Extensions of semi-Hölder real valued functions on a quasi-metric space

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

Abstract


In this note, the semi-Holder real valued functions on a quasi-metric (asymmetric metric) space are defined. An extension theorem for such functions and some consequences are presented.

Authors

Costica Mustata
Tiberiu Popoviciu Institute of Numerical analysis, Romania

Keywords

Semi-Holder functions; extensions.

Paper coordinates

C. Mustăţa, Extensions of semi-Hölder real valued functions on a quasi-metric space, Rev. Anal. Numer. Theor. Approx., 38 (2009), no. 2, pp. 164-169.

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Journal

Revue d’Analyse Numerique et de theorie de l’Approximation

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Publisher House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2009-vol38-no2-art6

Print ISSN

2501-059X

Online ISSN

2457-6794

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[6] Mustata, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001.

[7] Mustata, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, no. 3, pp. 222–230, 1977.

[8] Mustata, C., A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ. Baia Mare, Ser. B. Matematica-Informatica 18, pp. 275–280, 2002.

[9] Romaguera, S.andSanchis, M., Semi-Lipschitz functions and best approximation inquasi–metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.

[10] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions, Acta Math Hungar,108no. 1–2, pp. 55–70, 2005.

[11] Sanchez-Alvarez, J.M., On semi-Lipschitz functions with values in a quasi-normedlinear space, Applied General Topology,6, no. 2, pp. 216–228, 2005.

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[13]Wells, J.H. and Williams, L.R., Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975.

[14]* * *, The Otto Dunkel Memorial Problem Book,New York, 1957.

Paper (preprint) in HTML form

2009-Mustata-Extensions of semi-Hölder-JNAAT

EXTENSIONS OF SEMI-HÖLDER REAL VALUED FUNCTIONS ON A QUASI-METRIC SPACE

COSTICĂ MUSTĂŢA*

Abstract

In this note the semi-Hölder real valued functions on a quasi-metric (asymmetric metric) space are defined. An extension theorem for such functions and some consequences are presented.

MSC 2000. 26A16, 46A22.
Keywords. Semi-Hölder functions, extensions.

1. PRELIMINARIES

Let X X XXX be a non-empty set. 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,∞) is called a quasi-metric on X X XXX [9] (see also [1]) if the following conditions hold:
AM1) d ( x , y ) = d ( y , x ) = 0 ⇔ x = y d ( x , y ) = d ( y , x ) = 0 ⇔ x = y d(x,y)=d(y,x)=0<=>x=yd(x, y)=d(y, x)=0 \Leftrightarrow x=yd(x,y)=d(y,x)=0⇔x=y
AM2) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) d(x,z) <= d(x,y)+d(y,z)d(x, z) \leq d(x, y)+d(y, z)d(x,z)≤d(x,y)+d(y,z), for all x , y , z ∈ X x , y , z ∈ X x,y,z in Xx, y, z \in Xx,y,z∈X.
When X X XXX is non-empty set and d d ddd a quasi-metric on X X XXX, the pair ( X , d X , d X,dX, dX,d ) is called a quasi-metric space.
The function d ¯ : X × X → [ 0 , ∞ ) d ¯ : X × X → [ 0 , ∞ ) bar(d):X xx X rarr[0,oo)\bar{d}: X \times X \rightarrow[0, \infty)d¯:X×X→[0,∞) defined by d ¯ ( x , y ) = d ( y , x ) d ¯ ( x , y ) = d ( y , x ) bar(d)(x,y)=d(y,x)\bar{d}(x, y)=d(y, x)d¯(x,y)=d(y,x) for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X is also a quasi-metric on X X XXX, called the conjugate quasi-metric of d d ddd.
Obviously, the function d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d s ( x , y ) = max { d ( x , y ) , d ¯ ( x , y ) } d^(s)(x,y)=max{d(x,y), bar(d)(x,y)}d^{s}(x, y)=\max \{d(x, y), \bar{d}(x, y)\}ds(x,y)=max{d(x,y),d¯(x,y)} is a metric on X X XXX. Each quasi-metric d d ddd on X X XXX induces a topology τ ( d ) τ ( d ) tau(d)\tau(d)τ(d), which has as a base the family of balls (forward open balls [4]).
(1.1) B + ( x , ε ) : { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . (1.1) B + ( x , ε ) : { y ∈ X : d ( x , y ) < ε } , x ∈ X , ε > 0 . {:(1.1)B^(+)(x","epsi):{y in X:d(x","y) < epsi}","x in X","epsi > 0.:}\begin{equation*} B^{+}(x, \varepsilon):\{y \in X: d(x, y)<\varepsilon\}, x \in X, \varepsilon>0 . \tag{1.1} \end{equation*}(1.1)B+(x,ε):{y∈X:d(x,y)<ε},x∈X,ε>0.
This topology is called the forward topology of X ( [ 4 ] ) , [ 1 ] ) X ( [ 4 ] ) , [ 1 ] ) X([4]),[1])X([4]),[1])X([4]),[1]) and is denoted also by τ + τ + tau_(+)\tau_{+}τ+.
The topology induced by the quasi-metric d ¯ d ¯ bar(d)\bar{d}d¯ is called the backward topology and is denoted by τ − τ − tau_(-)\tau_{-}τ−.
The topology τ + τ + tau_(+)\tau_{+}τ+is a T 0 T 0 T_(0)T_{0}T0 topology. If the condition AM1) is replaced by the condition
AM0) d ( x , y ) ≥ 0 d ( x , y ) ≥ 0 quad d(x,y) >= 0\quad d(x, y) \geq 0d(x,y)≥0 and d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0, for all x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X then the topology τ + τ + tau_(+)\tau_{+}τ+is a T 1 T 1 T_(1)T_{1}T1 topology.
Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be quasi-metric space. A sequence ( x k ) k ≥ 1 d x k k ≥ 1 d (x_(k))_(k >= 1)d\left(x_{k}\right)_{k \geq 1} d(xk)k≥1d-converge to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, respectively d ¯ d ¯ bar(d)\bar{d}d¯-converge to x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, iff
(1.2) lim k → ∞ d ( x 0 , x k ) = 0 , respectively lim k → ∞ d ¯ ( x 0 , x k ) = lim k → ∞ d ( x k , x 0 ) = 0 . (1.2) lim k → ∞   d x 0 , x k = 0 ,  respectively  lim k → ∞   d ¯ x 0 , x k = lim k → ∞   d x k , x 0 = 0 . {:[(1.2)lim_(k rarr oo)d(x_(0),x_(k))=0","" respectively "],[lim_(k rarr oo) bar(d)(x_(0),x_(k))=lim_(k rarr oo)d(x_(k),x_(0))=0.]:}\begin{align*} & \lim _{k \rightarrow \infty} d\left(x_{0}, x_{k}\right)=0, \text { respectively } \tag{1.2}\\ & \lim _{k \rightarrow \infty} \bar{d}\left(x_{0}, x_{k}\right)=\lim _{k \rightarrow \infty} d\left(x_{k}, x_{0}\right)=0 . \end{align*}(1.2)limk→∞d(x0,xk)=0, respectively limk→∞d¯(x0,xk)=limk→∞d(xk,x0)=0.
A A AAA set K ⊂ X K ⊂ X K sub XK \subset XK⊂X is called d d ddd - compact if every open cover of K K KKK with respect to the topology τ + τ + tau_(+)\tau_{+}τ+has a finite subcover. We say that K K KKK is d d ddd-sequentially compact if every sequence in K K KKK has a d d ddd-converget subsequence with limit in K K KKK (Definition 4.1 in [1]).
Finally, the set Y Y YYY in ( X , d X , d X,dX, dX,d ) is called ( d , d ¯ d , d ¯ d, bar(d)d, \bar{d}d,d¯ )-sequentially compact if every sequence ( y n ) n ≥ 1 y n n ≥ 1 (y_(n))_(n >= 1)\left(y_{n}\right)_{n \geq 1}(yn)n≥1 in Y Y YYY has a subsequence ( y n k ) k ≥ 1 d y n k k ≥ 1 d (y_(n_(k)))_(k >= 1)d\left(y_{n_{k}}\right)_{k \geq 1} d(ynk)k≥1d-convergent to u ∈ Y u ∈ Y u in Yu \in Yu∈Y and d d ddd-convergent to v ∈ Y v ∈ Y v in Yv \in Yv∈Y. By Lemma 3.1 in [1] if the topology of X X XXX is T 1 T 1 T_(1)T_{1}T1, i.e. d d ddd verifies the axioms AM0) and AM2), it follows that u = v u = v u=vu=vu=v.
The following definition of a d d ddd-semi-Hölder function (of exponent α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) ) is inspired by the definition of semi-Lipschitz function in [9].
Definition 1. Let Y Y YYY be a non-empty subset of a quasi-metric space ( X , d X , d X,dX, dX,d ), and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) arbitrarily chosen, but fixed. A function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is called d d ddd-semi-Hölder (of exponent α α alpha\alphaα ) if there exists L = L ( f , Y ) ≥ 0 L = L ( f , Y ) ≥ 0 L=L(f,Y) >= 0L=L(f, Y) \geq 0L=L(f,Y)≥0 (named a d d ddd -semi-Hölder constant for f f fff ) such that
(1.3) f ( x ) − f ( y ) ≤ L d α ( x , y ) , (1.3) f ( x ) − f ( y ) ≤ L d α ( x , y ) , {:(1.3)f(x)-f(y) <= Ld^(alpha)(x","y)",":}\begin{equation*} f(x)-f(y) \leq L d^{\alpha}(x, y), \tag{1.3} \end{equation*}(1.3)f(x)−f(y)≤Ldα(x,y),
for all x , y ∈ Y x , y ∈ Y x,y in Yx, y \in Yx,y∈Y.
The smallest d d ddd-semi-Hölder constant for f f fff, verifying (1.3), is denoted by ‖ f | α , Y ‖ f α , Y ||f|_(alpha,Y)\|\left. f\right|_{\alpha, Y}‖f|α,Y and
(1.4) ‖ f | α , Y = sup { ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 , x , y ∈ Y } . (1.4) ‖ f α , Y = sup ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 , x , y ∈ Y . {:(1.4)||f|_(alpha,Y)=s u p{((f(x)-f(y))vv0)/(d^(alpha)(x,y)):d(x,y) > 0,quad x,y in Y}.:}\begin{equation*} \|\left. f\right|_{\alpha, Y}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d^{\alpha}(x, y)}: d(x, y)>0, \quad x, y \in Y\right\} . \tag{1.4} \end{equation*}(1.4)‖f|α,Y=sup{(f(x)−f(y))∨0dα(x,y):d(x,y)>0,x,y∈Y}.
This means that ‖ f | α , Y = inf { L ≥ 0 : L ‖ f α , Y = inf { L ≥ 0 : L ||f|_(alpha,Y)=i n f{L >= 0:L\|\left. f\right|_{\alpha, Y}=\inf \{L \geq 0: L‖f|α,Y=inf{L≥0:L verifying (1.3) } } }\}}.
The set of all d d ddd-semi-Hölder function f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R is denoted by Λ α ( Y ) Λ α ( Y ) Lambda_(alpha)(Y)\Lambda_{\alpha}(Y)Λα(Y), i.e.
(1.5) Λ α ( Y ) := { f : Y → R , f is d -semi-Hölder of exponent α } . (1.5) Λ α ( Y ) := { f : Y → R , f  is  d -semi-Hölder of exponent  α } . {:(1.5)Lambda_(alpha)(Y):={f:Y rarrR","f" is "d"-semi-Hölder of exponent "alpha}.:}\begin{equation*} \Lambda_{\alpha}(Y):=\{f: Y \rightarrow \mathbb{R}, f \text { is } d \text {-semi-Hölder of exponent } \alpha\} . \tag{1.5} \end{equation*}(1.5)Λα(Y):={f:Y→R,f is d-semi-Hölder of exponent α}.
This set is a cone in the linear space R Y R Y R^(Y)\mathbb{R}^{Y}RY of all functions f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R, i.e. Λ α ( Y ) Λ α ( Y ) Lambda_(alpha)(Y)\Lambda_{\alpha}(Y)Λα(Y) is closed with respect to pointwise operations of multiplication with nonnegative real numbers of a function in Λ α ( Y ) Λ α ( Y ) Lambda_(alpha)(Y)\Lambda_{\alpha}(Y)Λα(Y), and of addition of two functions in Λ α ( Y ) Λ α ( Y ) Lambda_(alpha)(Y)\Lambda_{\alpha}(Y)Λα(Y).
The functional ‖ ⋅ | α , Y : Λ α ( Y ) → [ 0 , ∞ ) ‖ ⋅ α , Y : Λ α ( Y ) → [ 0 , ∞ ) ||*|_(alpha,Y):Lambda_(alpha)(Y)rarr[0,oo)\|\left.\cdot\right|_{\alpha, Y}: \Lambda_{\alpha}(Y) \rightarrow[0, \infty)‖⋅|α,Y:Λα(Y)→[0,∞) is nonnegative and sublinear, and the pair ( Λ α ( Y ) , ‖ ⋅ | α , Y Λ α ( Y ) , ‖ ⋅ α , Y Lambda_(alpha)(Y),||*|_(alpha,Y)\Lambda_{\alpha}(Y), \|\left.\cdot\right|_{\alpha, Y}Λα(Y),‖⋅|α,Y ) is called the asymmetric normed cone of d d ddd-semi Hölder functions on Y Y YYY (compare with [10]).
The cone ( Λ α ( Y ) , ‖ ⋅ | α , Y Λ α ( Y ) , ‖ ⋅ α , Y Lambda_(alpha)(Y),||*|_(alpha,Y)\Lambda_{\alpha}(Y), \|\left.\cdot\right|_{\alpha, Y}Λα(Y),‖⋅|α,Y ) is different from the cone of d d ddd-semi-Lipschitz function ( α = 1 ) ( α = 1 ) (alpha=1)(\alpha=1)(α=1) considered in [9]. For example, if one considers Y = [ 0 , 1 ] d ( x , y ) = | x − y | Y = [ 0 , 1 ] d ( x , y ) = | x − y | Y=[0,1]d(x,y)=|x-y|Y=[0,1] d(x, y)=|x-y|Y=[0,1]d(x,y)=|x−y| and f : [ 0 , 1 ] → R , f ( x ) = x sin ⁡ 1 x , x ∈ ( 0 , 1 ] ; f ( 0 ) = 0 f : [ 0 , 1 ] → R , f ( x ) = x sin ⁡ 1 x , x ∈ ( 0 , 1 ] ; f ( 0 ) = 0 f:[0,1]rarrR,f(x)=x sin((1)/(x)),x in(0,1];f(0)=0f:[0,1] \rightarrow \mathbb{R}, f(x)=x \sin \frac{1}{x}, x \in(0,1] ; f(0)=0f:[0,1]→R,f(x)=xsin⁡1x,x∈(0,1];f(0)=0, then it is known that f ∈ Λ α ( X , d ) f ∈ Λ α ( X , d ) f inLambda_(alpha)(X,d)f \in \Lambda_{\alpha}(X, d)f∈Λα(X,d) if and only if α ∈ ( 0 , 1 / 2 ] α ∈ ( 0 , 1 / 2 ] alpha in(0,1//2]\alpha \in(0,1 / 2]α∈(0,1/2] (see[14], Problem 153) and in this case ‖ f | α , Y ≤ { 1 + 2 ln ⁡ ( 1 + 2 π ) + 2 π } 1 / 2 ‖ f α , Y ≤ { 1 + 2 ln ⁡ ( 1 + 2 π ) + 2 π } 1 / 2 ||f|_(alpha,Y) <= {1+2ln(1+2pi)+2pi}^(1//2)\|\left. f\right|_{\alpha, Y} \leq\{1+2 \ln (1+2 \pi)+2 \pi\}^{1 / 2}‖f|α,Y≤{1+2ln⁡(1+2π)+2π}1/2.

2. EXTENSIONS OF d-SEMI-HÖLDER FUNCTIONS

Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X and f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y). A function F ∈ Λ α ( X ) F ∈ Λ α ( X ) F inLambda_(alpha)(X)F \in \Lambda_{\alpha}(X)F∈Λα(X) is called an extension of f f fff (preserving the semi-Hölder constant L ( f , Y ) L ( f , Y ) L(f,Y)L(f, Y)L(f,Y) if
(2.1) F | Y = f and L ( F , X ) = L ( f , Y ) (2.1) F Y = f  and  L ( F , X ) = L ( f , Y ) {:(2.1)F|_(Y)=f" and "L(F","X)=L(f","Y):}\begin{equation*} \left.F\right|_{Y}=f \text { and } L(F, X)=L(f, Y) \tag{2.1} \end{equation*}(2.1)F|Y=f and L(F,X)=L(f,Y)
The existence of extension in Λ α ( X ) Λ α ( X ) Lambda_(alpha)(X)\Lambda_{\alpha}(X)Λα(X) for each f ∈ Λ α ( X ) f ∈ Λ α ( X ) f inLambda_(alpha)(X)f \in \Lambda_{\alpha}(X)f∈Λα(X) is assured by the following theorem.
Theorem 2. Let ( X , d X , d X,dX, dX,d ) be a quasi-metric space, Y ⊂ X Y ⊂ X Y sub XY \subset XY⊂X and f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y) with d d ddd-semi-Hölder constant L ( f , Y ) L ( f , Y ) L(f,Y)L(f, Y)L(f,Y). Then there exist F ∈ Λ α ( X ) F ∈ Λ α ( X ) F inLambda_(alpha)(X)F \in \Lambda_{\alpha}(X)F∈Λα(X) such that
F | Y = f and L ( F , X ) = L ( f , Y ) F Y = f  and  L ( F , X ) = L ( f , Y ) F|_(Y)=f" and "L(F,X)=L(f,Y)\left.F\right|_{Y}=f \text { and } L(F, X)=L(f, Y)F|Y=f and L(F,X)=L(f,Y)
Proof. Let G : X → R G : X → R G:X rarrRG: X \rightarrow \mathbb{R}G:X→R defined by
(2.2) G ( x ) = sup y ∈ Y { f ( y ) − L ( f , Y ) ⋅ d α ( y , x ) } , x ∈ X (2.2) G ( x ) = sup y ∈ Y   f ( y ) − L ( f , Y ) ⋅ d α ( y , x ) , x ∈ X {:(2.2)G(x)=s u p_(y in Y){f(y)-L(f,Y)*d^(alpha)(y,x)}","x in X:}\begin{equation*} G(x)=\sup _{y \in Y}\left\{f(y)-L(f, Y) \cdot d^{\alpha}(y, x)\right\}, x \in X \tag{2.2} \end{equation*}(2.2)G(x)=supy∈Y{f(y)−L(f,Y)⋅dα(y,x)},x∈X
Let y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y be a fixed element, and x ∈ X x ∈ X x in Xx \in Xx∈X. For every y ∈ Y y ∈ Y y in Yy \in Yy∈Y,
f ( y ) − L ( f , Y ) ⋅ d α ( y , x ) = f ( y ) − f ( y 0 ) − L ( f , Y ) d α ( y , x ) + f ( y 0 ) ≤ L ( f , Y ) d α ( y , y 0 ) − L ( f , Y ) d α ( y , x ) + f ( y 0 ) = f ( y 0 ) + L ( f , Y ) [ d α ( y , y 0 ) − d α ( y , x ) ] ≤ f ( y 0 ) + L ( f , Y ) d α ( x , y 0 ) f ( y ) − L ( f , Y ) ⋅ d α ( y , x ) = f ( y ) − f y 0 − L ( f , Y ) d α ( y , x ) + f y 0 ≤ L ( f , Y ) d α y , y 0 − L ( f , Y ) d α ( y , x ) + f y 0 = f y 0 + L ( f , Y ) d α y , y 0 − d α ( y , x ) ≤ f y 0 + L ( f , Y ) d α x , y 0 {:[f(y)-L(f","Y)*d^(alpha)(y","x)=f(y)-f(y_(0))-L(f","Y)d^(alpha)(y","x)+f(y_(0))],[ <= L(f","Y)d^(alpha)(y,y_(0))-L(f","Y)d^(alpha)(y","x)+f(y_(0))],[=f(y_(0))+L(f","Y)[d^(alpha)(y,y_(0))-d^(alpha)(y,x)]],[ <= f(y_(0))+L(f","Y)d^(alpha)(x,y_(0))]:}\begin{aligned} f(y)-L(f, Y) \cdot d^{\alpha}(y, x) & =f(y)-f\left(y_{0}\right)-L(f, Y) d^{\alpha}(y, x)+f\left(y_{0}\right) \\ & \leq L(f, Y) d^{\alpha}\left(y, y_{0}\right)-L(f, Y) d^{\alpha}(y, x)+f\left(y_{0}\right) \\ & =f\left(y_{0}\right)+L(f, Y)\left[d^{\alpha}\left(y, y_{0}\right)-d^{\alpha}(y, x)\right] \\ & \leq f\left(y_{0}\right)+L(f, Y) d^{\alpha}\left(x, y_{0}\right) \end{aligned}f(y)−L(f,Y)⋅dα(y,x)=f(y)−f(y0)−L(f,Y)dα(y,x)+f(y0)≤L(f,Y)dα(y,y0)−L(f,Y)dα(y,x)+f(y0)=f(y0)+L(f,Y)[dα(y,y0)−dα(y,x)]≤f(y0)+L(f,Y)dα(x,y0)
Then it follows that the set
{ f ( y ) − L ( f , Y ) d α ( y , x ) : y ∈ Y } f ( y ) − L ( f , Y ) d α ( y , x ) : y ∈ Y {f(y)-L(f,Y)d^(alpha)(y,x):y in Y}\left\{f(y)-L(f, Y) d^{\alpha}(y, x): y \in Y\right\}{f(y)−L(f,Y)dα(y,x):y∈Y}
is bounded from above, and G ( x ) G ( x ) G(x)G(x)G(x) exists for every x ∈ X x ∈ X x in Xx \in Xx∈X. By the definition of G ( x ) G ( x ) G(x)G(x)G(x), for every y ∈ Y y ∈ Y y in Yy \in Yy∈Y
G ( x ) ≥ f ( y ) − L ( f , Y ) d α ( y , x ) , x ∈ X G ( x ) ≥ f ( y ) − L ( f , Y ) d α ( y , x ) , x ∈ X G(x) >= f(y)-L(f,Y)d^(alpha)(y,x),x in XG(x) \geq f(y)-L(f, Y) d^{\alpha}(y, x), x \in XG(x)≥f(y)−L(f,Y)dα(y,x),x∈X
and for x = y x = y x=yx=yx=y one obtains
G ( y ) ≥ f ( y ) . G ( y ) ≥ f ( y ) . G(y) >= f(y).G(y) \geq f(y) .G(y)≥f(y).
On the other hand, for y ∈ Y y ∈ Y y in Yy \in Yy∈Y and every y ′ ∈ Y y ′ ∈ Y y^(')in Yy^{\prime} \in Yy′∈Y,
f ( y ′ ) − f ( y ) ≤ L ( f , Y ) ⋅ d α ( y ′ , y ) . f y ′ − f ( y ) ≤ L ( f , Y ) ⋅ d α y ′ , y . f(y^('))-f(y) <= L(f,Y)*d^(alpha)(y^('),y).f\left(y^{\prime}\right)-f(y) \leq L(f, Y) \cdot d^{\alpha}\left(y^{\prime}, y\right) .f(y′)−f(y)≤L(f,Y)⋅dα(y′,y).
It follows
f ( y ′ ) − L ( f , Y ) ⋅ d α ( y ′ , y ) ≤ f ( y ) f y ′ − L ( f , Y ) ⋅ d α y ′ , y ≤ f ( y ) f(y^('))-L(f,Y)*d^(alpha)(y^('),y) <= f(y)f\left(y^{\prime}\right)-L(f, Y) \cdot d^{\alpha}\left(y^{\prime}, y\right) \leq f(y)f(y′)−L(f,Y)⋅dα(y′,y)≤f(y)
and taking the supremum with respect to y ′ ∈ Y y ′ ∈ Y y^(')in Yy^{\prime} \in Yy′∈Y one obtains
G ( y ) ≤ f ( y ) , y ∈ Y G ( y ) ≤ f ( y ) , y ∈ Y G(y) <= f(y),y in YG(y) \leq f(y), y \in YG(y)≤f(y),y∈Y
Consequently G | Y = f G Y = f G|_(Y)=f\left.G\right|_{Y}=fG|Y=f.
Now, let u , v ∈ X u , v ∈ X u,v in Xu, v \in Xu,v∈X and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Choosing y ∈ Y y ∈ Y y in Yy \in Yy∈Y such that
G ( u ) ≤ f ( y ) − L ( f , Y ) d α ( y , u ) + ε G ( u ) ≤ f ( y ) − L ( f , Y ) d α ( y , u ) + ε G(u) <= f(y)-L(f,Y)d^(alpha)(y,u)+epsiG(u) \leq f(y)-L(f, Y) d^{\alpha}(y, u)+\varepsilonG(u)≤f(y)−L(f,Y)dα(y,u)+ε
it follows
G ( u ) − G ( v ) ≤ f ( y ) − L ( f , Y ) d α ( y , u ) + ε − f ( y ) + L ( f , Y ) d α ( y , v ) = L ( f , Y ) [ d α ( y , v ) − d α ( y , u ) ] + ε ≤ L ( f , Y ) d α ( u , v ) + ε G ( u ) − G ( v ) ≤ f ( y ) − L ( f , Y ) d α ( y , u ) + ε − f ( y ) + L ( f , Y ) d α ( y , v ) = L ( f , Y ) d α ( y , v ) − d α ( y , u ) + ε ≤ L ( f , Y ) d α ( u , v ) + ε {:[G(u)-G(v) <= f(y)-L(f","Y)d^(alpha)(y","u)+epsi-f(y)+L(f","Y)d^(alpha)(y","v)],[=L(f","Y)[d^(alpha)(y,v)-d^(alpha)(y,u)]+epsi],[ <= L(f","Y)d^(alpha)(u","v)+epsi]:}\begin{aligned} G(u)-G(v) & \leq f(y)-L(f, Y) d^{\alpha}(y, u)+\varepsilon-f(y)+L(f, Y) d^{\alpha}(y, v) \\ & =L(f, Y)\left[d^{\alpha}(y, v)-d^{\alpha}(y, u)\right]+\varepsilon \\ & \leq L(f, Y) d^{\alpha}(u, v)+\varepsilon \end{aligned}G(u)−G(v)≤f(y)−L(f,Y)dα(y,u)+ε−f(y)+L(f,Y)dα(y,v)=L(f,Y)[dα(y,v)−dα(y,u)]+ε≤L(f,Y)dα(u,v)+ε
Because ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 is arbitrarily chosen, one obtains:
G ( u ) − G ( v ) ≤ L ( f , Y ) d α ( u , v ) G ( u ) − G ( v ) ≤ L ( f , Y ) d α ( u , v ) G(u)-G(v) <= L(f,Y)d^(alpha)(u,v)G(u)-G(v) \leq L(f, Y) d^{\alpha}(u, v)G(u)−G(v)≤L(f,Y)dα(u,v)
for u , v ∈ X u , v ∈ X u,v in Xu, v \in Xu,v∈X, i.e. G ∈ Λ α ( X ) G ∈ Λ α ( X ) G inLambda_(alpha)(X)G \in \Lambda_{\alpha}(X)G∈Λα(X). Moreover L ( G , X ) ≤ L ( f , Y ) L ( G , X ) ≤ L ( f , Y ) L(G,X) <= L(f,Y)L(G, X) \leq L(f, Y)L(G,X)≤L(f,Y). Because G | Y = f G Y = f G|_(Y)=f\left.G\right|_{Y}=fG|Y=f one obtains also
L ( f , Y ) = L ( G | Y , Y ) ≤ L ( G , X ) L ( f , Y ) = L G Y , Y ≤ L ( G , X ) L(f,Y)=L(G|_(Y),Y) <= L(G,X)L(f, Y)=L\left(\left.G\right|_{Y}, Y\right) \leq L(G, X)L(f,Y)=L(G|Y,Y)≤L(G,X)
and consequently, L ( f , Y ) = L ( G , X ) L ( f , Y ) = L ( G , X ) L(f,Y)=L(G,X)L(f, Y)=L(G, X)L(f,Y)=L(G,X).
Remark 3. Observe that the function F : X → R F : X → R F:X rarrRF: X \rightarrow \mathbb{R}F:X→R,
(2.3) F ( x ) = inf y ∈ Y { f ( y ) + L ( f , Y ) d α ( x , y ) } , x ∈ X (2.3) F ( x ) = inf y ∈ Y   f ( y ) + L ( f , Y ) d α ( x , y ) , x ∈ X {:(2.3)F(x)=i n f_(y in Y){f(y)+L(f,Y)d^(alpha)(x,y)}","x in X:}\begin{equation*} F(x)=\inf _{y \in Y}\left\{f(y)+L(f, Y) d^{\alpha}(x, y)\right\}, x \in X \tag{2.3} \end{equation*}(2.3)F(x)=infy∈Y{f(y)+L(f,Y)dα(x,y)},x∈X
is another extension of f ∈ Λ α ( f , Y ) f ∈ Λ α ( f , Y ) f inLambda_(alpha)(f,Y)f \in \Lambda_{\alpha}(f, Y)f∈Λα(f,Y).
Moreover, if H H HHH is any extension of f f fff, i.e. H | Y = f H Y = f H|_(Y)=f\left.H\right|_{Y}=fH|Y=f and L ( H , X ) = L ( f , Y ) L ( H , X ) = L ( f , Y ) L(H,X)=L(f,Y)L(H, X)=L(f, Y)L(H,X)=L(f,Y) then
(2.4) G ( x ) ≤ H ( x ) ≤ F ( x ) , x ∈ X (2.4) G ( x ) ≤ H ( x ) ≤ F ( x ) , x ∈ X {:(2.4)G(x) <= H(x) <= F(x)","x in X:}\begin{equation*} G(x) \leq H(x) \leq F(x), x \in X \tag{2.4} \end{equation*}(2.4)G(x)≤H(x)≤F(x),x∈X
where G G GGG is defined by (2.2) and F F FFF is defined by (2.3).
From (2.4) it follows that G G GGG is the minimal extension of f f fff, and F F FFF is the maximal extension of f f fff.
Indeed let H H HHH an extension of f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y). Then, 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 ) ≤ L ( f , Y ) d α ( x , y ) H ( x ) − H ( y ) ≤ L ( f , Y ) d α ( x , y ) H(x)-H(y) <= L(f,Y)d^(alpha)(x,y)H(x)-H(y) \leq L(f, Y) d^{\alpha}(x, y)H(x)−H(y)≤L(f,Y)dα(x,y)
implying
H ( x ) ≤ H ( y ) + L ( f , Y ) d α ( x , y ) = f ( y ) + L ( f , y ) d α ( x , y ) H ( x ) ≤ H ( y ) + L ( f , Y ) d α ( x , y ) = f ( y ) + L ( f , y ) d α ( x , y ) {:[H(x) <= H(y)+L(f","Y)d^(alpha)(x","y)],[=f(y)+L(f","y)d^(alpha)(x","y)]:}\begin{aligned} H(x) & \leq H(y)+L(f, Y) d^{\alpha}(x, y) \\ & =f(y)+L(f, y) d^{\alpha}(x, y) \end{aligned}H(x)≤H(y)+L(f,Y)dα(x,y)=f(y)+L(f,y)dα(x,y)
Taking the infimum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y one obtain
H ( x ) ≤ F ( x ) , x ∈ X H ( x ) ≤ F ( x ) , x ∈ X H(x) <= F(x),x in XH(x) \leq F(x), x \in XH(x)≤F(x),x∈X
Analogously,
H ( y ) − H ( x ) ≤ L ( f , Y ) ⋅ d α ( y , x ) H ( y ) − H ( x ) ≤ L ( f , Y ) ⋅ d α ( y , x ) H(y)-H(x) <= L(f,Y)*d^(alpha)(y,x)H(y)-H(x) \leq L(f, Y) \cdot d^{\alpha}(y, x)H(y)−H(x)≤L(f,Y)⋅dα(y,x)
implies
f ( y ) − L ( f , Y ) d α ( y , x ) ≥ H ( x ) f ( y ) − L ( f , Y ) d α ( y , x ) ≥ H ( x ) f(y)-L(f,Y)d^(alpha)(y,x) >= H(x)f(y)-L(f, Y) d^{\alpha}(y, x) \geq H(x)f(y)−L(f,Y)dα(y,x)≥H(x)
and taking the supremum with respect to y ∈ Y y ∈ Y y in Yy \in Yy∈Y one obtain
G ( x ) ≥ H ( x ) , x ∈ X G ( x ) ≥ H ( x ) , x ∈ X G(x) >= H(x),x in XG(x) \geq H(x), x \in XG(x)≥H(x),x∈X
It follows (2.4).
REMARK 4. For f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y) denote by E ( f ) E ( f ) E(f)\mathcal{E}(f)E(f) the (non-empty) set of all extensions of f f fff i.e.
(2.5) E ( f ) := { H ∈ Λ α ( X ) : H | Y = f and L ( H , X ) = L ( f , Y ) } (2.5) E ( f ) := H ∈ Λ α ( X ) : H Y = f  and  L ( H , X ) = L ( f , Y ) {:(2.5)E(f):={H inLambda_(alpha)(X):H|_(Y)=f" and "L(H,X)=L(f,Y)}:}\begin{equation*} \mathcal{E}(f):=\left\{H \in \Lambda_{\alpha}(X):\left.H\right|_{Y}=f \text { and } L(H, X)=L(f, Y)\right\} \tag{2.5} \end{equation*}(2.5)E(f):={H∈Λα(X):H|Y=f and L(H,X)=L(f,Y)}
Obviously, E ( f ) E ( f ) E(f)\mathcal{E}(f)E(f) is a convex subset of the cone Λ α ( X ) Λ α ( X ) Lambda_(alpha)(X)\Lambda_{\alpha}(X)Λα(X).
Remark 5. Let f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y) and let ‖ f | α ⋅ Y ‖ f α ⋅ Y ||f|_(alpha*Y)\|\left. f\right|_{\alpha \cdot Y}‖f|α⋅Y be the smallest d d ddd-semi-Hölder constant for f f fff on Y Y YYY (see (1.4)). Then the functions G G GGG and F F FFF defined by (2.2) and (2.3), where L ( f , Y ) = ‖ f | α , Y L ( f , Y ) = ‖ f α , Y L(f,Y)=||f|_(alpha,Y)L(f, Y)=\|\left. f\right|_{\alpha, Y}L(f,Y)=‖f|α,Y are extensions for f f fff, preserving the constant ‖ f | α , Y ‖ f α , Y ||f|_(alpha,Y)\|\left. f\right|_{\alpha, Y}‖f|α,Y.
Consider a fixed element y 0 ∈ Y y 0 ∈ Y y_(0)in Yy_{0} \in Yy0∈Y, and let
(2.6) Λ α , 0 ( Y ) := { f ∈ Λ α ( Y ) : f ( y 0 ) = 0 } . (2.6) Λ α , 0 ( Y ) := f ∈ Λ α ( Y ) : f y 0 = 0 . {:(2.6)Lambda_(alpha,0)(Y):={f inLambda_(alpha)(Y):f(y_(0))=0}.:}\begin{equation*} \Lambda_{\alpha, 0}(Y):=\left\{f \in \Lambda_{\alpha}(Y): f\left(y_{0}\right)=0\right\} . \tag{2.6} \end{equation*}(2.6)Λα,0(Y):={f∈Λα(Y):f(y0)=0}.
Then the functional ‖ | α , Y : Λ α , 0 ( Y ) → [ 0 , ∞ ) ‖ α , Y : Λ α , 0 ( Y ) → [ 0 , ∞ ) |||_(alpha,Y):Lambda_(alpha,0)(Y)rarr[0,oo)\|\left.\right|_{\alpha, Y}: \Lambda_{\alpha, 0}(Y) \rightarrow[0, \infty)‖|α,Y:Λα,0(Y)→[0,∞) defined by
(2.7) ‖ f | α , Y = sup { ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y } (2.7) ‖ f α , Y = sup ( f ( x ) − f ( y ) ) ∨ 0 d α ( x , y ) : d ( x , y ) > 0 ; x , y ∈ Y {:(2.7)||f|_(alpha,Y)=s u p{((f(x)-f(y))vv0)/(d^(alpha)(x,y)):d(x,y) > 0;x,y in Y}:}\begin{equation*} \|\left. f\right|_{\alpha, Y}=\sup \left\{\frac{(f(x)-f(y)) \vee 0}{d^{\alpha}(x, y)}: d(x, y)>0 ; x, y \in Y\right\} \tag{2.7} \end{equation*}(2.7)‖f|α,Y=sup{(f(x)−f(y))∨0dα(x,y):d(x,y)>0;x,y∈Y}
is a quasi-norm on Λ α , 0 ( Y ) Λ α , 0 ( Y ) Lambda_(alpha,0)(Y)\Lambda_{\alpha, 0}(Y)Λα,0(Y), i.e. the following properties hold:
a) ‖ f | α , Y ≥ 0 ; f = 0 ‖ f α , Y ≥ 0 ; f = 0 ||f|_(alpha,Y) >= 0;f=0\|\left. f\right|_{\alpha, Y} \geq 0 ; f=0‖f|α,Y≥0;f=0 iff − f ∈ Λ α , 0 ( Y ) − f ∈ Λ α , 0 ( Y ) -f inLambda_(alpha,0)(Y)-f \in \Lambda_{\alpha, 0}(Y)−f∈Λα,0(Y) and ‖ f | α , Y = ‖ − f | α , Y = 0 , f ∈ Λ α , 0 ( Y ) ; f α , Y = − f α , Y = 0 , f ∈ Λ α , 0 ( Y ) ; ||f|_(alpha,Y)=||-f|_(alpha,Y)=0,f inLambda_(alpha,0)(Y);\left\|\left.f\right|_{\alpha, Y}=\right\|-\left.f\right|_{\alpha, Y}=0, f \in \Lambda_{\alpha, 0}(Y) ;‖f|α,Y=‖−f|α,Y=0,f∈Λα,0(Y);
b) ‖ a f | α , Y = a ‖ f | α , Y a f α , Y = a f α , Y ||af|_(alpha,Y)=a||f|_(alpha,Y)\left.\left\|\left.a f\right|_{\alpha, Y}=a\right\| f\right|_{\alpha, Y}‖af|α,Y=a‖f|α,Y for every f ∈ Λ α , 0 ( Y ) f ∈ Λ α , 0 ( Y ) f inLambda_(alpha,0)(Y)f \in \Lambda_{\alpha, 0}(Y)f∈Λα,0(Y) and a ≥ 0 a ≥ 0 a >= 0a \geq 0a≥0;
c) ‖ f + g | α , Y ≤ ‖ f | α , Y + ‖ g | α , Y f + g α , Y ≤ f α , Y + ‖ g α , Y ||f+g|_(alpha,Y) <= ||f|_(alpha,Y)+||g|_(alpha,Y)\left.\left\|f+\left.g\right|_{\alpha, Y} \leq\right\| f\right|_{\alpha, Y}+\|\left. g\right|_{\alpha, Y}‖f+g|α,Y≤‖f|α,Y+‖g|α,Y for all f , g ∈ Λ α , 0 ( Y ) f , g ∈ Λ α , 0 ( Y ) f,g inLambda_(alpha,0)(Y)f, g \in \Lambda_{\alpha, 0}(Y)f,g∈Λα,0(Y).
Remark 6. Let f ∈ Λ α ( X ) f ∈ Λ α ( X ) f inLambda_(alpha)(X)f \in \Lambda_{\alpha}(X)f∈Λα(X) and Y 1 ⊂ Y 2 ⊂ X Y 1 ⊂ Y 2 ⊂ X Y_(1)subY_(2)sub XY_{1} \subset Y_{2} \subset XY1⊂Y2⊂X. Suppose that q ≥ ‖ f | α , X q ≥ ‖ f α , X q >= ||f|_(alpha,X)q \geq \|\left. f\right|_{\alpha, X}q≥‖f|α,X and let
G 1 ( x ) = sup y ∈ Y 1 { f ( y ) − q d α ( y , x ) } , x ∈ X , G 2 ( x ) = sup y ∈ Y 2 { f ( y ) − q d α ( y , x ) } , x ∈ X G 1 ( x ) = sup y ∈ Y 1   f ( y ) − q d α ( y , x ) , x ∈ X , G 2 ( x ) = sup y ∈ Y 2   f ( y ) − q d α ( y , x ) , x ∈ X {:[G_(1)(x)=s u p_(y inY_(1)){f(y)-qd^(alpha)(y,x)}","x in X","],[G_(2)(x)=s u p_(y inY_(2)){f(y)-qd^(alpha)(y,x)}","x in X]:}\begin{aligned} & G_{1}(x)=\sup _{y \in Y_{1}}\left\{f(y)-q d^{\alpha}(y, x)\right\}, x \in X, \\ & G_{2}(x)=\sup _{y \in Y_{2}}\left\{f(y)-q d^{\alpha}(y, x)\right\}, x \in X \end{aligned}G1(x)=supy∈Y1{f(y)−qdα(y,x)},x∈X,G2(x)=supy∈Y2{f(y)−qdα(y,x)},x∈X
Then G 1 ( x ) ≤ G 2 ( x ) ≤ f ( x ) G 1 ( x ) ≤ G 2 ( x ) ≤ f ( x ) G_(1)(x) <= G_(2)(x) <= f(x)G_{1}(x) \leq G_{2}(x) \leq f(x)G1(x)≤G2(x)≤f(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X and G 1 | Y 1 = G 2 | Y 1 = f | Y 1 G 1 Y 1 = G 2 Y 1 = f Y 1 G_(1)|_(Y_(1))=G_(2)|_(Y_(1))=f|_(Y_(1))\left.G_{1}\right|_{Y_{1}}=\left.G_{2}\right|_{Y_{1}}=\left.f\right|_{Y_{1}}G1|Y1=G2|Y1=f|Y1.
Also, if
F 1 ( x ) = inf y ∈ Y 1 { f ( y ) + q d α ( x , y ) } , x ∈ X F 1 ( x ) = inf y ∈ Y 1   f ( y ) + q d α ( x , y ) , x ∈ X F_(1)(x)=i n f_(y inY_(1)){f(y)+qd^(alpha)(x,y)},x in XF_{1}(x)=\inf _{y \in Y_{1}}\left\{f(y)+q d^{\alpha}(x, y)\right\}, x \in XF1(x)=infy∈Y1{f(y)+qdα(x,y)},x∈X
and
F 2 ( x ) = inf y ∈ Y 2 { f ( y ) + q d α ( x , y ) } , x ∈ X F 2 ( x ) = inf y ∈ Y 2   f ( y ) + q d α ( x , y ) , x ∈ X F_(2)(x)=i n f_(y inY_(2)){f(y)+qd^(alpha)(x,y)},x in XF_{2}(x)=\inf _{y \in Y_{2}}\left\{f(y)+q d^{\alpha}(x, y)\right\}, x \in XF2(x)=infy∈Y2{f(y)+qdα(x,y)},x∈X
then
F 1 ( x ) ≥ F 2 ( x ) ≥ f ( x ) , for all x ∈ X F 1 ( x ) ≥ F 2 ( x ) ≥ f ( x ) ,  for all  x ∈ X F_(1)(x) >= F_(2)(x) >= f(x)," for all "x in XF_{1}(x) \geq F_{2}(x) \geq f(x), \text { for all } x \in XF1(x)≥F2(x)≥f(x), for all x∈X
and
F 1 | Y 1 = F 2 | Y 1 = f | Y 1 . F 1 Y 1 = F 2 Y 1 = f Y 1 . F_(1)|_(Y_(1))=F_(2)|_(Y_(1))=f|_(Y_(1)).\left.F_{1}\right|_{Y_{1}}=\left.F_{2}\right|_{Y_{1}}=\left.f\right|_{Y_{1}} .F1|Y1=F2|Y1=f|Y1.
Consequently, if ( Y n ) n ≥ 1 Y n n ≥ 1 (Y_(n))_(n >= 1)\left(Y_{n}\right)_{n \geq 1}(Yn)n≥1 is a sequence in 2 X 2 X 2^(X)2^{X}2X such that Y 1 ⊂ Y 2 ⊂ … ⊂ Y n ⊂ … Y 1 ⊂ Y 2 ⊂ … ⊂ Y n ⊂ … Y_(1)subY_(2)sub dots subY_(n)sub dotsY_{1} \subset Y_{2} \subset \ldots \subset Y_{n} \subset \ldotsY1⊂Y2⊂…⊂Yn⊂…, f ∈ Λ α , 0 ( X ) f ∈ Λ α , 0 ( X ) f inLambda_(alpha,0)(X)f \in \Lambda_{\alpha, 0}(X)f∈Λα,0(X) and q ≥ ‖ f | α , X q ≥ ‖ f α , X q >= ||f|_(alpha,X)q \geq \|\left. f\right|_{\alpha, X}q≥‖f|α,X then the sequences ( G n ) n ≥ 1 G n n ≥ 1 (G_(n))_(n >= 1)\left(G_{n}\right)_{n \geq 1}(Gn)n≥1 and ( F n ) n ≥ 1 F n n ≥ 1 (F_(n))_(n >= 1)\left(F_{n}\right)_{n \geq 1}(Fn)n≥1, where
G n ( x ) = sup y ∈ Y n { f ( y ) − q d α ( y , x ) } , x ∈ X G n ( x ) = sup y ∈ Y n   f ( y ) − q d α ( y , x ) , x ∈ X G_(n)(x)=s u p_(y inY_(n)){f(y)-qd^(alpha)(y,x)},x in XG_{n}(x)=\sup _{y \in Y_{n}}\left\{f(y)-q d^{\alpha}(y, x)\right\}, x \in XGn(x)=supy∈Yn{f(y)−qdα(y,x)},x∈X
and
F n ( x ) = inf y ∈ Y n { f ( y ) + q d α ( x , y ) } , x ∈ X F n ( x ) = inf y ∈ Y n   f ( y ) + q d α ( x , y ) , x ∈ X F_(n)(x)=i n f_(y inY_(n)){f(y)+qd^(alpha)(x,y)},x in XF_{n}(x)=\inf _{y \in Y_{n}}\left\{f(y)+q d^{\alpha}(x, y)\right\}, x \in XFn(x)=infy∈Yn{f(y)+qdα(x,y)},x∈X
are monotonically increasing, respectively decreasing, G n , F n ∈ Λ α ( X ) G n , F n ∈ Λ α ( X ) G_(n),F_(n)inLambda_(alpha)(X)G_{n}, F_{n} \in \Lambda_{\alpha}(X)Gn,Fn∈Λα(X), n = 1 , 2 , … n = 1 , 2 , … n=1,2,dotsn=1,2, \ldotsn=1,2,…, and G n ( x ) ≤ f ( x ) ≤ F n ( x ) G n ( x ) ≤ f ( x ) ≤ F n ( x ) G_(n)(x) <= f(x) <= F_(n)(x)G_{n}(x) \leq f(x) \leq F_{n}(x)Gn(x)≤f(x)≤Fn(x), for all x ∈ X x ∈ X x in Xx \in Xx∈X.
Because, for every y ∈ Y n y ∈ Y n y inY_(n)y \in Y_{n}y∈Yn
f ( y ) − q d α ( y , x ) ≤ G n ( x ) ≤ F n ( x ) ≤ f ( y ) + q d α ( x , y ) f ( y ) − q d α ( y , x ) ≤ G n ( x ) ≤ F n ( x ) ≤ f ( y ) + q d α ( x , y ) f(y)-qd^(alpha)(y,x) <= G_(n)(x) <= F_(n)(x) <= f(y)+qd^(alpha)(x,y)f(y)-q d^{\alpha}(y, x) \leq G_{n}(x) \leq F_{n}(x) \leq f(y)+q d^{\alpha}(x, y)f(y)−qdα(y,x)≤Gn(x)≤Fn(x)≤f(y)+qdα(x,y)
it follows that,
F n ( x ) − G n ( x ) ≤ q [ d α ( x , y ) + d α ( y , x ) ] . F n ( x ) − G n ( x ) ≤ q d α ( x , y ) + d α ( y , x ) . F_(n)(x)-G_(n)(x) <= q[d^(alpha)(x,y)+d^(alpha)(y,x)].F_{n}(x)-G_{n}(x) \leq q\left[d^{\alpha}(x, y)+d^{\alpha}(y, x)\right] .Fn(x)−Gn(x)≤q[dα(x,y)+dα(y,x)].
Taking the infimum with respect to y ∈ Y n y ∈ Y n y inY_(n)y \in Y_{n}y∈Yn one obtains:
(2.8) F n ( x ) − G n ( x ) ≤ q inf y ∈ Y n [ d α ( x , y ) + d α ( y , x ) ] (2.8) F n ( x ) − G n ( x ) ≤ q inf y ∈ Y n   d α ( x , y ) + d α ( y , x ) {:(2.8)F_(n)(x)-G_(n)(x) <= qi n f_(y inY_(n))[d^(alpha)(x,y)+d^(alpha)(y,x)]:}\begin{equation*} F_{n}(x)-G_{n}(x) \leq q \inf _{y \in Y_{n}}\left[d^{\alpha}(x, y)+d^{\alpha}(y, x)\right] \tag{2.8} \end{equation*}(2.8)Fn(x)−Gn(x)≤qinfy∈Yn[dα(x,y)+dα(y,x)]
for every x ∈ X x ∈ X x in Xx \in Xx∈X.
If Y n Y n Y_(n)Y_{n}Yn is d s d s d^(s)d^{s}ds-dense in X X XXX, where d s ( x , y ) = d ( x , y ) ∨ d ( y , x ) d s ( x , y ) = d ( x , y ) ∨ d ( y , x ) d^(s)(x,y)=d(x,y)vv d(y,x)d^{s}(x, y)=d(x, y) \vee d(y, x)ds(x,y)=d(x,y)∨d(y,x) for every x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X then, by (2.8) it follows that F n ( x ) = G n ( x ) , x ∈ X F n ( x ) = G n ( x ) , x ∈ X F_(n)(x)=G_(n)(x),x in XF_{n}(x)=G_{n}(x), x \in XFn(x)=Gn(x),x∈X. Consequently, a function f ∈ Λ α ( Y ) f ∈ Λ α ( Y ) f inLambda_(alpha)(Y)f \in \Lambda_{\alpha}(Y)f∈Λα(Y) where Y Y YYY is d s d s d^(s)d^{s}ds-dense in X X XXX has an unique extension F ∈ Λ α ( X ) F ∈ Λ α ( X ) F inLambda_(alpha)(X)F \in \Lambda_{\alpha}(X)F∈Λα(X).

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  1. *Institutul de Calcul "T. Popoviciu", Cluj-Napoca, Romania, e-mail: cmustata2001@ yahoo.com, cmustata@ictp.acad.ro.
2009

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