in the second x=y_(n)in Yx=y_{n} \in Y.It follows Y=XY=X, a contradiction.
Consider the function f:X rarrRf: X \rightarrow \mathbb{R} defined by
f(x)=d(x,y_(0))-d(x_(0),y_(0)),x in Xf(x)=d\left(x, y_{0}\right)-d\left(x_{0}, y_{0}\right), x \in X
We have
{:[f(y_(0))=d(y_(0),y_(0))-d(x_(0),y_(0))=-d(x_(0),y_(0)) < 0],[f(y_(n))=d(y_(n),y_(0))-d(x_(0),y_(0)) > -d(x_(0),y_(0))]:}\begin{aligned}
& f\left(y_{0}\right)=d\left(y_{0}, y_{0}\right)-d\left(x_{0}, y_{0}\right)=-d\left(x_{0}, y_{0}\right)<0 \\
& f\left(y_{n}\right)=d\left(y_{n}, y_{0}\right)-d\left(x_{0}, y_{0}\right)>-d\left(x_{0}, y_{0}\right)
\end{aligned}
for all n=1,2,dotsn=1,2, \ldots. Define the sequence of functions varphi_(n):f(X)rarr[0,1]\varphi_{n}: f(X) \rightarrow[0,1] by
varphi_(n)(t)={[1," if "t < f(y_(0))],[(f(y_(n))-t)/(f(y_(n))-f(y_(0))),t in[f(y_(0)),f(y_(n))]],[0,t > f(y_(n))]:}\varphi_{n}(t)= \begin{cases}1 & \text { if } t<f\left(y_{0}\right) \\ \frac{f\left(y_{n}\right)-t}{f\left(y_{n}\right)-f\left(y_{0}\right)} & t \in\left[f\left(y_{0}\right), f\left(y_{n}\right)\right] \\ 0 & t>f\left(y_{n}\right)\end{cases}
The function Psi_(n)=varphi_(n)@f:X rarrR,n=1,2,dots\Psi_{n}=\varphi_{n} \circ f: X \rightarrow \mathbb{R}, n=1,2, \ldots, satisfy
||Psi_(n)||_(d) >= ((varphi_(n)(f(y_(0)))-varphi_(n)(f(y_(n))))vv0)/(d(y_(0),y_(n)))=(1)/(d(y_(0),y_(n)))rarr oo\left\|\Psi_{n}\right\|_{d} \geq \frac{\left(\varphi_{n}\left(f\left(y_{0}\right)\right)-\varphi_{n}\left(f\left(y_{n}\right)\right)\right) \vee 0}{d\left(y_{0}, y_{n}\right)}=\frac{1}{d\left(y_{0}, y_{n}\right)} \rightarrow \infty
for n rarr oon \rightarrow \infty.
By the inequality (12)
By the inequality (12)
d(x,Y)+d^(-1)(x,Y) <= (1-0)/(||Psi_(n)||_(d))rarr0d(x, Y)+d^{-1}(x, Y) \leq \frac{1-0}{\left\|\Psi_{n}\right\|_{d}} \rightarrow 0
for n rarr oon \rightarrow \infty,showing that YY is dense in XX, with respect to the quasi-metric dd and with respect to d^(-1)d^{-1}, as well.
References
[1] Cobzas, S., Mustăta, C., Norm Preserving Extension of Convex Lipschitz Functions, J.A.T. 24(1978), 555-564.
[2] Fletcher, P., Lindgren, W.F., "Quasi-Uniform Spaces" Dekken, New York, 1982.
[3] McShane, J.A., Extension of range of functions, Bull.Amer.Math.Soc. 40(1934) 837-842.
[4] Mustăţa, C., Best Approximation and Unique Extension of Lipschitz Functions, J.A.T. 19(1977), 222-230.
[5] Mustăţa, C., On the Extension of Semi-Lipschitz Functions on QuasiMetric Space (to appear).
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces, J.A.T. 103 (2000) 292-301.
[7] Wells, J.H., Williams, L.R., Embeddings and Extensions in Analysis, Springer-Verlag, Berlin 1975.
[2] Fletcher, P., Lindgren, W.F., "Quasi-Uniform Spaces" Dekken, New York, 1982.
[3] McShane, J.A., Extension of range of functions, Bull.Amer.Math.Soc. 40(1934) 837-842.
[4] Mustăţa, C., Best Approximation and Unique Extension of Lipschitz Functions, J.A.T. 19(1977), 222-230.
[5] Mustăţa, C., On the Extension of Semi-Lipschitz Functions on QuasiMetric Space (to appear).
[6] Romaguera, S., Sanchis, M., Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces, J.A.T. 103 (2000) 292-301.
[7] Wells, J.H., Williams, L.R., Embeddings and Extensions in Analysis, Springer-Verlag, Berlin 1975.
Received 19.10.2000
"T. Popoviciu" Institute of Numerical Analysis str. Gh. Bilaşcu nr. 37
C.P. 68, O.P. 1
3400 Cluj-Napoca