"Graph Paper" Trace Characterizations of Functions of Finite Energy
““Graph Paper” Trace Characterizations of Functionsof Finite Energy
Robert S. Strichartz
Abstract.
We characterize functions of finite energy in the plane in terms of their traceson the lines that make up “graph paper” with squares of side length m n for all n , andcertain -order Sobolev norms on the graph paper lines. We also obtain analogous resultsfor functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinskicarpet.
1. Introduction
Functions of finite energy play an important role in analysis and probability. On Eu-clidean space or a domain in Euclidean space, these are just the functions whose gradient inthe distribution sense belongs to L , with the energy given by (cid:90) |∇ F | dx (1.1)As such they make up a homogeneous Sobolev space that we will denote here as H . Themore usual inhomogeneous Sobolev space is smaller, requiring that F ∈ L as well [
11, 15 ].There are many ways to generalize the notion of finite energy to other contexts. For example,as the functions in the domain of a Dirichlet form [ ]. In this paper we will only considerfunctions of finite energy in regions in the plane, and on two classical fractals, the Sierpinskigasket [
10, 17 ] and the Sierpinski carpet [
2, 3 ].It is well-known that functions of finite energy in the plane (or in higher dimensions) donot have to be continuous, so the value F ( x, y ) at a point is not meaningful. Nevertheless,the trace on a line, say T F ( x ) = F ( x, -orderhomogeneous Sobolev space that we will denote here by H / ( R ), defined by the finitenessof (cid:90) ∞−−∞ (cid:90) ∞−∞ | f ( x ) − f ( y ) | | x − y | dxdy , (1.2)with a corresponding norm estimate. Of course it is the norm estimate that is importantsince it implies the existence of the trace by routine arguments. The result is sharp, meaningthat there is an extension operator from H / ( R ) to H ( R ). There are in fact two rathernatural -order Sobolev spaces on R . The other one, which we denote by ˜ H / ( R ) is larger,and only requires the finiteness of an integral like (1.2) where the integration is restricted to Research supported in part by the National Science Foundation, grant DMS-1162045. a r X i v : . [ m a t h . F A ] J un he region | x − y | (cid:54)
1. We will show that the trace of a function F of finite energy in thestrip { ( x, y ) : 0 < y < } only belongs to ˜ H ( R ). In particular this implies that there doesnot exist a Sobolev extension theorem from H of the strip to H ( R ), even though such aresult for inhomogeneous Sobolev spaces is well-known and essentially trivial.The trace of a function of finite energy on a single line does not, of course, determine thefunction. What about the trace of an infinite collection of lines that together form a densesubset of the plane? A simple example is the set of lines of “graph paper,” where we takethe graph paper squares to have side length m n , where m is an integer ( m (cid:62)
2) and n variesover Z , so the graph papers GP m n are nested. The main results of this paper are first a tracetheorem that characterizes the traces of H ( R ) functions on GP m n in terms of a Sobolevspace H / ( GP m n ) with a given norm, and then the characterization of H ( R ) in terms ofa uniform bound on the norms of the traces on GP m n as n → −∞ . The trace theorem isdiscussed in section 3 in the context of Sobolev spaces H / on metric graphs (graphs whoseedges have specified length, [ ]), as discussed in section 2. Because the functions in thesespaces need not be continuous, the key issue is to understand a kind of “gluing” conditionat the vertices of the graph. It turns out that this condition was given in [ ]. For theconvenience of the reader we give all the proofs in section 2, although many of the results arealready known, because they are usually treated in the context of inhomogeneous Sobolevspaces. In section 4 we discuss the trace characterizations of H ( R ). In section 5 we discussthe analogous results on the two fractals. It turns out that the trace theorems are alreadyknown [
7, 8, 9 ], and the Sobolev spaces are H β for values satisfying < β <
1. The spacesof functions of finite energy on these fractals consist of continuous functions, as do the tracespaces, so there is no difficulty defining the traces, and the “gluing” condition at vertices issimply continuity. Thus the fractal analog of the trace characterization is perhaps simplerthan the theorem in the plane. We also characterize the traces on Julia sets of functions offinite energy in the unbounded component of the complement of the Julia set. We believestrongly that there is a great benefit to thinking about problems in both the smooth and thefractal contexts, and looking for interactions in the ideas that emerge. We hope this papergives some support to this point of view.
2. Metric Graphs A metric graph G = ( V, E, L e ) consists of a graph ( V, E ) with vertices V and edges E ,and a function that assigns a length L e in (0 , ∞ ] to each edge e ∈ E . Definition . For a metric graph G = ( V, E, L e ) , define the homogeneous Sobolevnorm (cid:107) f (cid:107) H / ( G ) = (cid:88) e ∈ E (cid:90) L e (cid:90) L e | f ( e ( x )) − f ( e ( y )) | | x − y | dxdy + (cid:88) e ∼ e (cid:48) (cid:90) L | f ( e ( x )) − f ( e (cid:48) ( x )) | x dx (2.1) (in the second sum L = min( L e , L e (cid:48) ) , and the parameterizations of e and e (cid:48) are chosen so that e (0) and e (cid:48) (0) correspond to the intersection point). We define the Sobolev space H / ( G ) tobe the equivalence classes (modulo constants) of locally L functions for which the norm isfinite. It is easy to see that H / ( G ) is a Hilbert space. xample . Let G = R , so G has no vertices and a single edge of infinite length. Weneed to modify (2.1) in this case to read (cid:107) f (cid:107) H / ( R ) = (cid:90) ∞−∞ (cid:90) ∞−∞ | f ( x ) − f ( y ) | | x − y | dxdy . (2.2)For this example we also want to consider the smaller norm (cid:107) f (cid:107) H / ( R ) = (cid:90) (cid:90) | x − y | (cid:54) | f ( x ) − f ( y ) | | x − y | dxdy (2.3)and corresponding larger Sobolev space ˜ H / ( R ).We note that the space H / ( R ) is M¨obius invariant, meaning that f ∈ H / ( R ) if andonly if f ◦ M ∈ H / ( R ) with equal norms, for M ( x ) = ax + bcx + d with ( a bc d ) ∈ SL(2 , R ). Indeed itsuffices to verify this for translations M ( x ) = x + b , dilations M ( x ) = ax and the inversion M ( x ) = x , where it follows by a simple change of variable in the integral defining the norm.We note that the same statement is false for ˜ H / ( R ).We may easily characterize these norms and spaces in terms of the Fourier transform ˆ f .The finiteness of the norm easily implies that f is a tempered distribution so ˆ f is well definedas a tempered distribution, and the equivalence of the functions that differ by a constantmeans ˆ f is only defined up to the addition of an arbitrary multiple of the delta function.Note that there is no “canonical” choice of f and ˆ f within each equivalence class. Theorem . a) f ∈ H / ( R ) if and only if ˆ f may be identified with a function that islocally in L in the complement of the origin with (cid:90) ∞−∞ | ˆ f ( ξ ) | | ξ | dξ < ∞ , (2.4) and (2.4) is in fact a constant multiple of (2.2)b) f ∈ ˜ H / ( R ) if and only if ˆ f may be identified with a function that is locally in L inthe compliment of the origin, with (cid:90) | ξ | (cid:62) | ˆ f ( ξ ) | | ξ | dξ + (cid:90) | ξ | (cid:54) | ˆ f ( ξ ) | | ξ | dξ < ∞ , (2.5) and (2.5) is bounded above and below by a multiple of (2.3). Proof. a) is of course well-known, and follows from the formal computation (cid:107) f (cid:107) H / ( R ) = (cid:90) ∞−∞ (cid:90) ∞−∞ | f ( x + t ) − f ( x ) dx dtt = (cid:90) ∞−∞ (cid:90) ∞−∞ | ˆ f ( ξ ) | | e πiξt − | dξ dtt = c (cid:90) ∞−∞ | ˆ f ( ξ ) | | ξ | dξ for c = (cid:82) ∞−∞ | e πit − | t dt . o prove b) we similarly compute (cid:107) f (cid:107) H / ( R ) = (cid:90) − (cid:90) ∞−∞ | f ( x + t ) − f ( x ) dx dtt = (cid:90) ∞−∞ | ˆ f ( ξ ) | (cid:18)(cid:90) − | e πiξt − | dtt (cid:19) dξ Now (cid:90) − | e πiξt − | dtt = | ξ | (cid:90) | ξ |−| ξ | | e πit − | dtt ,and for | ξ | (cid:62) | ξ | (cid:54)
1, the integrand is bounded above and below by a constant, so the integral isbounded above and below by the length of the interval. This shows the equivalence of (2.3)and (2.5).The formal computation easily implies that any f ∈ ˜ H / ( R ) has a Fourier transformsatisfying (2.5). To complete the proof we need to show that any locally L function g ( ξ )with (cid:90) | ξ | (cid:62) | g ( ξ ) | | ξ | dξ + (cid:90) | ξ | (cid:54) | g ( ξ ) | | ξ | dξ < ∞ (2.6)is in fact the Fourier transform of a function in ˜ H / ( R ). Since the only problem is nearthe origin, we may assume that g is supported in [ − , h ( ξ ) = ξg ( ξ ). Note that h ∈ L by (2.6). We define a distribution ˜ g associated to g as follows. Note that (cid:104) ˜ g, ϕ (cid:105) = (cid:82) h ( ξ ) ϕ ( ξ ) dξξ is well-defined for any ϕ ∈ S with ϕ (0) = 0. Choose ψ ∈ S with ψ (0) = 1.Then ϕ ( ξ ) = ( ϕ ( ξ ) − ϕ (0) ψ ( ξ )) + ϕ (0) ψ ( ξ ), with the first summand vanishing at the origin.We will choose to have (cid:104) ˜ g, ψ (cid:105) = 0, so our definition of ˜ g is (cid:104) ˜ g, ϕ (cid:105) = (cid:90) h ( ξ ) ( ϕ ( ξ ) − ϕ (0) ψ ( ξ )) dξξ . (2.7)It follows that h ( ξ ) = ξ ˜ g ( ξ ) in the distribution sense. The inverse Fourier transform of ˜ g is the function f . Note that f has a derivative in L so it is continuous, and the formalcomputation shows that f ∈ ˜ H / ( R ). (cid:3) A trivial consequence of the theorem is that the space ˜ H / ( R ) is strictly larger than H / ( R ). On the other hand, L ( R ) ∩ ˜ H / ( R ) = L ( R ) ∩ H / ( R ). Example . Let G be the graph with one vertex and two edges of infinite length meetingat the vertex. We may realize G as the real line with edges ( −∞ ,
0] and [0 , ∞ ), and we writeit as R − ∪ R + . We see that (2.1) explicitly is (cid:107) f (cid:107) H / ( R − ∪ R + ) = (cid:90) −∞ (cid:90) −∞ | f ( x ) − f ( y ) || x − y | dxdy + (cid:90) ∞ (cid:90) ∞ | f ( x ) − f ( y ) | | x − y | dxdy + (cid:90) ∞ | f ( x ) − f ( − x ) | x dx (2.8) heorem . The spaces H / ( R − ∪ R + ) and H / ( R ) are identical with equivalentnorms. Proof.
This result is essentially contained in [ ], section III.3. Let x, y stand forvariables that are always positive. Since (cid:82) ∞ dy ( x + y ) = x we have (cid:90) ∞ | f ( x ) − f ( − x ) | x dx = (cid:90) ∞ (cid:90) ∞ | f ( x ) − f ( − x ) | ( x + y ) dydx .Writing f ( x ) − f ( − x ) = ( f ( x ) − f ( y )) + ( f ( y ) − f ( − x )), we have by the triangle inequality (cid:18)(cid:90) ∞ | f ( x ) − f ( − x ) | x dx (cid:19) / (cid:54) (cid:18)(cid:90) ∞ (cid:90) ∞ | f ( x ) − f ( y ) | ( x + y ) dydx (cid:19) / + (cid:18)(cid:90) ∞ (cid:90) ∞ | f ( y ) − f ( − x ) | | x + y | dydx (cid:19) / (cid:54) (cid:107) f (cid:107) H / ( R ) since x + y ) (cid:54) x − y ) . This yields the bound of (2.8) by a multiple of (cid:107) f (cid:107) H / ( R ) . A similarargument gives (cid:18)(cid:90) ∞ (cid:90) ∞ | f ( y ) − f ( − x ) | | x + y | dydx (cid:19) / (cid:54) (cid:18)(cid:90) ∞ (cid:90) ∞ f ( x ) − f ( y ) | | x + y | dydx (cid:19) / + (cid:18)(cid:90) ∞ | f ( x ) − f ( − x ) | x dx (cid:19) / for the bound in the other direction. (cid:3) Example . Let G be the graph Z ; in other words the vertices are the integers and theedges are [ k, k + 1] for k ∈ Z of length 1. Then (2.1) is explicitly (cid:107) f (cid:107) H / ( Z ) = (cid:88) k ∈ Z (cid:90) k +1 k (cid:90) k +1 k | f ( x ) − f ( y ) | | x − y | dxdy + (cid:88) k ∈ Z (cid:90) | f ( k + t ) − f ( k − t ) | t dt . (2.9) Theorem . The spaces H / ( Z ) and ˜ H / ( R ) are identical with equivalent norms. Proof.
The first term on the right side of (2.9) is clearly bounded by (cid:107) f (cid:107) H / ( R ) . Forthe second term we note that an argument as in the proof of Theorem 2.3 gives the estimate (cid:90) | f ( k + t ) − f ( k − t ) | t dt (cid:54) c (cid:90) k +1 k − (cid:90) k +1 k − | f ( x ) − f ( y ) | | x − y | dxdy ,and summing over k ∈ Z we obtain (cid:88) k ∈ Z (cid:90) | f ( k + t ) − f ( k − t ) | t dt (cid:54) c (cid:90) (cid:90) | x − y | (cid:54) | f ( x ) − f ( y ) | | x − y | dxdy .A straightforward estimate controls the integral over 1 (cid:54) | x − y | (cid:54) | x − y | (cid:54)
1, so we have (cid:107) f (cid:107) H / ( Z ) (cid:54) c (cid:107) f (cid:107) H / ( R ) . or the reverse estimate we use an argument in the proof of Theorem 2.3 to obtain (cid:90) kk − (cid:90) k +1 k | f ( x ) − f ( y ) | | x − y | dxdy (cid:54) (cid:90) kk − (cid:90) kk − | f ( x ) − f ( y ) | | x − y | dxdy + (cid:90) k +1 k (cid:90) k +1 k | f ( x ) − f ( y ) | | x − y | dxdy + (cid:90) t | f ( k + t ) − f ( k − t ) | t dt and then sum over k ∈ Z . (cid:3) Example . Let G be the square graph SQ δ with side length δ . So SQ δ has 4 verticesthat we will identify with the points (0 , δ, δ, δ ), (0 , δ ) in the plane and 4 edges oflength δ . Then (cid:107) f (cid:107) H / ( SQ δ ) = (cid:90) δ (cid:90) δ | f ( x, − f ( y, | | x − y | dxdy + (cid:90) δ (cid:90) δ | f ( δ, x ) − f ( δ, y ) | | x − y | dxdy + (cid:90) δ (cid:90) δ | f ( x, δ ) − f ( y, δ ) | | x − y | dxdy + (cid:90) δ (cid:90) δ | f (0 , x ) − f (0 , y ) | | x − y | dxdy + (cid:90) δ | f ( x, − f (0 , x ) | dxx + (cid:90) δ | f ( x, − f ( δ, δ − x ) | dxx + (cid:90) δ | f ( x, δ ) − f ( δ, x ) | dxx + (cid:90) δ | f (0 , x ) − f ( δ − x, δ ) | dxx . (2.10)Although the H / ( SQ δ ) norm does not involve comparisons between values on oppositeedges, it is not difficult to show bounds (cid:90) | f ( x, − f ( x, δ ) | dx (cid:54) cδ (cid:107) f (cid:107) H / ( SQ δ ) (cid:90) | f (0 , y ) − f ( δ, y ) | dy (cid:54) cδ (cid:107) f (cid:107) H / ( SQ δ ) . (2.11) Example . Let G be the graph paper graph GP δ with vertices at { ( jδ, kδ ) } , j, k ∈ Z and horizontal and vertical edges of length δ joining ( jδ, kδ ) with (( j + 1) δ, kδ ) and ( jδ, kδ ) ith ( jδ, ( k + 1) δ ). The norm is given by (cid:107) f (cid:107) H / ( GP δ ) = (cid:88) j (cid:88) k (cid:90) δ (cid:90) δ | f ( jδ + x, kδ ) − f ( jδ + y, kδ ) | | x − y | dxdy + (cid:88) j (cid:88) k (cid:90) δ (cid:90) δ | f ( jδ, kδ + x ) − f ( jδ, kδ + y ) | | x − y | dxdy + (cid:88) j (cid:88) k (cid:90) δ − δ | f ( jδ + x, kδ ) − f ( jδ, kδ + x ) | dx | x | + (cid:88) j (cid:88) k (cid:90) δ | f ( jδ + x, kδ ) − f ( jδ − x, kδ ) | dxx + (cid:88) j (cid:88) k (cid:90) δ | f ( jδ, kδ + x ) − f ( jδ, kδ − x ) | dxx . (2.12)Of course we could get an equivalent norm by deleting the last two sums in (2.12), as theyare controlled by the third sum. We may regard GP δ as a countable union of square graphs SQ δ ,and it is easily seen that f ∈ H / ( GP δ ) if and only if the restriction of f to each of thesquare graphs is in H / ( SQ δ ) with the sum of the squares of the norms (cid:107) f (cid:107) H / ( SQ δ ) finite,and this gives an equivalent norm.
3. Traces of functions of finite energy
Consider the homogeneous Sobolev space H ( R ) of functions with finite energy (cid:107) F (cid:107) H ( R ) = (cid:90) R |∇ F ( x, y ) | dxdy . (3.1)These form a Hilbert space modulo constants. Functions of finite energy do not have to becontinuous, as the example F ( x, y ) = log | log( x + y ) | (multiplied by an appropriate cutofffunction) shows. However, it is well-known that these functions have well-defined traceson straight lines that are in H / ( R ), and H / ( R ) is the exact space of traces. Since theusual treatment of traces involves inhomogeneous Sobolev spaces we give the proof for theconvenience of the reader. We omit the routine step of actually defining the traces and justprove the norm estimates. Theorem . The trace map T : H ( R ) → H / ( R ) given formally by T F ( x ) = F ( x, is continuous, (cid:107) T F (cid:107) H / ( R ) (cid:54) c (cid:107) F (cid:107) H ( R ) . (3.2) Moreover there exists a continuous extension map E : H / ( R ) → H ( R ) with T Ef = f and (cid:107) Ef (cid:107) H ( R ) (cid:54) c (cid:107) f (cid:107) H / ( R ) (3.3) roof. We work on the Fourier transform side, where (cid:107) F (cid:107) H ( R ) = (cid:90) R ( ξ + η ) | ˆ F ( ξ, η ) | dξdη and (3.4)( T f ) ∧ ( ξ ) = (cid:90) ∞−∞ ˆ F ( ξ, η ) dη . (3.5)By Theorem 2.2 we have (cid:107) T f (cid:107) H / ( R ) = (cid:90) ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ ˆ F ( ξ, η ) dη (cid:12)(cid:12)(cid:12)(cid:12) | ξ | dξ .By Cauchy-Schwarz we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞−∞ ˆ F ( ξ, η ) dη (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:18)(cid:90) ∞−∞ ( ξ + η ) | ˆ F ( ξ, η ) | dη (cid:19) (cid:18)(cid:90) ∞−∞ ξ + η dη (cid:19) = π | ξ | (cid:18)(cid:90) ∞−∞ ( ξ + η ) | ˆ F ( ξ, η ) | dη (cid:19) so (cid:107) T f (cid:107) H / ( R ) (cid:54) π (cid:90) ∞−∞ (cid:90) ∞−∞ ( ξ + η ) | ˆ F ( ξ, η ) | dη = π (cid:107) F (cid:107) H ( R ) proving (3.2).Conversely, given f ∈ H / ( R ) define Ef = F by the Poisson integral F ( x, y ) = | y | π (cid:90) f ( x − t ) t + y dt (3.6)so that T F = f . Then ˆ F ( ξ, η ) = 1 π ˆ f ( ξ ) | ξ | η + | ξ | . (3.7)By (3.4) we have (cid:107) F (cid:107) H ( R ) = 1 π (cid:90) R | ˆ f ( ξ ) | | ξ | η + ξ dξdη = 1 π (cid:90) ∞−∞ | ˆ f ( ξ ) | | ξ | dξ so we obtain (3.3) by Theorem 2.2. (cid:3) Note that we define the extension Ef to be harmonic in each half-plane y > y <
0. Since harmonic functions minimize energy, our extension achieves the minimum H ( R ) norm.There is a virtually identical trace theorem for functions of finite energy in the half-plane,say y > R . To see this we only have to observe that an even reflection RF ( x, y ) = F ( x, − y ) for y < H ( R ) continuously to H ( R ). heorem . The trace map T : H ( R ) → H / ( R ) given formally by T F ( x ) = F ( x, is well-defined and bounded, and there exists a bounded extension map E : H / ( R ) → H ( R ) with T Ef = f , and the analogues of (3.2) and (3.3) hold. If we combine this with the well-known observation that energy is conformally invariantin the plane (not true in other dimensions, however), we obtain a powerful tool for obtainingtrace theorems for other domains: find a conformal map between the domain and the half-space R , and transfer the H / ( R ) norm from the boundary of R to the boundary of thedomain, assuming the conformal map extends continuously to the boundary.A simple example is the strip S = { ( x, y ) : 0 < y < π } . In complex notation ϕ ( z ) = log z is the conformal map from R to S , with ψ ( z ) = e z its inverse. So F ∈ H ( S ) if and onlyif F ◦ ϕ ∈ H ( R ) with equal norms. Then f ( t ) = F ( ϕ ( t )) ∈ H / ( R ). Using Theorem 2.2this means (cid:90) ∞ (cid:90) ∞ | F (log t ) − F (log s ) | | t − s | dtds + (cid:90) ∞ (cid:90) ∞ | F (log t + iπ ) − F (log s + iπ ) | | t − s | dtds + (cid:90) ∞ | F (log t ) − F (log t + iπ ) | dtt (cid:54) c (cid:107) F (cid:107) H ( S ) . (3.9)The change of variable x = log t , y = log s transforms the let hand side of (3.9) into (cid:90) ∞−∞ (cid:90) ∞−∞ | F ( x ) − F ( y ) | e x e y | e x − e y | dxdy + (cid:90) ∞−∞ (cid:90) ∞−∞ | F ( x + y ) − F ( x + iπ ) | e x e y | e x − e y | dxdy + (cid:90) ∞−∞ | F ( x ) − F ( x + iπ ) | dx . (3.10)To simplify the notation we split the trace of F on the boundary of S into two pieces, T F ( x ) = F ( x ) and T F ( x ) = F ( x + iπ ), so that T F and T F are functions on R . Theorem . If F ∈ H ( S ) then T F and T F are in ˜ H / ( R ) and T F − T F ∈ L ( R ) ,with (cid:107) T F (cid:107) H / ( R ) + (cid:107) T F (cid:107) H / ( R ) + (cid:107) T F − T F (cid:107) (cid:54) c (cid:107) F (cid:107) H ( S ) . (3.11) Conversely, given f and f in ˜ H / ( R ) with f − f ∈ L ( R ) , there exists F = E ( f , f ) with T F = f , T F = f , F ∈ H ( S ) with the reverse estimate of (3.11) holding. Proof.
In view of (3.10) it suffices to show that (cid:90) ∞−∞ (cid:90) ∞−∞ | F ( x ) − F ( y ) | e x e y | e x e y | | e x − e y | dxdy (3.12)is bounded above and below by a constant multiple of (cid:90) (cid:90) | x − y | (cid:54) | F ( x ) − F ( y ) | | x − y | dxdy = (cid:107) F (cid:107) ˜ H / ( R ) . (3.13)Note that we may rewrite (3.12) as14 (cid:90) ∞−∞ (cid:90) ∞−∞ | F ( x ) − F ( y ) | (cid:12)(cid:12) sinh (cid:0) x − y (cid:1)(cid:12)(cid:12) dxdy . (3.14) t is clear that (3.14) is bounded below by a multiple of (3.13), and for the upper boundwe need (cid:82)(cid:82) | x − y | (cid:62) | f ( x ) − f ( y ) | | sinh(( x − y ) / | bounded above by a multiple of (3.13), but this is a routineexercise because of the exponential decay of (cid:12)(cid:12) sinh (cid:0) x − y (cid:1)(cid:12)(cid:12) − . (cid:3) It might seem perplexing that the trace space on each of the lines is larger than H / ( R ),since in particular this implies that there are functions in H ( S ) that do not extend to H ( R ). However, it is easy to give an example of such a function: just take F ( x, y ) = g ( x )where g (0) = 0 for x (cid:54) g ( x ) = 1 for x (cid:62) g is smooth in [0 , ∇ F hascompact support in S so F ∈ H / ( S ), but g / ∈ H / ( R ).Another simple example is the first quadrant Q = { ( x, y ) : x > y > } . Then ϕ ( z ) = √ z is the conformal map of R to Q , with inverse ψ ( z ) = z . Again it is convenientto split the trace into two parts mapping to functions on R + , namely T F ( x ) = F ( x,
0) and T F ( x ) = F (0 , x ). Since F ∈ H ( Q ) if and only if F ◦ ϕ ∈ H ( R ), again by Theorem 2.2we have the expression (cid:90) ∞ (cid:90) ∞ | T F ( √ t ) − T F ( √ s ) | | t − s | dsdt + (cid:90) ∞ (cid:90) ∞ | T F ( √ t ) − T F ( √ s ) | | t − s | dsdt + (cid:90) ∞ | T F ( √ t ) − T F ( √ t ) | dtt (3.15)for the trace norm. With the substitutions t = x , s = y this becomes4 (cid:90) ∞ (cid:90) ∞ | T F ( x ) − T F ( y ) | | x − y | xy | x + y | dxdy + 4 (cid:90) ∞ (cid:90) ∞ | T F ( x ) − T F ( y ) | | x − y | xy | x + y | dxdy + 2 (cid:90) ∞ | T F ( x ) − T F ( x ) | dxx . (3.16)It is easy to see that if f , f ∈ H / ( R + ) and (cid:90) ∞ | f ( x ) − f ( x ) | dxx < ∞ (3.17)then there exists F ∈ H ( Q ) with T F = f and T F = f , because xy | x + y | is bounded. Inother words, the function f ( x ) = (cid:40) f ( x ) if x > f ( x ) if x < H / ( R ), and (cid:107) F (cid:107) H ( Q ) (cid:54) c (cid:107) f (cid:107) H / ( R ) . It is possible to show the converse statement aswell, but this involves some technicalities since xy | x + y | is not bounded below. It is easier toobserve that F ∈ H ( Q ) may be extended by even reflection across the axes to a functionin H ( R ), so the even reflections of T F and T F must be in H / ( R ), so T F and T F must be in H / ( R + ), and we already have (3.17) for f = T F , f = T F . A direct proof of(3.17) is possible but involves technicalities.Another simple example is the unit disk D , with ϕ ( z ) = − z z the conformal mapping of R to D . The trace space of H ( D ) is H / ( C ) for C the unit circle with norm (cid:107) f (cid:107) H / ( C ) = (cid:90) π (cid:90) π | f ( e iθ ) − f ( e iθ (cid:48) ) | (cid:12)(cid:12) sin ( θ − θ (cid:48) ) (cid:12)(cid:12) dθdθ (cid:48) . (3.18) f course 2 (cid:12)(cid:12) sin ( θ − θ (cid:48) ) (cid:12)(cid:12) is exactly the chordal distance | e iθ − e iθ (cid:48) | . It is interesting to observethat exactly the same trace space arises from the exterior of the circle {| z | > } , as z (cid:55)→ / ¯ z is an anticonformal map of D to this exterior domain that agrees with ϕ ( z ) on the circle.Similarly, for a circle C r of radius r , the analog of (3.18) is (cid:107) f (cid:107) H / ( C r ) = (cid:90) π (cid:90) π | f ( re iθ − f ( re iθ (cid:48) ) | (cid:12)(cid:12) r sin ( θ − θ (cid:48) ) (cid:12)(cid:12) rdθrdθ (cid:48) . (3.19)Of course it is not necessary to use a conformal map ϕ . A Lipschitz map or even aquasiconformal map changes the H norm by a bounded amount. So for SQ ◦ δ , the interior ofthe square SQ δ , the trace space of H ( SQ ◦ δ ) is H / ( SQ δ ) with norm given by (2.10), sinceone can “square the circle” with a Lipschitz map.Next we consider traces on infinite collections of lines. First consider the horizontal linecollection HLC = { ( x, nπ ) : x ∈ R , n ∈ Z } . For a function F in H ( R ) define the traces T n F ( x ) = F ( x, πn ). Theorem . A set of functions { f n } on R are the traces f n = T n F for F ∈ H ( R ) ifand only if f n ∈ ˜ H / ( R ) and f n − f n +1 ∈ L ( R ) with (cid:88) n (cid:107) f n (cid:107) H / ( R ) + (cid:88) n (cid:107) f n − f n +1 (cid:107) L ( R ) < ∞ , (3.20) and the corresponding norm equivalence holds. Proof.
Basically we just have to apply Theorem 3.3 to each of the strips { nπ < y < ( n + 1) π } and sum (3.11) over all the strips. To do this we just have to observe that afunction belongs to H ( R ) if and only if its restriction to each strip is in H of that strip,the traces agree on neighboring strips, and the sum of the energies is finite. (cid:3) There is something a bit unsettling about this result. We know that f n = T n F actuallybelongs to the smaller space H / ( R ) for F ∈ H ( R ), yet this space plays no role in thecharacterization (3.20). It is an indirect consequence of the theorem that if { f n } is a familyof functions satisfying (3.20), then each f n is indeed in H / ( R ). It should be possible toprove this directly, but again this seems rather technical. Note that we only get a uniformbound for (cid:107) f n (cid:107) H / ( R ) . The following example shows that we can’t do too much better thanthis (most likely (cid:107) f n (cid:107) H / ( R ) = o (1)).Consider the function F ( x, y ) = (1 + x + y ) − α for α >
0. A direct computation showsthat |∇ F ( x, y ) | (cid:54) α (1 + x + x y ) − α − , so F ∈ H ( R ). Now T n F ( x ) = (1 + π n + x ) − α = (1 + π n ) − α g (cid:18) x √ π n (cid:19) for g ( x ) = (1 + x ) − α . It is easy to see that g ∈ H / ( R ), so by dilation invariance of the H / ( R ) norm we see that (cid:107) T n F (cid:107) H / ( R ) = c (1 + π n ) − α so (cid:80) (cid:107) T n F (cid:107) H / ( R ) = ∞ for α (cid:54) .Next we consider the trace on the graph paper graph GP δ . Theorem . The trace space of H ( R ) on GP δ is exactly H ( GP δ ) with norm givenby (2.12). Proof.
We simply use the trace theorem of H on each δ -square that makes up GP δ and add. (cid:3) n place of square graph paper we could consider triangular graph paper TGP δ consistingof the tiling of the plane by equilateral triangles of side length δ . Then the analog of Theorem3.5 holds with essentially the same proof.
4. The graph paper trace characterization
In this section we fix an integer m (cid:62)
2, and consider the sequence of graph paper graphs GP m n , thought of as the unions of the edges, or equivalently the countable union of horizontaland vertical lines in the plane with m n separation. These are nested subsets of the plane, GP m n ⊂ GP m n (cid:48) if n (cid:48) < n and we are interested in the limit as n → −∞ , so the graph papergets increasingly finer.We let T n denote the trace map from functions defined on R to GP m n . By the nestingproperty we may also consider T n to be defined on functions on GP m n (cid:48) with n (cid:48) < n . Ourgoal is to characterize functions in H ( R ) by their traces T n F . Theorem . a) Let F ∈ H ( R ) . Then T n F ∈ H / ( GP m n ) for all n with uniformlybounded norms, and sup n ∈ Z (cid:107) T n F (cid:107) H / ( GP mn ) (cid:54) c (cid:107) F (cid:107) H ( R ) (4.1) b) Let f n ∈ H / ( GP m n ) be a sequence of functions with uniformly bounded norms sat-isfying the consistency condition T n f n (cid:48) = f n if n (cid:48) < n . Then there exists F ∈ H ( R ) suchthat T n F = f n and (cid:107) F (cid:107) H ( R ) (cid:54) c sup n ∈ Z (cid:107) f n (cid:107) H / ( GP mn ) (4.2) Proof.
Part a) is an immediate consequence of Theorem 3.5. To prove b) we define F n to be the harmonic extension of f n into each of the graph paper spaces. Since these harmonicextensions minimize energy, we have F n ∈ H ( R ) and (cid:107) F n (cid:107) H ( R ) (cid:54) c (cid:107) f n (cid:107) H / ( GP mn ) ,again by Theorem 3.5. Thus there exists a subsequence n j → −∞ such that F n j convergesin the weak topology of H ( R ) to a function F satisfying (4.2). It remains to show that theweak convergence respects traces, so that T n F n j = f n for all n j implies T n F = f n .But the equality of traces on GP m n is the same as equality of traces on each of the linesthat make up GP m n ; and since all lines are essentially equivalent, it suffices to show that F n j ( x,
0) converges weakly in H / ( R ) to F ( x, H ( R ) and H / ( R ) are just weighted L spaces.The weak convergence F n j → F in H ( R ) says (cid:90) (cid:90) ˆ F n j ( ξ, η ) G ( ξ, η )( ξ + η ) dξdη → (cid:90) (cid:90) ˆ F ( ξ, η ) G ( ξ, η )( ξ + η ) dξdη (4.3)for every G ∈ L (( ξ + η ) dξdη ). The weak convergence F n j ( x, → F ( x,
0) requires thatwe show (cid:90) (cid:18)(cid:90) ˆ F n j ( ξ, η ) dη (cid:19) H ( ξ ) | ξ | dξ → (cid:90) (cid:18)(cid:90) ˆ F ( ξ, η ) dη (cid:19) H ( ξ ) | ξ | dξ (4.4)for every H ∈ L ( | ξ | dξ ). So given H , choose G ( ξ, η ) = | ξ | H ( ξ ) ξ + η . (4.5) ince (cid:90) (cid:90) | G ( ξ, η ) | ( ξ + η ) dξdη = (cid:90) (cid:18)(cid:90) | ξ | ξ + η dη (cid:19) | H ( ξ ) | dξ = π (cid:90) | H ( ξ ) | | ξ | dξ we may use the choice of G in (4.2). But then (4.3) and (4.4) are identical. (cid:3) This result localizes in several ways. For example, if F ∈ H ( R ) and we wish to estimatethe amount of energy that is contained in an open set Ω, that is (cid:90) Ω |∇ F | dxdy , (4.6)we just have to take the sum of the terms in (2.12) that correspond to edges contained inΩ. Denote this sum by (cid:107) T n F (cid:107) H / (Ω ∩ GP mn ) . Then (4.6) is bounded above and below by aconstant times sup n ∈ Z (cid:107) T n F (cid:107) H / (Ω ∩ GP mn ) . (4.7)We obtain the same norm equivalence if we only assume F ∈ H (Ω), meaning (4.6) is finite.(Note that this does not say anything about the trace of F on the boundary of Ω.) Also,we may start by assuming that F ∈ H ( R ), meaning that (4.6) is finite whenever Ω isbounded, and obtain the norm equivalence of (4.6) and (4.7).The same result will also hold if we replace GP m n by the triangular TGP m n .It is clear that we may replace the sup in (4.2) and (4.7) by the lim sup as n → −∞ . Itis not clear that a limit has to exist, however, since we only have estimates above and below,rather than identity, for our norms.We can also characterize functions of finite energy by their traces on pencils of parallellines of equal separation; in other words, the horizontal lines in GP m n . Denote this by PP m n .We will use Theorem 3.4, but the norms defined by (3.20) are not dilation invariant. Thatmeans we want to define ˜ H / ( PP m n ) by the finiteness of (cid:88) k ∈ Z (cid:90) (cid:90) | x − y | (cid:54) m n | f ( x, km n ) − f ( y, km n ) | | x − y | dxdy + (cid:88) k ∈ Z m − n (cid:90) ∞−∞ | f ( x, ( k + 1) m n ) − f ( x, km n ) | dx , (4.8)and we define this to be (cid:107) f (cid:107) H / ( PP mn ) . Then the analog of Theorem 4.1 holds with T n F equalto the trace on PP m n and H / ( GP m n ) replaced by ˜ H / ( PP m n ). The proof is essentially thesame, using the scaled version of Theorem 3.4 with (4.8) in place of (3.20).
5. Fractals
The Sierpinski gasket ( SG ) is the self-similar fractal defined by the identity SG = (cid:91) i =0 Φ i ( SG ) (5.1) here Φ i are the homothety maps of the plane Φ i ( x ) = x + q i and { q , q , q } are thevertices of an equilateral triangle with side length 1. SG is the unique nonempty compactsubset of the plane satisfying (5.1). The mappings { Φ i } comprise what is called an iteratedfunction system , and the iterates of the mappings are denoted Φ w = Φ w ◦ · · · ◦ Φ w m where w = ( w , . . . , w m ) is a word of length | w | = m and each w j = 0, 1, or 2. Then by iterating(5.1) we obtain SG = (cid:91) | w | = m Φ w ( SG ) (5.2)expressing SG as a union of 3 m miniature gaskets (called m -cells ) that are similar to SG with similarity ratio 2 − m . Note that SG has the post-critically finite (PCF) property thatdistinct m -cells can intersect only at the vertices Φ w q i . For this reason we refer to { q i } asthe boundary of SG , and { Φ w q i } as the boundary of the m -cell Φ m ( SG ), although these arenot boundaries in the topological sense.We may approximate SG by the metric graphs SG m = SG ∩ TG − m . So the vertices are { Φ w q i } , for | w | = m and i = 0 , ,
2, the edges are { Φ w e ij } for | w | = m and e ij is the edge ofthe original triangle joining q i and q j , and Φ w e ij has length 2 − m . Let E m ( f ) = (cid:88) i (cid:54) = j (cid:88) | w | = m | f (Φ w q i ) − f (Φ w q j ) | (5.3)denote the unrenormalized graph energy on SG m . Kigami (see [
10, 17 ]) defines an energyon SG by E ( f ) = lim m →∞ (cid:18) (cid:19) m E m ( f ). (5.4)The renormalization factor (5 / m may be explained as follows: the sequence (5 / m E m ( f )is always nondecreasing, and there exists a 3-dimensional space of harmonic functions forwhich it is constant. We can then define dom E , the space of functions of finite energy, asthose functions for which (5.4) is finite. This is a space of continuous functions on SG thatforms an infinite dimensional Hilbert space (after modding out by the constants) with norm E ( f ) / . This energy satisfies the self-similar identity E ( f ) = (cid:88) i =0 (cid:18) (cid:19) E ( f ◦ Φ i ) (5.5)and satisfies the axioms for a local regular Dirichlet form ([ ]). Up to a constant multipleit is the only Dirichlet form with those properties. It is also symmetric with respect to the D symmetry group of the triangle. This energy forms the basic building block for a wholetheory of analysis on SG , including a theory of Laplacians. We will not be using this widertheory here, but direct the curious reader to [
10, 17 ] for details.Since the functions in dom E are continuous, there is no problem defining traces T m on SG m . The problem of characterizing the trace space T n ( SG ) on the boundary of the trianglehas been solved by Jonsson [
8, 9 ] (see [ ] for a different proof) in terms of Sobolev spacesof order β , with β = + log 5 / . Note that < β <
1. For any metric graph G we define β ( G ) (for any β in the above range) to be the space of continuous functions such that (cid:107) F (cid:107) H β ( G ) = (cid:88) e ∈ E (cid:90) L e (cid:90) L e | F ( e ( x )) − F ( e ( y )) | | x − y | β dxdy (5.6)is finite. Note that in contrast to (2.1), there is no term comparing values on intersectingedges, since the continuity condition takes care of the comparison (this idea is also used in[ ]). We then have the following result analogous to Theorem 3.1. Proposition
7, 8, 9 ]) . The trace map T is continuous from dom E to H β ( SG ) with β = + log 5 / with (cid:107) T F (cid:107) H β ( SG (cid:54) c E ( F ) . (5.7) Moreover, there exists a continuous linear extension map E : H β ( SG ) → dom E with T E f = f and E ( E f ) (cid:54) c (cid:107) f (cid:107) H β ( SG ) . (5.8)We note that [
7, 8, 9 ] use a slightly different, but equivalent norm for H β ( SG ).Next we need to obtain the analogous statement for the trace map T m to SG m . Wenote that energy is additive for continuous functions, and in view of the self-similarity (5.5)iterated, E ( F ) = (cid:88) | w | = m (cid:18) (cid:19) m E ( F ◦ Φ w ), (5.9)and if we apply (5.8) to F ◦ Φ w we have (cid:88) | w | = m (cid:18) (cid:19) m (cid:107) T F ◦ Φ w (cid:107) H β ( SG ) (cid:54) c (cid:88) | w | = m (cid:18) (cid:19) m E ( F ◦ Φ w ) = c E ( F ) (5.10)by (5.9). Now we observe that SG m = (cid:83) | w | = m Φ w ( SG ), and this is a disjoint union of edges,since each edge is just a side of a triangle Φ w ( SG ) for some w with | w | = m .So consider one of these edges, Φ w ( e ij ). It is parameterized by x in the interval [0 , − m ],and the contribution (5.6) is (cid:90) − m (cid:90) − m | F ( e ( x )) − F ( e ( y )) | | x − y | β dxdy = 4 m β (cid:90) (cid:90) | F (Φ w ( e ij ( x ))) − F (Φ w ( e ij ( y ))) | | x − y | β dxdy (5.11)after a change of variables. Summing all the contributions over all the edges in SG m yields (cid:107) T m F (cid:107) H β ( SG m ) = (cid:88) | w | = m m (1+2 β ) m (cid:107) T F ◦ Φ w (cid:107) H β ( SG ) (5.12)by (5.11). But the choice of β makes β = , so (5.12) combined with (5.10) yields (cid:107) T m F (cid:107) H β ( SG m ) (cid:54) c E ( F ). (5.13)This is the exact analog of (5.7). heorem . The trace map T m is continuous from dom E to H β ( SG m ) for β as inProposition 5.1 and the estimate (5.13) holds. Moreover, there exists a continuous linearextension map E m : H β ( SG m ) → dom E with T m E m f = f and E ( E m f ) (cid:54) c (cid:107) f (cid:107) H β ( SG m ) . (5.14) Proof.
We have already established (5.13). To define the extension map E m we set E m ( f ) = Φ − w E ( f ◦ Φ w ) on Φ w ( SG ). (5.15)Note that E m ( f ) is continuous, because at the boundary points of the m -cells that make up SG m we have E m ( f ) = f . The same reasoning that obtains (5.13) from (5.7) also leads from(5.8) to (5.14). (cid:3) Next we have the analog of Theorem 4.1.
Theorem . a) Let F ∈ dom E . Then T m F ∈ H β ( SG m ) for all m with uniformlybounded norms, and sup m (cid:107) T m F (cid:107) H β ( SG m ) (cid:54) c E ( F ) (5.16) b) Let f m ∈ H β ( SG m ) be a sequence of functions with uniformly bounded norms satisfyingthe consistency condition T m f m (cid:48) = f m if m (cid:54) m (cid:48) . Then there exists F ∈ dom E such that T m f = f m and E ( F ) (cid:54) c sup m (cid:107) f m (cid:107) H β ( SG m ) . (5.17) Proof. (5.16) is an immediate ff consequence of (5.13). To prove b) construct a sequenceof functions F m by taking the harmonic (energy minimizing) extension of f m from SG m to SG .Then by (5.14), the sequence { F m } is uniformly bounded in dom E . A quantitative versionof the continuity of functions in dom E implies that the sequence { F m } is also uniformlyequicontinuous. Thus by passing to a subsequence twice we can find a subsequence { F m j } that converges both weakly in the Hilbert space dom E and uniformly to a function F indom E with the estimate (5.17) holding. Because the convergence is pointwise and theconsistency condition holds we have T m j F = F m j = f m j on SG m j , so T m F = f m . (cid:3) The second example of a fractal we consider is the Sierpinski carpet ( SC ), again definedby a self-similar identity SC = (cid:91) i =1 Φ i ( SC ) (5.18)where now Φ i are the homothety maps of the plane with contraction ratio 1 / / ]were given in the late 1980’s, and recently in [ ] it was shown that up to a constant multiplethere is a unique self-similar energy, so both approaches yield the same energy. Once again,all functions in dom E are continuous. The self-similar identity for the energy here is E ( F ) = (cid:88) i =1 r E ( F ◦ Φ i ), (5.19) here r is a constant whose exact value has not been determined ( r is slightly larger than1 . SC by a sequence of metric graphs, { SC m } , with SC m = SC ∩ GP − m . Thus, the edges of SC m have length 3 − m and are of the form Φ w ( e i ) with | w | = m , where e , e , e , e are the boundary edges of the unit square. Again let T m denotethe trace map onto SC m . The trace space for T has been identified by Hino and Kumagai[ ] as the Sobolev space H β ( SC ) with β = + log r log 9 . Note that again < β < Proposition ]) . The trace map T is continuous from dom E to H β ( SC ) for β = + log r log 9 with (cid:107) T F (cid:107) H β ( SC ) (cid:54) c E ( F ) . (5.20) Moreover, there exists a continuous linear extension map E : H β ( SC ) → dom E with T E f = f and E ( E f ) (cid:54) c (cid:107) f (cid:107) H β ( SC ) .. (5.21)We now claim that the analogs of Theorem 5.2 and 5.3 hold for SC in place of SG , withessentially the same proof. The only detail that needs to be checked is the dilation argument.In this case the contribution to (5.6) from the edge e = F w ( e ) is (cid:90) − m (cid:90) − m | F ( e i ( x )) − F ( e i ( y )) | | x − y | β dxdy = 9 m β (cid:90) (cid:90) | F (Φ w ( e i ( x ))) − F (Φ w ( e i ( y ))) | | x − y | β dxdy (5.22)after a change of variable, as the analog of (5.11). We note that β = r in this case,so summing (5.22) yields the analog of (5.13) as a subsequence of (5.20). The rest of thearguments are the same.For our final fractal example we consider the classical Julia sets of complex polynomials.Fix a polynomial P ( z ) (of degree at least two) and let J denote its Julia set. We assume J is connected. In many cases (see [ ]) it is possible to parameterize J by the unit circle asfollows. Let Ω denote the unbounded component of the complement of J in C , so Ω ∪ {∞} is simply connected, and let ϕ be a conformal map from { z : | z | > } to Ω. In many cases ϕ extends continuously to the boundary circle, and this maps C onto J (usually not one-to-one). Although there is usually no useful formula for ϕ , in many cases it is possible todescribe explicitly the points on C that are identified under ϕ . There have been a numberof papers that utilize this parametrization to construct an energy on J [
13, 1, 5, 14 ].Here we deal with a different question: how to characterize the traces on J of functionsof finite energy on Ω. The answer is almost immediate using the methods of section 3. Weknow that F ∈ H (Ω) if and only if F ◦ ϕ ∈ H ( | z | > F ◦ ϕ on C is exactly H / ( C ). Thus the space of traces of F on J , that we should denote H / ( J ),is characterized by the finiteness of (cid:107) F (cid:107) H / ( J ) = (cid:90) π (cid:90) π | F ( ϕ ( e iθ )) − F ( ϕ ( e iθ (cid:48) )) | ( θ − θ (cid:48) ) dθdθ (cid:48) . (5.23)One could perhaps hope for a more direct characterization in terms of an integral involving | F ( z ) − F ( z (cid:48) ) | as z and z (cid:48) vary over J . This would involve choosing a measure on J (there re more than one natural choices) and finding the appropriate denominator in terms of adistance from z to z (cid:48) on J . Good luck! References [1] T. Aougab, S. Dong, R. Strichartz,
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