Applications to Analysis on Quasimetric Spaces
In this chapter we study the implications of our general metrization theory at the level of quasimetric spaces, with special emphasis on analytical aspects. More specifically, we study the nature of Hölder functions on quasimetric spaces by proving densit
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Applications to Analysis on Quasimetric Spaces
The metrization result contained in Theorem 3.46 leads to natural improvements for a large number of basic results in the area of analysis on quasimetric spaces, and in this chapter we single out several topics for which we intend to show that our work so far has a significant impact. The presentation here underscores the thesis that a considerable portion of the well-established body of results in the area of analysis on metric spaces may be quite successfully developed in the more general context of quasimetric spaces. This is relevant since not only is the category of quasimetric spaces more inclusive than the category of metric spaces but, at the same time, the former constitutes a more natural and flexible setting than the latter. For example, the category of quasimetric spaces contains the family of all quasi-Banach spaces, which in turn encompasses a multitude of function spaces (measuring smoothness, on various scales) that are of fundamental importance in analysis. Also, as opposed to the case of metric spaces, the category of quasimetric spaces is stable under any positive “power dilation,” 7! ˛ , of the quasimetric. Lastly, we wish to note that while our results here are either generalizations of known facts or altogether new, in all cases the emphasis is on the optimality of smoothness measured on the H¨older scale.
4.1 Category of Quasimetric Spaces We start with some preliminary considerations. For the remainder of our work, if X is a given set of cardinality 2, then we denote by Q.X / the collection of all quasidistances on X . The reader is reminded that, as in (1.1), a quasidistance on X is a function W X X ! Œ0; C1/ with the property that there exists a finite constant c 1 such that for every x; y; z 2 X one has .x; y/ D 0 ” x D y;
.x; y/ D .y; x/;
.x; y/ c..x; z/ C .z; y//: (4.1)
D. Mitrea et al., Groupoid Metrization Theory, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8397-9 4, © Springer Science+Business Media New York 2013
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4 Applications to Analysis on Quasimetric Spaces
Next, for each 2 Q.X / define C WD sup x;y;z2X
.x; y/ ; max f.x; z/; .z; y/g
(4.2)
not all equal
and note that 8 2 Q.X / H) .x; y/ C max f.x; z/; .z; y/g; 8 x; y; z 2 X;
(4.3)
8 2 Q.X / H) C 2 Œ1; C1/;
(4.4)
8 2 Q.X /; 8 ˇ 2 .0; C1/ H) ˇ 2 Q.X / and Cˇ D .C /ˇ ;
(4.5)
8 2 Q.X /; 8 ˛ 2 .0; C1 H) ˛ 2 Q.X / and C˛ C ;
(4.6)
where the last inequality is a consequence of (3.107). Remark 4.1. (i) Recall that a distance d on the set X is called an ultrametric provided the stronger version of the triangle inequality d.x; y/ max fd.x; z/; d.z; y/g
for all x; y; z 2 X
(4.7)
holds. Hence, ultrametric on X ” 2 Q.X / and C D 1:
(4.8)
In light of this observation, it is natural to refer to an inequality of the type (4.3) as a quasi-ultrametric condition for . Thus, C from (4.2) is the optimal constant appearing in a quasi-ultrametric condition for a given 2 Q.X /. (ii) If .X; d / is
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