Some remarks about metric spaces, spherical mappings, functions and their derivatives

Publicacions Matem`atiques, Vol 40 (1996), 411–430. SOME REMARKS ABOUT METRIC SPACES, SPHERICAL MAPPINGS, FUNCTIONS AND THEIR DERIVATIVES Stephen Semmes Abstract If p∈Rn, then we have the radial projection map from Rn\{p} onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive Sobolev inequalities, etc. In this paper we shall discuss these topics mostly in the context of metric doubling measures, which provides a nontrivial setting in which these mappings exist and can be used. This provides an alternative approach (or understanding) of the results in [DS], and a variation on the themes of [Se4]. 1. Introduction. Consider Euclidean space Rnfor a moment, and let pbe any point in Rn. Define a mapping πp:Rn\{p}→Sn−1by (1.1) πp(x)= x−p |x−p|. Now suppose that (M,d(x, y)) is a metric space. Assume that Mis homeomorphic to Rn, but we want to permit it to have very different geometry. Given a point p∈M, let us ask ourselves whether we can find a mapping θp:M\{p}→Sn−1such that θpsatisfies the “Lipschitz” condition (1.2) |θp(x)−θp(y)|≤Cd(x, y) min(d(x, p),d(y,p)) The author is supported by the U.S. National Science Foundation. 412 S. Semmes for some constant Cand all x, y ∈M\{p}, and such that θpis topologically nondegenerate in the sense of having nonzero degree, or degree 1, or being homotopic to πp. (To talk about the degree in this case we use the fact that M\{p}has the same homotopy type as Sn−1.) The Lipschitz condition in (1.2) is the natural one in this context, it amounts to saying that θpis locally Lipschitz with norm bounded by a constant divided by the distance to the singularity p. Note that πphas this property (with respect to the Euclidean metric). The existence of such a mapping θpindicates approximate Euclidean behavior of the geometry of Mnear p. For Euclidean space itself we can simply take θp=πp, and if Mis bilipschitz equivalent to a Euclidean space then we also get mappings like θp. We shall see that this is also true if Mis quasisymmetrically equivalent to Euclidean space, although this case is a bit trickier. Note however that there are plenty of examples of metric spaces whose geometry is very much like Euclidean geometry in some ways but which are still very different from Euclidean spaces. Different in the absence of a bilipschitz or quasisymmetric parameterization, for instance. See [Se2], [Se3]. The existence of a mapping θpcan capture some geometric structure that one might want even when there is not a good parameterization. The existence of mappings like θpenjoy a naive geometric appeal, but they also have a concrete usefulness. Under suitable conditions they can permit us to control the value of a function at the point pin terms of the gradient of the function on the whole space, in a manner analogous to classical formulas on Euclidean space. Once one has such estimates one can then try to derive Sobolev inequalities, etc. As a practical matter we shall focus here on the metric spaces which arise from metric doubling measures. This concept will be recalled in the next section. In [DS] it was proved that one has for metric doubling measures practically the same Sobolev and Poincar´e inequalities as for Euclidean spaces. The observations of this paper provide a different approach to this result, or rather a better understanding of the approach in [DS], with clearer general principles. This paper is also a variation on the theme of [Se4]. In [Se4]too we were constructing special mappings on metric spaces and then using these mappings to derive Sobolev inequalities. The present paper and [Se4] share common underlying principles, but the special mappings in [Se4] were slightly less beautiful than the ones given here, while the constructions in [Se4] were more flexible if more complicated and less sharp. The present paper will also be a kind of complement to [Se4], in that Metric spaces and derivatives 413 [Se4] was long and pretty thorough about technicalities and generality, whereas here we shall try to present the main points clearly and concisely in a special case while skipping technical but fairly straightforward discussions of the various consequences of the method. We review metric doubling measures in the next section, and then discuss special mappings into spheres in Section 3 for non-Euclidean metrics. This construction is applied in Section 5 to control the value of a function in terms of an integral of its gradient, with bounds that are appropriate for the geometry of a metric doubling measure. Section 4 before that addresses a technical point about smoothing the maps obtained in Section 3. In Section 6 we explain the principles of this method more succinctly in terms of differential forms, and we describe some other ways to use these principles. Some readers might find it pleasant to skip to Section 6 first. I would like to thank Juha Heinonen and the referee for their comments and suggestions. 2. Metric doubling measures. Definition 2.1. Adoubling measure on Rnis a positive Borel measure µwhich is finite on compact sets, not identically zero, and which satisfies (2.2) µ(2B)≤Cµ(B) for some constant C>0 and all (Euclidean) balls Bin Rn. Here 2B denotes the ball with the same center as Bbut twice the radius. Given a doubling measure µon Rnwe can associate a quasimetric δ(x, y)=δµ(x, y)onRnby (2.3) δ(x, y)=µ(Bx,y)1 n, where Bx,y is the smallest (closed) ball which contains xand y.Tosay that δ(x, y) is a quasimetric means that it is nonnegative and symmetric, that it vanishes exactly on the diagonal, and that it satisfies (2.4) δ(x, z)≤K(δ(x, y)+δ(y,z)) for some constant Kand all x, y, z ∈Rn. It is not hard to check (2.4) using the doubling condition on µ, and also that this quasimetric defines the standard topology on Rn. Definition 2.5. A doubling measure µon Rnis said to be a metric doubling measure if there is a metric δ(x, y)onRnand a constant C>0 so that (2.6) C−1δ(x, y)≤δ(x, y)≤Cδ(x, y) 414 S. Semmes for all x, y ∈Rn. For the record, a metric is a symmetric nonnegative function which vanishes exactly on the diagonal and which satisfies the triangle inequality. It is the latter which really matters here, i.e., replacing Kwith 1 in (2.4). As a practical matter the triangle inequality is very useful because it permits one to construct plenty of Lipschitz functions. On a quasimetric space there may be no nonconstant Lipschitz functions, e.g., (Rn,|x−y|2). When n=1,δ(x, y) itself satisfies the triangle inequality, but this can fail in a strong way in higher dimensions. For instance, the measure µ(x)=|x1|dx on R2is a doubling measure but not a metric doubling measure. The failure of the metric doubling measure condition stems from the fact that the density |x1|vanishes on the x1-axis, which would force all points on the x1-axis to have δ(·,·) distance zero from each other if δ(·,·) were equivalent in size to a true metric. More generally if µis a metric doubling measure of the form µ=ω(x)dx, where ω(x) is, say, continuous, then ωcannot be identically zero on a rectifiable curve. On the other hand, metric doubling measures can have a density ωthat vanishes to large order, even on a large set. The zero set of ωhas to be highly disconnected or crinkled. The notion of a metric doubling measure was introduced in [DS], in a slightly different but equivalent form and with a different name (“strong A∞weights”). See also [Se1], [Se2], for more examples especially. Note that if d(x, y) is any quasimetric on a set M, then there exists a metric ρ(x, y)onMand constants C>0 and s≥1 so that (2.7) C−1ρ(x, y)s≤d(x, y)≤Cρ(x, y)s for all x, y ∈M. This is essentially the content of the proof of Theorem 2 in [MS]. Thus the quasimetric associated to a metric doubling measure always has a metric around, the issue is whether we can take s=1in (2.7). The following is a basic and gorgeous fact about metric doubling measures. Theorem 2.8. If n>1and µis a metric doubling measure on Rn, then µand Lebesgue measure are mutually absolutely continuous. Thus every metric doubling measure is of the form µ=ω(x)dx for a locally integrable function ω(x). The weights ω(x) that arise this way are called strong A∞weights. It turns out that they are always A∞weights in the sense of harmonic analysis (see [Ga], [J] for the definition). Metric spaces and derivatives 415 Note that doubling measures can be singular with respect to Lebesgue measure. It is quite amusing that the geometric property of being a metric doubling measure is sufficient to ensure absolute continuity. Theorem 2.8 was observed in [DS], which is to say that it was observed in [DS] that Theorem 2.8 follows from an argument of Gehring [Ge]. See also Proposition 3.4 in [Se1]. If µis a metric doubling measure on Rn, then we get a metric space (Rn,δ(x, y)) determined by µ. (Actually the metric δ(·,·) is not determined uniquely, but it is determined up to size, meaning that any other such metric must be bounded above and below by constant multiples of δ(·,·), and that is sufficient for our purposes.) What can we say about the geometry of this metric space? We can view Theorem 2.8 as a rectifiability result for this metric space, because it implies that this metric space is asymptotically Euclidean at almost all points (the Lebesgue points of ω(x)). The A∞condition that ωsatisfies actually implies a kind of uniform rectifiability property of this metric space. The main result in [DS] said that one has the same kind of Sobolev-Poincar´e inequalities for this space as if it were bilipschitz equivalent to Rnequipped with the standard Euclidean metric. One of the main points of [DS] was to shed some light on the question of whether these metric spaces always are bilipschitz equivalent to standard Euclidean spaces. This is trivially true in dimension 1, but it turns out to be false in dimensions greater than or equal to 3. Dimension 2 is unknown. See [Se2]. Note that (Rn,δ(x, y)) is bilipschitz equivalent to (Rn,|x−y|)ifand only if there is a quasiconformal mapping f:Rn→Rnwhose Jacobian is bounded above and below by constant multiples of ω. This is well known and not difficult to verify using the definitions. 3. Constructing special mappings. Let d(x, y) be a metric on Rn. In this section we shall assume that the identity is a quasisymmetric mapping from (Rn,d(x, y)) onto Rn equipped with the Euclidean metric. This means that there is a homeomorphism η:[0,∞)→[0,∞) such that (3.1) d(x, y)≤td(x, z) implies |x−y|≤η(t)|x−z| for all x, y, z ∈Rnand t>0. This implies in particular that d(x, y) determines the same topology as the Euclidean metric (and indeed it implies a similar condition with the roles of d(x, y) and the Euclidean metric reversed). Examples 3.2. (a) If d(x, y)=|x−y|sfor some 0 <s≤1, then d(x, y) satisfies (3.1). (It also satisfies (3.1) when s>1, but it is not a metric then.) 416 S. Semmes (b) If µis a doubling measure on Rn, then the associated quasidistance δ(x, y) satisfies the quasisymmetry property (3.1). This is well known and not hard to derive from the definitions. It will not be a metric in general, though, but in the case of a metric doubling measure it will be equivalent in size to a metric. As indicated in the introduction, we would like to build mappings which approximate πpas defined in (1.1) but which respect the d(x, y)-geometry as in (1.2). Theorem 3.3. Suppose that d(x, y)is a metric on Rnwhich satisfies the quasisymmetry condition (3.1). Then for each p∈Rnand >0we can find a mapping θ:Rn\{p}→Sn−1which satisfies  θ(x)−x−p |x−p| ≤and(3.4) |θ(x)−θ(y)|≤C()d(x, y) min(d(x, p),d(y,p)) (3.5) for all x, y ∈Rn\{p}. Here C()depends also on the function η(t)in (3.1) and the dimension nbut nothing else. It is important here that d(x, y) be a metric and not a quasimetric, we need to have the triangle inequality in order to have an adequate supply of Lipschitz functions with respect to d(x, y). Note that (3.4) (for a reasonable choice of ) implies that θis homotopic to πpthrough continuous mappings from Rn\{p}into Sn−1. To prove Theorem 3.3 we use the techniques of the Whitney Extension Theorem ([St]), namely Whitney decompositions and partitions of unity. All constants in the argument that follows are permitted to depend on the function η(t) from (3.1). Let p∈Rnand >0 be given, with <.01, say. We may as well assume that is as small as we want, the conclusion gets stronger as  gets smaller. Let {Qi}be a listing of the maximal dyadic cubes in Rn\{p}such that (3.6) diam Qi≤dist(Qi,p), where “dist” is the distance in the Euclidean metric. The maximality of these cubes ensures that (3.7) 3 diam Qi>dist(Qi,p). Metric spaces and derivatives 417 By standard reasoning the cubes have disjoint interiors and their union is all of Rn\{p}. (Note that we interpret “cube” to mean closed cube.) Let qidenote the center of Qi. We want to have a partition of unity associated to {Qi}, but we want this partition of unity to be well-behaved with respect to the geometry of d(x, y) rather than Euclidean geometry. We want the functions in the partition of unity to be Lipschitz with respect to d(x, y), for instance. We start by constructing auxiliary functions ψi:Rn→Rwith the properties that 0 ≤ψi≤Con Rn,ψi≥1onQi,ψi≡0onRn\2Qi, and the Lipschitz condition (3.8) |ψi(x)−ψi(y)|≤Cd(x, y) min(d(x, p),d(y,p)). This constant Cis permitted to depend on as well as η(t). The denominator on the right side may look a bit odd, but it is convenient for deriving (3.5). We define ψi(x) in terms of the metric d(·,·) as follows. Given x∈Rnwe start by taking its distance to Rn\2Qiwith respect to d(·,·). This gives a function αi(x) which is Lipschitz with respect to d(·,·) with norm ≤1 and which vanishes on Rn\2Qibut which has the wrong size. We then divide αi(x) by the distance Diwith respect to d(·,·) between Qiand Rn\2Qiand set ψi(x)=αi(x)/Di. This function is at least 1 on Qi. It is not hard to see that ψi(x) is also bounded, using the quasisymmetry hypothesis. It remains to check the Lipschitz condition (3.8). If xand yboth lie outside 2Qi, then the left side of (3.8) vanishes, and there is nothing to check. If exactly one of xand ylies in 2Qiand the other lies outside 3Qi—say x∈2Qiand y∈Rn\3Qi— then |x−y|≥C−1|x−p|and the right side of (3.8) is bounded from below, because of the quasisymmetry condition. In this case (3.8) is automatic since ψiis bounded. Finally, if both xand ylie in 3Qi, then |x−p|and |y−p|are both comparable in size to the Euclidean distance between Qiand Rn\2Qi(by (3.6), (3.7), and the requirement that  be small), and so d(x, p) and d(y,p) are both comparable in size to the distance Diwith respect to d(·,·) between Qiand Rn\2Qi, because of the quasisymmetry property. (Note that the constants here depend on , but we do not mind.) From here the Lipschitz condition (3.8) follows from ψi=αi/Diand the fact that αiis Lipschitz with norm ≤1 with respect to d(·,·). Thus we get our functions ψi. For the partition of unity we take (3.9) φi=ψi jψj . More precisely, although the denominator on the right hand side vanishes at p, we interpret φito also vanish at p.Thus0≤φi≤ψi≤C, since 418 S. Semmes jψj≥1onRn\{p}. Notice that ψj(x)= 0 for only finitely many j for any given x, that φi≡0onRn\2Qi, and that (3.10)  i φi≡1onRn\{p}. Let us check that φienjoys the Lipschitz condition (3.11) |φi(x)−φi(y)|≤Cd(x, y) min(d(x, p),d(y,p)). We want to use (3.8). To do this we need to check that for each fixed point zthe number of j’s for which ψj(z)= 0 is uniformly bounded. If ψj(z)= 0, then z∈2Qj, and (3.6), (3.7) imply that diam Qjis comparable in size to |z−p|. Thus all the Qj’s with z∈2Qjhave approximately the same diameter, and there can only be boundedly many of them because they have disjoint interiors. Thus ψj(z)= 0 for only a bounded number of j’s for any given z, and then (3.11) follows from (3.8) and the definition (3.9) of φiby a simple computation. Let us now define a mapping h:Rn\{p}→Rnby (3.12) h(x)= i qi−p |qi−p|φi(x). The idea is that this is a good approximation to x−p |x−p|. Specifically, we have that (3.13)  h(x)−x−p |x−p| ≤20  for all x∈Rn\{p}. To see this we write (3.14)  h(x)−x−p |x−p| ≤ i qi−p |qi−p|−x−p |x−p| φi(x). If φi(x)= 0, then x∈2Qi, whence |x−qi|≤2 diam Qi≤2dist(Qi,p)≤ 2|qi−p|, by (3.6). Since is small we conclude that (3.15)  qi−p |qi−p|−x−p |x−p| ≤20 . (There is nothing special about 20 here, but there is no point in worrying about the right constant.) This implies (3.13), because of (3.10). Metric spaces and derivatives 419 Define θ:Rn\{p}→Sn−1by θ(x)= h(x) |h(x)|. This is well defined, because (3.13) ensures that |h(x)|is close to 1. From (3.13) we also get (3.4), but with an extra constant factor on the right hand side of (3.4) that does not bother us. It remains to verify the Lipschitz condition (3.5). It is enough to show that the analogous estimate holds for h instead of θ.Forhwe have that (3.16) |h(x)−h(y)|≤ i |φi(x)−φi(y)|. If φi(x)= 0, then x∈2Qi, and for each xthis can hold only for a bounded number of i’s. Thus there are only boundedly many nonzero terms in (3.16). We can use (3.11) for each of them, and in the end we obtain the analogue of (3.5) for h. This completes the proof of Theorem 3.3. 4. Metric doubling measures, special mappings, and smooth approximations. Let µ=ω(x)dx be a metric doubling measure on Rn,n>1. (Remember Theorem 2.8. We restrict ourselves to n>1 here not because the n= 1 case does not work, but because it works in a degenerate manner that is not very interesting.) From µwe get a quasimetric δ(x, y)on Rnas in (2.3), and the assumption that µis a metric doubling measure implies that δ(x, y) is comparable in size to a true metric, as in (2.6). This metric satisfies the quasisymmetry condition (3.1), because δ(x, y) does, and so we may apply Theorem 3.3 to it. We conclude that for each p∈Rnwe can find a mapping θas in Theorem 3.3, but with d(x, y) replaced with δ(x, y). Fix p∈Rnand >0, and let θbe as in Theorem 3.3. In practical computation we would be much happier if θwere C1, so that we would not have to think about applying standard rules of calculus, but of course we cannot take θto be C1in general. We can approximate θby smooth functions in the usual way, and this provides enough control for many applications. We shall deal with this technical point in this section. Fix a nonnegative smooth function φ(x)onRnwith support contained in the unit ball and Rnφ(x)dx = 1. Define φt(x) for t>0byφt(x)= t−nφ(x/t). Set θt=φt∗θ, where ∗denotes convolution. This is a smooth function, but it is only a good approximation to θaway from the point p. That is, we are interested in topological properties of θ, its topological nontriviality as a mapping into the sphere in particular, and so we really want an approximation to θwhich takes values in Rn\{0}. Precisely because of this topological nontriviality we cannot do this across p, make a smooth approximation across pwith values in Rn\{0}, and so this approximation should be taken seriously away from ponly. 426 S. Semmes the previous section. There we were working with the mapping θinstead of πp, and although θis not given so explicitly as πp, it does enjoy many of the same properties. For θwe have the minor inconvenience of its lack of smoothness, but this can be managed with little difficulty, by smoothing it for instance, as we did before. We never quite said before that what we were doing was deriving versions of (6.2) and (6.3) for θinstead of πp, but that is really what was happening. For conceptual purposes this language is clearer, although in practical use it is sometimes simpler to work a little differently, as in Section 5. Let us remain in this language for the moment and consider some variations on the theme. Instead of looking for a formula for f(p) in terms of the gradient of fwe can take two points pand qand look for a formula for f(p)−f(q). Of course one can simply take the difference of two formulas like (6.3), but it is nicer to have a formula that does not require fto have compact support, indeed a formula which is local and works for any smooth function defined on a ball that contains pand q. How can we get such a formula, along the lines of the construction just described? Instead of building a mapping πpas above we should look for a mapping from Rn\{p, q}into Sn−1which is constant outside of a ball that contains pand q, which behaves like πpnear p, and which behaves in a manner like that of πqnear q, but with an extra reflection included in order to reverse the local degree at q. It is not difficult to construct such mappings, although they are not so easily written as πp. Once one has such a mapping one can pull back the volume form λon Sn−1to get a smooth (n−1)-form βon Rn\{p, q}which has singularities at each of pand qlike αhas at p(6.1). This form βis also closed, and if we extend it as a current across the singularities we get that dβ =[p]−[q] and(6.4) Rn df ∧β=f(p)−f(q),(6.5) modulo sign errors. Because our mapping is chosen to be constant outside a certain ball, βvanishes off of the same ball, and for (6.5) we do not need fto be defined on all of Rn, etc. Metric spaces and derivatives 427 Again this construction does not really depend on specific choices or explicit computations, just some basic properties of the mapping in Sn−1 that we build. Instead of a smooth mapping adapted to Euclidean geometry we can build mappings adapted to the geometry of a metric doubling measure on Rn. That is, we approximate the mapping for Euclidean geometry by one which is locally Lipschitz with respect to the new metric, with suitable estimates near the singularities. We can make this approximation in the same manner as in Section 3, and with the same estimates at the singularities. This approximation will have the same topological properties as the one for Euclidean geometry, because we can build it in such a way as to be a small perturbation of the mapping for Euclidean geometry in the supremum norm, so as to be homotopic to the Euclidean mapping through mappings into Sn−1. Thus we can maintain the local degrees near the singularities, and this nondegeneracy gives rise to formulas like (6.5), estimates like (5.2), etc., in the same manner as in Section 5. This story of the two singular points is closer to the development given in [DS] and [Se4]. For applications to analysis the two-point formulas are generally more useful than the one-point formulas, and more flexible. For instance one can derive Poincar´e-type inequalities by averaging the two-point formulas, and the locality of the two-point formulas is very convenient. The one-point estimates can be derived from the two-point estimates by sending the other point to infinity, but the direct constructions are cleaner. This construction for the two-point formulas gives practically the same result as the construction in [DS]. The constructions themselves are not quite the same though, and the approach here isolates better the topological properties needed and the principles involved. This makes it easier to make similar constructions in other contexts. For instance the constructions in [Se4] are based on similar principles, but this similarity is less obvious if one tries to compare [Se4] and [DS] directly. In the setting of [Se4], the geometry was not given so explicitly, it was not such a simple perturbation of Euclidean geometry, and the mappings could not be obtained as simple perturbations. It was also better to work there with one-parameter families of nontrivial mappings to Snwhich were constant off of balls, rather than individual radial projections into Sn−1, but this is more of a variation on the theme rather than a fundamentally different principle. (One can also see it in terms of stabilizing the space by taking the Cartesian product with the line or an interval.) As another variation on the theme it is nicer to keep track of the dependence on the point p, in the following sense. Instead of fixing p and defining the mapping πp:Rn\{p}→Sn−1as before, let us define a 428 S. Semmes mapping π:Rn×Rn\∆→Sn−1, ∆ = the diagonal = {(x, x):x∈Rn}, by (6.6) π(x, y)= x−y |x−y|. Again we can pull back the volume form λto a smooth (n−1)-form on Rn×Rn\∆. This form is closed, and it has a singularity at the diagonal like that of αnear p(6.1). We can extend it to a locally integrable current across the singularity, and when we do that we get a current γ such that (6.7) dγ = [∆] (modulo a possible sign error), where [∆] denotes the current of integration over the diagonal ∆. Keep in mind that the exterior derivative d here involves now derivatives in both sets of variables. This current γ provides the “kernel” for a familiar “homotopy operator” for d; that is an operator Tacting on differential forms which satisfies dT +Td=I. (I learned this view of homotopy operators from Reese Harvey and John Polking [HP].) Remember that one cannot solve the equation dφ =ψ in general, one must require the compatibility condition that dψ =0, and homotopy operators provide solutions to the d-equation when the compatibility condition is fulfilled. Again this construction does not really depend on the precise formula for π, only on properties that are more stable than that. Given a metric on Rnwhich is quasisymmetrically equivalent to the Euclidean metric we can build an analogous mapping which approximates πin the supremum norm, hence is homotopic to πthrough mappings from Rn×Rn\∆into Sn−1and so has approximately the same topological properties as π, and which has approximately the same analytical properties as πwith respect to the new metric. In particular one can do this for metrics associated to metric doubling measures. One can then build a homotopy operator for dwhose kernel satisfies the right bounds associated to the given metric doubling measure, analogous to (5.2). Once one has a kernel with the right bounds, one can get bounds on the corresponding operator. In summary, in analysis we are often looking for operators which provide some kind of inverse to a differential operator, like the exterior derivative d.OnRnwe can sometimes see these operators as being constructed from differential forms which satisfy certain properties, and these differential forms can sometimes be obtained from special mappings. In this way problems of calculus become problems of geometry. The problems of geometry are frequently stable under perturbations, Metric spaces and derivatives 429 their topological aspects being stable in particular, so that if we change the geometry we might be able to modify the mapping, in such a way as to be able to do everything for the new geometry. And in such a way as to solve problems of calculus for the new geometry which are much more mysterious when seen in purely analytical terms. References [DS] G. David and S. Semmes, Strong A∞-weights, Sobolev inequalities, and quasiconformal mappings, in “Analysis and Partial Differential Equations,” (C. Sadosky, ed.), Lecture Notes in Pure and Applied Mathematics 122, Marcel Dekker, 1990, pp. 101–111. [Ga] J. Garnett,“Bounded Analytic Functions,” Academic Press, 1981. [Ge] F. W. Gehring, The Lpintegrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. [HP] R. Harvey and J. Polking, Fundamental solutions in complex analysis Part I. The Cauchy Riemann operator, Duke Math. J. 46 (1979), 253–300. [J] J. L. Journ´ e,“Calder´on-Zygmund Operators, Pseudodifferential Operators, and the Cauchy Integral of Calder´on,” Lecture Notes in Math. 994, Springer-Verlag, 1983. [MS] R. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257–270. [N] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences (1974). [Se1] S. Semmes, Bilipschitz mappings and strong A∞weights, Ann. Acad. Sci. Fenn. Ser. A I 18 (1993), 211–248. [Se2] S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about A∞weights, Rev. Mat. Iberoamericana (1996). [Se3] S. Semmes, Good metric spaces without good parameterizations, Rev. Mat. Iberoamericana 12 (1996), 187–275. [Se4] S. Semmes, Finding Curves on General Spaces through Quantitative Topology with Applications for Sobolev and Poincar´e Inequalities, Selecta Math. (1996), (to appear). 430 S. Semmes [St] E. M. Stein,“Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, 1970. Department of Mathematics Rice University Houston, TX 77251-1892 U.S.A. Primera versi´o rebuda el 21 de Desembre de 1995, darrera versi´o rebuda el 10 de Juny de 1996