Happy fractals and some aspects of analysis on metric spaces

Publ. Mat. 47 (2003), 261–309 HAPPY FRACTALS AND SOME ASPECTS OF ANALYSIS ON METRIC SPACES Stephen Semmes∗ Dedicated to Leon Ehrenpreis and Mitchell Taibleson Abstract There has been a lot of interest and activity along the general lines of “analysis on metric spaces” recently, as in [2], [3], [26], [40], [41], [46], [48], [49], [51], [82], [83], [89], for instance. Of course this is closely related to and involves ideas concerning “spaces of homogeneous type”, as in [18], [19], [66], [67], [92], as well as subRiemannian spaces, e.g., [8], [9], [34], [47], [52], [53], [54], [55], [68], [70], [72], [73], [84], [86], [88]. In the present survey we try to give an introduction to some themes in this general area, with selections related to several points of view. Let us also mention [39], [93], [97], [98], [99] for topics dealing with nonstandard analysis, where one might think of a continuous metric space as something like a nonstandard graph. Contents 1. Graphs 263 2. Finitely-generated groups 265 3. Happy fractals 266 4. Lipschitz retracts 269 5. The Sierpinski gasket and carpet 274 6. Heisenberg groups 275 2000 Mathematics Subject Classification. 42-02. Key words. Graphs, happy fractals, Lipschitz classes. ∗This survey was prepared partially in connection with the trimester “Heat kernels, random walks, and analysis on manifolds and graphs” at the Centre ´ Emile Borel, Institut Henri Poincar´e, in the Spring of 2002. This trimester was organized by P. Auscher, G. Besson, T. Coulhon, and A. Grigor’yan, and the author was fortunate to be a participant. The proceedings will be published in the Contemporary Mathematics series of the American Mathematical Society, and a report on the trimester can be found in [82]. Another survey on related themes is [83]. The author is grateful to an unnamed reader for many helpful comments and suggestions. 262 S. Semmes 7. Some happy fractals from Helsinki 279 8. More on Lipschitz functions 280 9. Lipschitz functions of order α283 10. Some functions on the real line 284 11. Sums on general metric spaces 288 12. The Zygmund class on R289 13. Approximation operators, 1 291 14. Approximation operators, 2 293 15. A kind of Calder´on-Zygmund decomposition related to Lipschitz functions 295 16. A brief overview of “atoms” 296 References 302 As usual, to say that (M,d(x, y)) is a metric space means that Mis a nonempty set and that d(x, y)isanonnegative real-valued function on M×Msuch that d(x, y)=0if and only if x=y,d(x, y)=d(y,x) for all x, y ∈M, and d(x, z)≤d(x, y)+d(y,z)(0.1) for all x, y, z ∈M(the triangle inequality). Here we shall make the standing assumption that Mhas at least 2 elements,(0.2) to avoid degeneracies. If Eis a nonempty subset of M, then diam Edenotes the diameter of E, defined by diam E= sup{d(u, v):u, v ∈E}.(0.3) Given xin Mand a positive real number r,welet B(x, r) and B(x, r) denote the open and closed balls in Mwith center xand radius r,so that B(x, r)={y∈M:d(x, y)<r}, B(x, r)={y∈M:d(x, y)≤r}. (0.4) Sometimes there might be another metric space (N,ρ(u, v)) in play, and we may introduce a subscript as in BN(w,s)toindicate in which metric space the ball is defined. Of course the n-dimensional Euclidean space Rnwith the standard metric |x−y|is a basic example of a metric space, which is always good to keep in mind. Metric spaces associated to connected graphs will Happy Fractals and Analysis on Metric Spaces 263 be discussed in Section 1, and the special case of Cayley graphs from finitely-generated groups will be reviewed in Section 2. In Section 3 we consider some notions that apply to any metric space, concerning rectifiable paths in particular. Sections 4–7 deal with related notions and examples. The remaining sections deal with various general aspects of analysis on metric spaces. 1. Graphs Suppose that we have a graph consisting of a nonempty set Vof vertices and a set Eof edges. An element of Ecan be described by an unordered pair of distinct elements of V;wedonot wish to consider edges which form loops by themselves, or multiple edges between the same pair of vertices. Two vertices connected by an edge are said to be adjacent. Let us assume that our graph is connected, which is to say that every pair of vertices can be connected by a finite path. The length of a path is defined to be the number of edges that the path traverses. Thus the length of a path is a nonnegative integer, which is 0 in the case of a path that consists of a single vertex and traverses no edges. We define a metric d(v,w)onVby taking d(v,w)tobethe length of the shortest path between vand w.Itiseasy to see that (V,d(v,w)) is indeed then a metric space. Let us also assume that the graph is locally finite, which is to say that there are only finitely many vertices adjacent to a given vertex. For each pin Vand each positive integer mone can show that there are only finitely many vertices whose distance to pis at most m. In fact, let us assume that there is a nonnegative integer ksuch that for every vertex vin Vthere are at most kvertices win Vwhich are adjacent to v.Ifk=0then Vcontains only one vertex and there are no edges, and if k=1then Vhas either one or two vertices, with no edges if there is only one vertex and exactly one edge when there are two vertices. For simplicity let us assume that k≥2. If pis an element of Vand mis a nonnegative integer, then we define Am(p)tobethe number of vertices vin Vwhose distance to pis exactly equal to m.ThusA0(p)=1, since pis the only vertex at distance 0 from itself, and A1(p)≤k, since A1(p)isthe same as the number of vertices in Vwhich are adjacent to p.Form≥2wehave that Am(p)≤(k−1)Am−1(p).(1.1) Indeed, suppose that vis an element of Vwhose distance to pis exactly equal to m. Then there is vertex win Vsuch that vis adjacent to wand 264 S. Semmes the distance from wto pis exactly m−1. Since m≥2, there is also a vertex uin Vsuch that wis adjacent to uand the distance from uto p is exactly m−2. The total number of vertices in Vwhich are adjacent to wand which have distance to pequal to mis at most k−1, because there are at most kvertices which are adjacent to wat all, and uis adjacent to wand has distance to pequal to m−2. There are Am−1(p) vertices wwhose distance to pis equal to m−1, and hence there are at most (k−1)Am−1(p)vertices whose distance to pis equal to m,as desired. Thus Am(p) grows at most exponentially in min general, and exponential growth is certainly possible, at least when k≥3. Of course there are many interesting situations where the growth is in fact bounded by a polynomial. In this survey we shall focus on situations with polynomial growth, and the doubling condition described in Section 3 gives a nice version of this which makes sense in any metric space. Instead of looking at rates of growth in terms of Am(p), one also frequently considers the quantity m  j=0 Aj(p),(1.2) which is the same as the number of elements of Vwhose distance to p is at most equal to m. As a basic example, fix a positive integer n, and consider the set Znof points in Rnwith integer coordinates as a set of vertices. Two points v,w in Zncan be defined to be adjaced if v−whas n−1coordinates equal to 0 and the remaining coordinate equal to ±1. This is the same as saying that v,ware adjacent if and only if |v−w|=1.Inthis case it is not difficult to determine the metric on Zncoming from paths in the graph, namely d(v,w)= n  j=1 |vj−wj|,(1.3) where vj,wjdenote the jth coordinates of v,w, respectively. This is often called the taxicab metric, and it satisfies the following comparison with the Euclidean distance: |v−w|≤d(v,w)≤√n|v−w|.(1.4) The first inequality can be derived from the triangle inequality for the standard distance, since each step of size 1 in the graph metric is also a step of size 1 in the Euclidean distance. The second inequality is a consequence of the Cauchy-Schwarz inequality. Happy Fractals and Analysis on Metric Spaces 265 In this case the growth is polynomial, with the number of points at distance to a fixed point pless than or equal to ris on the order of rn. Notice that this number does not depend on p,because of translationinvariance. Concerning analysis and geometry on graphs and related matters, see [7], [23], [77], [96] and the article by Coulhon in [2], for instance. 2. Finitely-generated groups Avery interesting special case of graphs and their geometry comes from Cayley graphs of finitely generated groups. Let Γ be a group with a finite set Fof generators. Thus every element of Γ can be expressed as a product of elements of Fand their inverses, with the identity element viewed as an empty product of generators. For the Cayley graph of Γ we use Γ as the set of vertices, and define two elements γ1,γ2ofΓtobe adjacent if one of them can be written as the product of the other times an element of F, where the group operation is applied in that order. From this it follows that the graph is invariant under left-translations, which is to say that γ1,γ2are adjacent if and only if αγ 1,αγ 2for any α in Γ. Every pair of elements of Γ can be joined by a path in the Cayley graph, because of the assumption that every element of Γ can be expressed as a product of generators and their inverses. If d(γ1,γ 2) denotes the distance function on Γ coming from the Cayley graph, then we have that d(αγ 1,αγ 2)=d(γ1,γ 2)(2.1) for all α,γ1, and γ2in Γ, by left-invariance of the Cayley graph. If γis an element of Γ, then the number of elements of Γ which are adjacent to γis at most twice the number of elements of F,byconstruction. As in the preceding section, this leads to a simple exponential bound on the growth of the Cayley graph of Γ. Exponential growth occurs for free groups with at least two generators, and more generally for nonelementary hyperbolic groups in the sense of Gromov, as in [21], [22], [38], [42], [44]. Hyperbolic groups have very interesting spaces at infinity associated to them which satisfy the doubling property described in the next section. In addition to the references already mentioned, see [20], [45], [75], [76]inthis regard. Note that fundamental groups of compact Riemannian manifolds without boundary and with strictly negative sectional curvatures are nonelementary hyperbolic groups. Simply-connected symmetric spaces always have compact quotients by a well-known result of Borel [14], [78], and for symmetric spaces 266 S. Semmes of noncompact type and rank 1 the sectional curvatures are strictly negative. The graph associated to Znin the previous section is exactly its Cayley graph as group with the nstandard generators, where each generator has one coordinate equal to 1 and the others equal to 0. This graph has polynomial growth, as we saw, and more generally it is a well-known result that the Cayley graph of a finitely-generated group has polynomial growth when the group is virtually nilpotent, which means that the group contains a nilpotent subgroup of finite index. A famous theorem of Gromov [43] states that the converse is true. 3. Happy fractals Let us say that a metric space (M,d(x, y)) is a happy fractal if the following three conditions are satisfied. First, Mis complete as a metric space. Second, there is a constant C1>0sothat for each pair of points x,yin Mthere is a path in Mconnecting xto ywith length at most C1d(x, y). Third, Msatisfies the doubling property that there is a constant C2so that any ball Bin Mcan be covered by a family of balls with half the radius of Band at most C2elements. One might prefer the name happy metric space, since the metric space need not be fractal, as in the case of ordinary Euclidean spaces. There are plenty of examples which are more intricate and not fractal, such as domains or surfaces with cusps. There can be interesting fractal behavior at some kind of boundary, if not for the space itself. Cantor sets and snowflake curves give examples of self-similar fractals which satisfy the doubling condition but are not happy fractals, because every curve of finite length in these spaces is constant. Some basic examples of happy fractals will be discussed in the next few sections. It does not seem to be known whether every compact connected 4-dimensional topological manifold can be realized as a happy fractal, i.e., whether every compact Hausdorff topological space which is locally homeomorphic to the open unit ball in R4has a topologically-equivalent metric in which it becomes a happy fractal. This is true for dimensions not equal to 4, since n-dimensional topological manifolds admit unique smooth structures when n≤3 and they admit unique Lipschitz structures when n≥5. See [10], [28], [35], [71], [91] concerning these topics. In general dimensions there are plenty of questions about noncompact spaces. For instance, in this connection one might consider conditions of bounded local geometry, with the happy fractal aspect being concerned with larger scales. In dimension 4, let us recall a well-known result of Happy Fractals and Analysis on Metric Spaces 267 Quinn that every connected 4-dimensional topological manifold can be smoothed in the complement of a single point. Of course, near that point there can be a lot of complications, although there are also topological restrictions since that point is a topological manifold point. To be more precise, a path in Mwhich goes from a point xto a point y is a continuous mapping p(t) defined on a closed interval [a, b]inthe real line and with values in Msuch that p(a)=xand p(b)=y.If a=t0<t 1<t 2<···<t m=b(3.1) is a partition of [a, b], then we can associate to this partition the quantity m  j=1 d(p(tj),p(tj−1)),(3.2) which is the approximation to the length of pcorresponding to this partition. The length of the path is defined to be the supremum of (3.2) over all partitions of [a, b]. In general this can be infinite. A standard observation is that the quantity (3.2) can only increase as points are added to the partition, because of the triangle inequality. Any two partitions admit a common refinement, for which the approximation to the length is then greater than or equal to the approximations to the length associated to the original refinements. Suppose that the length of the path p(t)isfinite. Then the length of the restriction of pto any subinterval of [a, b]isalso finite, and is less than or equal to the length of the whole path. Let us define a function L(u, v) for u, v ∈[a, b], u≤v,tobethe length of the restriction of p(t)to[u, v]. Of course a constant path has length 0, which includes the case where the domain has one element. Note that d(p(u),p(v)) ≤L(u, v)(3.3) for all u, v ∈[a, b] with u≤v.If a≤u≤v≤w≤b,(3.4) then it is not hard to verify that L(u, w)=L(u, v)+L(v,w),(3.5) using the monotonicity properties of the length, and the possibility of taking refinements of the partitions in particular. Fix t∈[a, b]. If t>a, then lim s→t−L(s, t)=0.(3.6) 268 S. Semmes This is equivalent to saying that lim s→t−L(a, s)=L(a, t).(3.7) From the definition we know that L(a, s)ismonotone increasing in s,so that the limit on the left side exists and is less than or equal to the right side. To show that equality holds, one can choose a partition of [a, t]so that the approximation to the length of p(u) along this partition is close to L(a, t), and then check that L(a, s)isgreater than or equal to this approximation minus a small number when sis sufficiently close to t. This employs the continuity of p(u)att,tomove the last point in the partition from tto swithout making more than a small change to the approximation to the length. If t<b, then lim s→t+L(t, s)=0.(3.8) This is equivalent to lim s→t+L(s, b)=L(t, b),(3.9) which can be verified in the same manner as before. Set λ=L(a, b), and consider the real-valued function σ(t) defined on [a, b]by σ(t)=L(a, t).(3.10) Thus σ(t)ismonotone increasing (and not necessarily strictly increasing), σ(0) = 0, σ(b)=λ, and σ(t)iscontinuous by the preceding remarks. There is a mapping p:[0,λ]→Msuch that p(σ(t)) = p(t)(3.11) for all t∈[a, b]. In other words, if s, t ∈[a, b], s<t, and σ(s)=σ(t), then L(s, t)=0,sothat pis constant along [s, t], and (3.11) leads to a single value for pat σ(s)=σ(t). Moreover, (3.3) implies that d(p(r),p(w)) ≤|r−w|(3.12) for all r, w ∈[0,λ]. On the other hand, if q:[c, d]→Mis a path such that d(q(s),q(t)) ≤k|s−t|(3.13) for some constant kand all s, t ∈[c, d], then it is easy to check that the length of qon [c, d]isatmost k|c−d|. One can trade between kand |c−d|by rescaling in the domain. Happy Fractals and Analysis on Metric Spaces 269 Thus there is a path in Mfrom xto ywith length less than or equal to a constant Aif and only if there is a mapping q:[0,1] →Msuch that q(0) = x,q(1) = y, and (3.13) holds for all s, t ∈[0,1] with k≤A. Assuming that there is a path in Mfrom xto ywith finite length and that closed and bounded subsets of Mare compact, one can use the Arzela-Ascoli theorem to find such a mapping qwith kas small as possible, and this minimal kis the same as the length of the shortest path in Mfrom xto y. Awell-known result in basic analysis states that if (M,d(x, y)) is a complete metric space, then a closed subset Kof Mis compact if and only if Kis totally bounded, which means that for every >0 there is a finite family of balls in Mwith radius whose union contains K.Thus, if (M,d(x, y)) is complete, then closed and bounded subsets of Mare compact if and only if all balls in Mare totally bounded. It is easy to verify that the latter holds when Msatisfies the doubling property. In short, closed and bounded sets are compact in a happy fractal (or happy metric space). 4. Lipschitz retracts Suppose that (M,d(x, y)) is a metric space, and that Aand Eare subsets of M, with E⊆A.Amapping φ:A→Eis said to be a Lipschitz retract of Aonto Eif φ(x)=xfor all x∈E(4.1) and φis Lipschitz, so that there is a constant k≥0 such that d(φ(y),φ(z)) ≤kd(y,z)(4.2) for all y,z ∈A. Note that if Mis complete and Eis a closed subset of M, then one can always take Ato be closed, because any Lipschitz mapping from Ainto Ecan be extended to a Lipschitz mapping from the closure of Ainto E, and with the same Lipschitz constant k. Let us say that a complete metric space (N,ρ(u, v))isaLipschitz extension space with constant s≥1iffor every separable metric space (M,d(x, y)) and every mapping ffrom a subset Zof Minto Nwhich is Lipschitz with constant L,sothat ρ(f(x),f(y)) ≤Ld(x, y)(4.3) for all x, y ∈Z, there is an extension of fto a Lipschitz mapping from M into Nwith constant sL. 276 S. Semmes One can check that these dilations define group automorphisms of Hn, i.e., δr(w,s)◦(z,t)=δr(w,s)◦δr(z,t).(6.4) Also, for r1,r 2>0wehave that δr1(δr2(w,s)) = δr1r2(w,s).(6.5) Let us note that the group law and the dilations are compatible with the standard Euclidean topology on Hn, i.e., they define continuous mappings. Let us call a nonnegative real-valued function N(·)onHnanorm if it satisfies the following conditions: (a) Nis continuous; (b) Ntakes the value 0 at the origin and is strictly positive at other points in Hn; (c) N(w,s)−1=N(w,s) for all (w,s)∈Hn; (d) N(δr(w,s)) = rN(w, s) for all r>0 and (w, s)∈Hn; and (e) Nsatisfies the triangle inequality with respect to the group structure on Hn, which is to say that N(w,s)◦(z,t)≤N(w,s)+N(z,t)(6.6) for all (w,s),(z,t)∈Hn. In many situations it is sufficient to work with a weaker notion, in which (6.6) is replaced by the “quasitriangle inequality” which says that there is a positive constant C>0sothat the left side is less than or equal to Ctimes the right side. It is very easy to write down explicit formulae for “quasinorms” which satisfy conditions (a)–(d) and this weaker version of (e), and in fact this weaker version of (e) is implied by the other conditions. Also, any two quasinorms are comparable, which is to say that each is bounded by a constant multiple of the other. Indeed, because of the homogeneity condition (d), this statement can be reduced to one on a compact set not containing the origin, where it follows from the continuity and positivity of the quasinorms. Actual norms can be written down explicitly through simple but carefully-chosen formulae, as in [52]. Another aspect of this will be mentioned in a moment, but first let us define the distance function associated to a norm or quasinorm. If Nis a norm or quasinorm on Hn, then we can define an associated distance function dN(·,·)onHnby dN(w,s),(z,t)=N(w,s)−1◦(z,t).(6.7) Happy Fractals and Analysis on Metric Spaces 277 By construction, this distance function is automatically invariant under left translations on Hn, i.e., dN(y,u)◦(w,s),(y,u)◦(z,t)=dN(w,s),(z,t) (6.8) for all (y,u),(w,s),(z,t)∈Hn, simply because (y,u)◦(w,s)−1◦(y,u)◦(z,t)=(w, s)−1◦(z,t).(6.9) We also have that d(·,·)isnonnegative, equal to 0 when the two points in Hnare the same, and is positive otherwise, because of the corresponding properties of N. Similarly, dN(w,s),(z,t)=dN(z,t),(w, s),(6.10) because of the symmetry property N(w, s)−1=N(w,s)ofN, and dNδr(w,s),δ r(z,t)=rd N(w,s),(z,t) (6.11) by the homogeneity property of N. If Nis a norm, then (6.6) implies that dNsatisfies the usual triangle inequality for metrics. If Nis a quasinorm, then dNsatisfies the weaker version for quasimetrics, in which the right side is multiplied by a fixed positive constant. Just as different quasinorms on Hnare comparable, the corresponding distance functions are too, i.e., they are each bounded byaconstant times the other. A basic and remarkable feature of the Heisenberg groups with this geometry is that they are happy fractals. In fact one can define the distance between two points in terms of the infimum of the lengths of certain paths between the two points, where the family of paths and the notion of length enjoy left-invariance and homogeneity properties which lead to the same kind of properties for the distance function as above. This kind of distance function can also be shown to be compatible with the Euclidean topology on Hn. These features imply that this distance function is of the form dNfor some Nas above. The triangle inequality for the distance function is a consequence of its definition, and this leads to the triangle inequality for the corresponding N.Akey subtlety in this approach is that there is a sufficiently-ample supply of curves used in the definition of the distance to connect arbitrary points in Hn,because the curves are required to satisfy nontrivial conditions on the directions of their tangent vectors. 278 S. Semmes Let us return to the setting of an arbitrary norm Non Hn. The triangle inequality can be rewritten as N(w,s)≤N(z,t)+dN(w,s),(z,t), N(z,t)≤N(w, s)+dN(w,s),(z,t) (6.12) for all (w,s),(z,t)∈Hn.Thus |N(w,s)−N(z,t)|≤dN(w,s),(z,t) (6.13) for all (w,s),(z,t)∈Hn. For (w,s)=0,define φ(w,s)by φ(w,s)=δN(w,s)−1(w,s).(6.14) Thus Nφ(w,s)=1(6.15) by definition. If (w,s), (z,t) are both nonzero elements of Hn, then (6.16) dNφ(w,s),φ(z,t) ≤dNφ(w,s),δ N(w,s)−1(z,t)+dNδN(w,s)−1(z,t),φ(z,t). The first term on the right can be rewritten as dNδN(w,s)−1(w,s),δ N(w,s)−1(z,t)=N(w,s)−1dN(w,s),(z,t),(6.17) which is reasonable and nice for our purposes. The second term on the right can be rewritten as dNδN(z,t)N(w,s)−1(φ(z,t)),φ(z,t).(6.18) Let us think of this as being of the form dNδr(y,u),(y,u),(6.19) where ris a positive real number and (y,u)∈Hnsatisfies N(y,u)=1. Of course this expression is equal to 0 when r=1,and one can be interested in getting a bound for it in terms of r−1. Unfortunately one does not get a bound for (6.19) like O(|r−1|)in general, but more like O(|r−1|) for rreasonably close to 1. The bottom line is that the retraction φonto the unit sphere for Nis not Lipschitz, even in a small neighborhood of the sphere. To look at it another way, although the dilation mapping δris Lipschitz with constant rwith respect to dNon Hn,itdoesnothave good Lipschitz properties as a function of r, except on a small set. This is in contrast to the case of Euclidean geometry, where dilation by ris uniformly Lipschitz as a function of ron bounded subsets. Happy Fractals and Analysis on Metric Spaces 279 A closely related point is that while there are curves of finite length joining the origin in Hnto arbitrary elements of Hn, the trajectories of the dilations do not have this property. Certainly one can expect that it is more difficult to have Lipschitz retractions in the Heisenberg group than in Euclidean spaces, and this indicates that this is so even for relatively simple cases. Another basic mapping to consider is ψ(w,s)=δN(w,s)−2(w,s),(6.20) which takes Hnminus the origin to itself. This mapping is a reflection about the unit sphere for N, i.e., ψ(w,s)=(w,s) when N(w,s)=1, Nψ(w,s)=N(w,s)−1, and ψ(ψ(w,s)) = (w,s). Unlike the Euclidean case, there is once again trouble with the Lipschitz condition even on a small neighborhood of the unit sphere for N. 7. Some happy fractals from Helsinki There are clearly numerous variations for the type of construction about to be reviewed. We shall focus on a simple family with a lot of self-similarity. Let Nbe an odd integer greater than or equal to 5, and let Σ0denote the boundary of the unit cube in R3.ThusΣ 0consists of 6 twodimensional squares, each with sidelength 1. In the first stage of the construction, we subdivide each of these 6 squares into N2squares with sidelength 1/N .For each of the original 6 squares, we make a modification with the square of size 1/N in the middle. The “middle” makes sense because Nis odd. Specifically, we remove the middle squares, and replace each one with the union of the other 5 squares in the boundary of the cube with one face the middle square in question and which lies outside the unit cube with which we started. The surface that results from Σ0by making these modifications is denoted Σ1. This procedure can also be described as follows. Let R0denote the unit cube, so that Σ0=∂R0.Now define R1to be the union of R0and the 6 cubes with sidelength 1/N whose interiors are outside R0and which have a face which is a middle square of a face of R0. The surface Σ1is the boundary of R1. Using the decomposition of the boundary described in the first step, we can think of Σ1as the union of a bunch of two-dimensional squares of sidelength 1/N . Namely, there are 6 ·(N2−1) + 6 ·5 such squares. For each of these squares, we apply the same procedure as before. That is, we divide each square into N2squares of sidelength 1/N times the 280 S. Semmes sidelength of the squares that we have, so that the new squares have sidelength 1/N 2in this second step. For each of the squares from the first step, we make modifications only at the middle smaller squares just described, one middle small square for each square from the second step. Each of these middle small squares is removed and replaced with the union of 5 squares of the same sidelength which are in the boundary of the cube with interior outside R1and with one face being the small middle square in question. The result is a surface Σ2consisting of a bunch of squares of sidelength 1/N 2. The condition N≥5ishelpful for keeping the modifications at different places from bumping into each other or getting too close to doing that. One can also describe this in terms of adding a bunch of cubes of sidelength 1/N 2to R1, each with a face which is a middle square of a square from the first step, to get a new region R2. The surface Σ2is the boundary of R2. This process can be repeated indefinitely to get regions Rjand surfaces Σj=∂Rjfor all nonnegative integers j.Inthe limit we can take R to be the union of the Rj’s, and Σ to be the boundary of R, which is the same as the Hausdorff limit of the Σj’s. Of course this procedure is completely analogous to ones in the plane for producing snowflake curves. However, one does not get snowballs in the technical sense introduced by Pekka Koskela, because there are a lot of curves of finite length. Indeed, whenever a square is introduced in the construction, its four boundary segments are kept intact for all future stages, and hence in the limit. One can verify that Σ is a happy fractal. 8. More on Lipschitz functions Let (M,d(x, y)) be a metric space. Suppose that f(x)isareal or complex-valued function on M, and that Lis a nonnegative real number. We say that fis L-Lipschitz if |f(x)−f(y)|≤Ld(x, y)(8.1) for all x, y ∈M.Thusfis Lipschitz if it is L-Lipschitz for some L.Iff is Lipschitz, then we define fLip to be the supremum of |f(x)−f(y)| d(x, y) (8.2) Happy Fractals and Analysis on Metric Spaces 281 over all x, y ∈M, where this ratio is replaced with 0 when x=y.In other words, fis fLip-Lipschitz when fis Lipschitz, and fLip is the smallest choice of Lfor which fis L-Lipschitz. Note that · Lip is a seminorm, so that af +bgLip ≤|a|fLip +|b|gLip (8.3) for all constants a,band Lipschitz functions f,gon M. Also, fLip =0 if and only if fis a constant function on M. If fand gare real-valued L-Lipschitz functions on M, then the maximum and minimum of f,g, which are denoted max(f,g) and min(f,g), are L-Lipschitz functions too. Let us check this for max(f,g). It is enough to show that max(f,g)(x)−max(f,g)(y)≤Ld(x, y)(8.4) for all x, y ∈M, since one can interchange the roles of xand yto get a corresponding lower bound for max(f,g)(x)−max(f,g)(y). Assume, for the sake of definiteness, that max(f,g)(x)=f(x). Then we have max(f,g)(x)=f(x)≤f(y)+Ld(x, y) ≤max(f,g)(y)+Ld(x, y), (8.5) which is what we wanted. Here is a generalization of this fact. Lemma 8.6. Let {fσ}σ∈Abeafamily of real-valued functions on M which are all L-Lipschitz for some L≥0. Assume also that there is point pin Msuch that the set of real numbers {fσ(p):σ∈A}is bounded from above. Then the set {fσ(x):σ∈A}is bounded from above for every xin M(but not uniformly in xin general), and sup{fσ(x):σ∈A} is an L-Lipschitz function on M. Indeed, because fσis L-Lipschitz for all σin A,wehave that fσ(x)≤fσ(y)+Ld(x, y)(8.7) for all x,yin M. Applying this to y=p,wesee that {fσ(x):σ∈A} is bounded from above for every x,because of the analogous property for p.IfF(x)=sup{fσ(x):σ∈A}, then F(x)≤F(y)+Ld(x, y)(8.8) for all x,yin M,sothat Fis L-Lipschitz on M. For the record, let us write down the analogous statement for infima of L-Lipschitz functions. 282 S. Semmes Lemma 8.9. Let {fσ}σ∈Abeafamily of real-valued functions on M which are all L-Lipschitz for some L≥0. Assume also that there is point qin Msuch that the set of real numbers {fσ(q):σ∈A}is bounded from below. Then the set {fσ(x):σ∈A}is bounded from below for every xin M, and inf{fσ(x):σ∈A}is an L-Lipschitz function on M. For any point win M,d(x, w) defines a 1-Lipschitz function of x on M. This can be shown using the triangle inequality. Suppose now that f(x)isanL-Lipschitz function on M.Foreach w∈M, define fw(x)=f(w)+Ld(x, w). The fact that fis L-Lipschitz implies that f(x)≤fw(x) for all x, w ∈M.(8.10) Of course fx(x)=x, and hence f(x)=inf{fw(x):w∈M}.(8.11) Each function fw(x)isL-Lipschitz in x, since d(x, w)is1-Lipschitz in w. Similarly, we can set  fw(x)=f(x)−Ld(x, w), and then we have that f(x)=sup{ fw(x):w∈M},(8.12) and that  fw(x)isanL-Lipschitz function of xfor every w. Here is a variant of these themes. Let Ebe a nonempty subset of M, and suppose that fis a real-valued function on Ewhich is L-Lipschitz, so that |f(x)−f(y)|≤Ld(x, y)(8.13) for all x,yin M.Foreach win E, set fw(x)=f(x)+Ld(x, w) and  fw(x)=f(x)−Ld(x, w). Consider F(x)=inf{fw(x):w∈E},  F(x)=sup{ fw(x):w∈E}, (8.14) for xin M.For the same reasons as before, F(x)=  F(x)=f(x) when x lies in E. Using Lemmas 8.6 and 8.9, one can check that Fand  Fare L-Lipschitz real-valued functions on all of M, i.e., they are extensions of ffrom Eto Mwith the same Lipschitz constant L. If H(x)isany other real-valued function on Mwhich agrees with f on Eand is L-Lipschitz, then  fw(x)≤H(x)≤fw(x)(8.15) Happy Fractals and Analysis on Metric Spaces 283 for all win Eand xin M, and hence  F(x)≤H(x)≤F(x)(8.16) for all xin M. Remark 8.17.If Sis any nonempty subset of M, define dist(x, S) for x in Mby dist(x, S)=inf y∈Sd(x, y).(8.18) This function is always 1-Lipschitz in x,asinLemma 8.9. 9. Lipschitz functions of order α Let (M,d(x, y)) be a metric space, and let αbe apositive real number. A real or complex-valued function fon Mis said to be Lipschitz of order αif there is nonnegative real number Lsuch that |f(x)−f(y)|≤Ld(x, y)α (9.1) for all x, y ∈M. This reduces to the Lipschitz condition discussed in Section 8 when α=1.Weshall sometimes write Lip αfor the collection of Lipschitz functions of order α, which might be real or complex valued, depending on the context. One also sometimes refers to these functions as being “H¨older continuous of order α”. If fis Lipschitz of order α, then we define fLip αto be the supremum of |f(x)−f(y)| d(x, y)α (9.2) over all x, y ∈M, where this quantity is replaced with 0 when x=y.In other words, fLip αis the smallest choice of Lso that (9.1) holds for all x, y ∈M. This defines a seminorm on the space of Lipschitz functions of order α,asbefore, with fLip α=0if and only if fis constant. Of course fLip 1 is the same as fLip from Section 8. If fand gare real-valued functions on Mwhich are Lipschitz of order αwith constant L, then max(f,g) and min(f,g) are also Lipschitz of order αwith constant L. This can be shown in the same manner as for α=1. Similarly, the analogues of Lemmas 8.6 and 8.9 for Lipschitz functions of order αhold for essentially the same reasons as before. However, if α>1, it may be that the only functions that are Lipschitz of order αare the constant functions. This is the case when M=Rn, for instance, equipped with the standard Euclidean metric, because a function in Lip αwith α>1 has first derivatives equal to 0 everywhere. Instead of using derivatives, it is not hard to show that the function 284 S. Semmes has to be constant through more direct calculation too. On any metric space M,afunction which is Lipschitz or order αwith α>1isconstant on every path of finite length. This problem does not occur when α<1. Lemma 9.3. If 0<α≤1and a,bare nonnegative real numbers, then (a+b)α≤aα+bα. To see this, observe that max(a, b)≤(aα+bα)1/α,(9.4) and hence a+b≤max(a, b)1−α(aα+bα) ≤(aα+bα)1+(1−α)/α =(aα+bα)1/α. (9.5) Corollary 9.6. If (M,d(x, y)) is a metric space and αis a real number such that 0<α≤1, then d(x, y)αalso defines a metric on M. This is easy to check. The main point is that d(x, y)αsatisfies the triangle inequality, because of Lemma 9.3 and the triangle inequality for d(x, y). A function fon Mis Lipschitz of order αwith respect to the original metric d(x, y)ifand only if it is Lipschitz of order 1 with respect to d(x, y)α, and with the same norm. In particular, for each win M, d(x, w)αsatisfies (9.1) with L=1when 0 <α≤1, because of the triangle inequality for d(u, v)α. 10. Some functions on the real line Fix α,0<α≤1. For each nonnegative integer n, consider the function 2−nα exp(2nix)(10.1) on the real line R, where exp udenotes the usual exponential eu. Let us estimate the Lip αnorm of this function. Recall that |exp(iu)−exp(iv)|≤|u−v|(10.2) for all u, v ∈R. Indeed, one can write exp(iu)−exp(iv)asthe integral between uand vof the derivative of exp(it), and this derivative is iexp(it), which has modulus equal to 1 at every point. Thus, for any x, y ∈R,wehave that |2−nα exp(2nix)−2−nα exp(2niy)|≤2n(1−α)|x−y|.(10.3) Happy Fractals and Analysis on Metric Spaces 285 Of course (10.4) |2−nα exp(2nix)−2−nα exp(2niy)| ≤2−nα|exp(2nix)|+2 −nα|exp(2niy)|=2 −nα+1 as well. As a result, (10.5) |2−nα exp(2nix)−2−nα exp(2niy)| ≤2n(1−α)|x−y|α2−nα+11−α=2 1−α|x−y|α. This shows that the function (10.1) has Lip αnorm (with respect to the standard Euclidean metric on R) which is at most 21−α.Inthe opposite direction, if 2n(x−y)=π, then |2−nα exp(2nix)−2−nα exp(2niy)| =2 −nα|exp(2nix)|+2 −nα|exp(2niy)| =2 −nα+1 =2π−α|x−y|α, (10.6) so that the Lip αnorm is at least 2π−α. Now suppose that f(x)isacomplex-valued function on Rof the form f(x)= ∞  n=0 an2−nα exp(2nix),(10.7) where the an’s are complex numbers. We assume that the an’s are bounded, which implies that the series defining f(x) converges absolutely for each x. Set A= sup n≥0|an|.(10.8) Let mbeanonnegative integer. For each xin Rwe have that  ∞  n=m an2−nα exp(2nix)≤ ∞  n=m A2−nα =A(1 −2−α)−12−mα.(10.9) 292 S. Semmes Because we can take w=xin the infimum, we automatically have that AL(f)(x)≤f(x)(13.5) for all xin M.Inthe other direction, (13.2) and (13.4) lead to AL(f)(x)≥f(x)−fLip αfLip α Lα/(1−α) =f(x)−f1/(1−α) Lip αL−α/(1−α). (13.6) We also have that AL(f)isL-Lipschitz on M,asinLemma 8.9. Suppose that h(x)isareal-valued function on Mwhich is L-Lipschitz and satisfies h(x)≤f(x) for all xin M. Then h(x)≤h(w)+Ld(x, w)≤f(w)+Ld(x, w)(13.7) for all x,win M. Hence h(x)≤AL(f)(x)(13.8) for all xin M. Similarly, one can consider BL(f)(x)=sup{f(w)−Ld(x, w):w∈M},(13.9) and show that (13.10) BL(f)(x) = sup{f(w)−Ld(x, w):w∈M, Ld(x, w)1−α≤fLip α}. This makes it clear that the supremum is finite. As before, f(x)≤BL(f)(x)≤f(x)+f1/(1−α) Lip αL−α/(1−α),(13.11) and BL(f)isL-Lipschitz. If h(x)isareal-valued function on Mwhich is L-Lipschitz and satisfies f(x)≤h(x) for all xin M, then BL(f)(x)≤h(x)(13.12) for all xin M. Happy Fractals and Analysis on Metric Spaces 293 14. Approximation operators, 2 Let (M,d(x, y)) be a metric space, and let µbe apositive Borel measure on M.Weshall assume that µis a doubling measure, which means that there is a positive real number Csuch that µ(B(x, 2r)) ≤Cµ(B(x, r))(14.1) for all xin Mand positive real numbers r, and that the µ-measure of any open ball is positive and finite. Let tbeapositive real number. Define a function pt(x, y)onM×M by pt(x, y)=1−t−1d(x, y) when d(x, y)≤t =0 when d(x, y)>t, (14.2) and put ρt(x)=M pt(x, y)dµ(y).(14.3) This is positive for every xin M,because of the properties of µ. Also put φt(x, y)=ρt(x)−1pt(x, y),(14.4) so that M φt(x, y)dµ(y)=1(14.5) for all xin Mby construction. Fix a real number α,0<α≤1, and let fbe a complex-valued function on Mwhich is Lipschitz of order α. Define Pt(f)onMby Pt(f)(x)=M φt(x, y)f(y)dµ(y).(14.6) Because of (14.5), Pt(f)(x)−f(x)=M φt(x, y)(f(y)−f(x)) dµ(y),(14.7) and hence |Pt(f)(x)−f(x)|≤M φt(x, y)|f(y)−f(x)|dµ(y) ≤M φt(x, y)fLip αtαdµ(y)=fLip αtα. (14.8) In the second step we employ the fact that φt(x, y)=0when d(x, y)≥t. 294 S. Semmes Suppose that xand zare elements of M, and consider |Pt(f)(x)−Pt(f)(z)|.(14.9) If d(x, z)≥t, then |Pt(f)(x)−Pt(f)(z)| ≤|Pt(f)(x)−f(x)|+|f(x)−f(z)|+|Pt(f)(z)−f(z)| ≤fLip α(2 tα+d(x, z)α)≤3tα−1fLip αd(x, z). (14.10) Assume instead that d(x, z)≤t.Inthis case we write Pt(f)(x)−Pt(f)(z) as (14.11) M (φt(x, y)−φt(z,y)) f(y)dµ(y) =M (φt(x, y)−φt(z,y)) (f(y)−f(x)) dµ(y), using (14.5). This yields |Pt(f)(x)−Pt(f)(z)| ≤M|φt(x, y)−φt(z,y)||f(y)−f(x)|dµ(y) ≤(2t)αfLip αB(x,2t)|φt(x, y)−φt(z,y)|dµ(y), (14.12) where the second step relies on the observation that φt(x, y)−φt(z,y) is supported, as a function of y,inthe set B(x, t)∪B(z,t)⊆B(x, 2t).(14.13) Of course (14.14) φt(x, y)−φt(z,y) =(ρt(x)−1−ρt(z)−1)pt(x, y)+ρt(z)−1(pt(x, y)−pt(z,y)). Notice that |pt(x, y)−pt(z,y)|≤t−1d(x, z)(14.15) for all yin M.Tosee this, it is convenient to write pt(u, v)asλt(d(u, v)), where λt(r)isdefined for r≥0byλt(r)=1−t−1rwhen 0 ≤r≤t, and λt(r)=0when r≥t.Itiseasy to check that λtis t−1-Lipschitz, and hence λt(d(u, v)) is t−1-Lipschitz on Mas a function of ufor each fixed v, since d(u, v)is1-Lipschitz as a function of ufor each fixed v. Happy Fractals and Analysis on Metric Spaces 295 These computations and the doubling condition for µpermit one to show that B(x,2t)|φt(x, y)−φt(z,y)|dµ(y)≤C1t−1d(x, z)(14.16) for some positive real number C1which does not depend on x,z,ort. (Exercise.) Altogether, we obtain that Pt(f)Lip 1 ≤max(3,2αC1)tα−1fLip α.(14.17) 15. A kind of Calder´on-Zygmund decomposition related to Lipschitz functions Let (M,d(x, y)) be a metric space, and let fbe a real-valued function on M. Consider the associated maximal function N(f)(x)= sup y∈M y=x |f(y)−f(x)| d(y,x),(15.1) where this supremum may be +∞. Let Lbe apositive real number, and put FL={x∈M:N(f)(x)≤L}.(15.2) We shall assume for the rest of this section that FL=∅.(15.3) As in Section 13, define AL(f)by AL(f)(x)=inf{f(w)+Ld(x, w):w∈M}.(15.4) We shall address the finiteness of this infimum in a moment. As before, AL(f)(x)≤f(x)(15.5) for all xin M. If uis any element of FL, then |f(y)−f(u)|≤Ld(y,u)(15.6) for all yin M. Let xand wbe arbitrary points in M. The preceding inequality implies that f(u)≤f(w)+Ld(u, w),(15.7) and hence f(u)−Ld(x, u)≤f(w)+L(d(u, w)−d(x, u)) ≤f(w)+Ld(x, w), (15.8) 296 S. Semmes by the triangle inequality. This yields f(u)−Ld(x, u)≤AL(f)(x),(15.9) which includes the finiteness of AL(f)(x). If we take x=u, then we get f(u)≤AL(f)(u), so that f(u)=AL(f)(u) for all u∈FL.(15.10) For x∈ FL,weobtain f(x)−2Ld(x, u)≤AL(f)(x)(15.11) for all uin FL,bycombining (15.9) and (15.6) with y=x.Inother words, f(x)−AL(f)(x)≤2Ldist(x, FL).(15.12) Note that AL(f)isL-Lipschitz on M,byLemma 8.9. In the same way, if BL(f)(x)=sup{f(w)−Ld(x, w):w∈M},(15.13) then f(x)≤BL(f)(x)≤f(x)+2Ldist(x, FL)(15.14) for all xin M, and BL(f)isL-Lipschitz. 16. A brief overview of “atoms” Let (M,d(x, y)) be a metric space, and let sbe apositive real number. We say that (M,d(x, y)) is Ahlfors-regular of dimension sif Mis complete as a metric space, and if there is a positive Borel measure µ on Msuch that C−1 1rs≤µ(B(x, r)) ≤C1rs (16.1) for some positive real number C1, all xin M, and all r>0 such that r≤diam Mif Mis bounded. As a basic example, if Mis n-dimensional Euclidean space Rnwith the standard metric, and if µis Lebesgue measure, then in fact µ(B(x, r)) is equal to a constant times rn, where the constant is simply the volume of the unit ball. More exotically, one can consider simply-connected nonabelian nilpotent Lie groups, such as the Heisenberg groups. For these spaces one still has natural dilations as on Euclidean spaces, and Lebesgue measure is compatible with both the group structure and the dilations, in such a way that the measure of a ball of radius ris equal to a constant times rs, where sis now a geometric dimension that is larger Happy Fractals and Analysis on Metric Spaces 297 than the topological dimension. Other examples include fractals such as the Sierpinski gasket and carpet. Fix a metric space (M,d(x, y)) and a measure µon Msatisfying the conditions in the definition of Ahlfors-regularity, with dimension s. The following fact is sometimes useful: there is a constant k1≥1sothat if x is an element of Mand r,Rare positive numbers, with r≤R, then the ball B(x, R) can be covered by a collection of at most k1(R/r)sclosed balls of radius r.IfMis bounded, then we may as well assume that r<diam Mhere, because Mis automatically contained in a single ball with radius diam M.Wemay also assume that R≤diam M, since we could simply replace Rwith diam Mif Ris initially chosen to be larger than that. To establish the assertion in the preceding paragraph, let us begin with a preliminary observation. Suppose that Ais a subset of B(x, R) such that d(x, y)>rfor all x,yin A. Then the number of elements of Ais at most k1(R/r)s,ifwechoose k1large enough (independently of x,R, and r). Indeed,  a∈A µ(B(a, r/2)) = µ a∈A B(a, r/2)≤µ(B(x, 3R/2)),(16.2) where the first equality uses the disjointness of the balls B(a, r/2), a∈A. The Ahlfors-regularity property then applies to give a bound on the number of elements of Aof the form k1(R/r)s.Now that we have such a bound, suppose that Ais also chosen so that the number of its elements is maximal. Then B(x, R)⊆ a∈A B(a, r).(16.3) In other words, if zis an element of B(x, R), then d(z,a)≤rfor some a in A,because otherwise we could add zto Ato get a set which satisfies the same separation condition as A, but which has 1 more element. This yields the original assertion. In particular, closed and bounded subsets of Mare compact. This uses the characterization of compactness in terms of completeness and total boundedness, where the latter holds for bounded subsets of Mby the result just discussed. Let us look at some special families of functions on M, called atoms, as in [19]. For the sake of definiteness, we make the convention that a “ball” in Mmeans a closed ball (with some center and radius), if nothing else is specified. Suppose that pis a real number and ris an extended 298 S. Semmes real number such that 0<p≤1,1≤r≤∞,p<r.(16.4) An integrable complex-valued function a(x)onMwill be called a (p, r)-atom if it satisfies the following three conditions: first, there is a ball Bin Msuch that the support of ais contained in B, i.e., a(x)=0 when x∈M\B; second, M a(x)dµ(x)=0;(16.5) and third, 1 µ(B)M|a(x)|rdµ(x)1/r ≤µ(B)−1/p.(16.6) If r=∞, then (16.6) is interpreted as meaning that the supremum (or essential supremum, if one prefers) of ais bounded by µ(B)−1/p. The size condition (16.6) may seem a bit odd at first. A basic point is that it implies M|a(x)|pdµ(x)≤1,(16.7) by Jensen’s inequality. The index rreflects a kind of regularity of the atom, and notice that a (p, r1)-atom is automatically a (p, r2)-atom when r1≥r2. There are versions of this going in the other direction, from r2 to r1, and we shall say more about this soon. Suppose that a(x)isa(p, r)-atom on Mand that φ(x) lies in Lip α on Mfor some α. Consider the integral M a(x)φ(x)dµ(x).(16.8) Let B=B(z,t)bethe ball associated to a(x)asinthe definition of an atom. The preceding integral can be written as B(z,t) a(x)(φ(x)−φ(z)) dµ(x),(16.9) using also (16.5). Thus M a(x)φ(x)dµ(x)≤B(z,t)|a(x)||φ(x)−φ(z)|dµ(x) ≤µ(B(z,t))1−(1/p)tαφLip α. (16.10) Happy Fractals and Analysis on Metric Spaces 299 Ahlfors-regularity implies that M a(x)φ(x)dµ(x)≤C1−(1/p) 1t(1−(1/p))s+αφLip α.(16.11) In particular, M a(x)φ(x)dµ(x)≤C1−(1/p) 1φLip α (16.12) when α= ((1/p)−1) s. If we want to be able to choose α= ((1/p)−1) sand have α≤1, then we are lead to the restriction p≥s s+1.(16.13) Indeed, this condition does come up for some results, even if much of the theory works without it. There can also be some funny business at the endpoint, so that one might wish to assume a strict inequality in (16.13), or some statements would have to be modified when equality holds. In some situations this type of restriction is not really necessary, perhaps with some adjustments. Let us mention two basic scenarios. First, suppose that our metric space Mis something like a self-similar Cantor set, such as the classical “middle-thirds” Cantor set. In this case there are a lot of Lip αfunctions for all α>0, and, for that matter, there are a lot of functions which are locally constant. The computation giving (16.12) still works when α>1, and this is true in general. On the other hand, if M=Rnwith the standard Euclidean metric, then there other ways to define classes of more smooth functions, through conditions on higher derivatives. In connection with this, one can strengthen (16.5) by asking that the integral of an atom times a polynomial of degree at most some number is equal to 0. If one does this, then there are natural extensions of (16.12) for α>1, obtained by subtracting a polynomial approximation to φ(x). A basic manner in which atoms can be used is to test localization properties of linear operators. Suppose that Tis a bounded linear operator on L2(M), and that ais a (p, 2)-atom on M. Consider T(a)(16.14) (as well as T∗(a), for that matter). This is well-defined as an element of L2(M), since alies in L2(M). If B=B(z,t)isthe ball associated 300 S. Semmes to ain the definition of an atom, then the estimate 1 µ(B)M|T(a)(x)|2dµ(x)1/2 ≤T2,21 µ(B)M|a(x)|2dµ(x)1/2 ≤T2,2µ(B)−1/p (16.15) provides about as much information about T(a) around B,on2B= B(z,2t), say, as one might reasonably expect to have. However, in many situations one can expect to have decay of T(a)away from B,insuch a way that T(a)p≤k(16.16) for some constant kwhich does not depend on a. In this argument it is natural to take r=2,but a basic result in the theory is that one has some freedom to vary r.Specifically, if bis a (p, r)-atom on M, then it is possible to write bas b= i βibi,(16.17) where each biisa(p, ∞)-atom, each βiis a complex number, and i|βi|p is bounded by a constant that does not depend on b(but which may depend on por r). Let us give a few hints about how one can approach this. As an initial approximation, one can try to write bas b=βb+ j γjcj,(16.18) where bisa(p, ∞)-atom, βis a complex number such that |β|is bounded by a constant that does not depend on b, each cjisa(p, r)-atom, and j|γj|p≤1/2, say. If one can do this, then one can repeat the process indefinitely to get a decomposition as in (16.17). In order to derive (16.18), the method of Calder´on-Zygmund decompositions can be employed. Happy Fractals and Analysis on Metric Spaces 301 Recall that  k τkp ≤ k τp k (16.19) for nonnegative real numbers τkand 0 <p≤1. As a consequence, if {fk}is a family of measurable functions on Msuch that M|fk(x)|pdµ(x)≤1 for all k,(16.20) and if {θk}is a family of constants, then M k θkfk(x) p dµ(x)≤ k|θk|p.(16.21) Because of this, bounds on l|αl|pare natural when considering sums of the form lαlal, where the al’s are (p, r)-atoms and the αl’s are constants. A fundamental theorem concerning atoms is the following. Suppose that Tis a bounded linear operator on L2(M) again. (One could start as well with a bounded linear operator on some other Lvspace, with suitable adjustments.) Suppose also that there is a constant kso that (16.16) holds for all (p, 2)-atoms, where 0 <p≤1, as before, or even simply for all (p, ∞)-atoms. Then Tdetermines a bounded linear operator on Lqfor 1 <q<2. This indicates how atoms are sufficiently abundant to be useful. The proof of this theorem relies on an argument like the one in Marcinkeiwicz interpolation. In the traditional setting, one of the main ingredients is to take a function fin Lqon M, and, for a given positive real number λ, write it as f1+f2, where f1(x)=f(x) when |f(x)|≤λ, f1(x)=0when |f(x)|>0, f2(x)=f(x) when |f(x)|>λ, and f2(x)=0 when |f(x)|≤λ. Notice in particular that f1lies in Lwfor all w≥q, and that f2lies in Lufor all u≤q.For the present purposes, the idea is to use decompositions which are better behaved, with f2having a more precise form as a sum of multiples of atoms. The Calder´on-Zygmund method is again applicable, although it should be mentioned that one first works with (p, r)-atoms with one choice of r, and then afterwards makes a conversion to a larger rusing the results described before. In addition to considering the effect of Ton atoms, one can consider the effect of T∗on atoms, and this leads to conclusions about Ton Lq for q>2, by duality. 308 S. Semmes [85] E. M. 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