ESTIMATES FOR THE RESOLVENT KERNEL OF THE LAPLACIAN ON P.C.F. SELF SIMILAR FRACTALS AND BLOWUPS.

Transcription

1 ESTIMATES FOR THE RESOLVENT KERNEL OF THE LAPLACIAN ON PCF SELF SIMILAR FRACTALS AND BLOWUPS LUKE G ROGERS 1 Introduction One of the main features of analysis on ost-critically finite self-similar cfss) sets is that it is ossible to understand the behavior of the Lalacian and its inverse, the Green oerator, in terms of the self-similar structure of the set Indeed, a maor ste in the aroach to analysis on self-similar fractals via Dirichlet forms was Kigami s roof [8, 10] that for a self-similar Dirichlet form the Green kernel can be written exlicitly as a series in which each term is a rescaling of a single exression via the self-similar structure In [6] this result was extended to show that the resolvent kernel of the Lalacian, meaning the kernel of z ), can also be written as a self-similar series for suitable values of z C Part of the motivation for that work was that it gives a new understanding of functions of the Lalacian such as the heat oerator e t ) by writing them as integrals of the resolvent The urose of the resent work is to establish estimates that ermit the above aroach to be carried out We study the functions occurring in the series decomosition from [6] see Theorem 33 below for this decomosition) and give estimates on their decay From this we determine estimates on the resolvent kernel and on kernels of oerators defined as integrals of the resolvent kernel In articular we recover the shar uer estimates for the heat kernel see Theorem 10) that were roved for cfss sets by Hambly and Kumagai [4] by robabilistic methods see also [1, 13, 3] for earlier results of this tye on less general classes of sets) It is imortant to note that the receding authors were able to rove not ust uer estimates but also lower bounds for the heat kernel, and therefore were able to rove sharness of their bounds Our methods ermit shar bounds for the resolvent kernel on the ositive real axis, but we do not know how to obtain these globally in the comlex lane or how to obtain lower estimates for the heat kernel from them Therefore in this direction our results are not as strong as those obtained in [4] However in other directions we obtain more information than that known from heat kernel estimates, and we hoe that our aroach will comlement the existing robabilistic methods In articular we are able to obtain resolvent bounds on any ray in C other than the negative real axis where the sectrum lies), while standard calculations from heat kernel bounds only give these estimates in a half-lane A further consequence of our aroach is that we extend in Theorem 97) the decomosition from [6] to the case of blowus, which are non-comact sets with local structure equivalent to that of the underlying self-similar sets The blowu of a cfss set bears the same relation to the original set as the real line bears to the unit interval, see [17] for details The structure of the aer is as follows In Section 3 we recall some basic features of analysis on cfss sets, as well as the main result of [6], which is the decomosition of the resolvent as a weighted sum of iecewise eigenfunctions Section 4 then discusses Date: January 7, Mathematics Subect Classification Primary 8A80, 60J35 1

2 LUKE G ROGERS the natural decomosition of the cfss set according to a self-similar harmonic structure The next two sections deal with shar estimates on the real axis In Section 5 we obtain estimates of iecewise eigenfunctions for which the eigenvalues are real To show these we decomose into ieces in the manner of Section 4, rove that each iece has one normal derivative that is larger than the others by a factor, and use this to show that the only way to glue ieces together smoothly is by requiring the size to decay by this factor each time we cross a cell of the decomosition This imlies that iecewise eigenfunctions have sub- Gaussian decay From these estimates it is routine, though somewhat long and technical, to estimate the kernel of λ ) for λ on the ositive real axis and to rove that the decomosition of [6] is valid on blowus in this setting The former is in Section 6 and the latter in Section 7 In Section 8 we switch gears and rove some comlex analytic estimates related to the Phragmen-Lindelöf theorem Our goal is a method for obtaining resolvent estimates on sectors in the comlex lane from our decay estimates on the real axis This method is not restricted to the setting of cfss sets, and may be of interest for roving resolvent estimates in more general settings, such as metric measure saces We find that combining decay estimates on the real axis with some weak estimate on a sector away from the sectrum roves decay on any ray in the sector In Section 9 we show that the required weak estimates can be roved on cfss sets by a generalization of some arguments from Chater 4 of [9] As a result we obtain our main result, Theorem 96, giving decay estimates for iecewise eigenfunctions and the resolvent kernel away from the negative real axis in C Section 10 contains some examles, including a roof of the uer bounds for the heat kernel Acknowledgements This aer relies heavily on its recursor [6]; the author thanks his co-authors on that aer, esecially Bob Strichartz, whose suggestion that there might be an aroach to heat kernel estimates via Kigami s theory and the results of [6] insired the resent work 3 PCFSS Fractals: Energy, Lalacian, Metric, Eigenvalue Counting and Resolvent kernel We work on a cfss fractal X corresonding to an iterated function system {F 1, F J } In full generality this could be a comact metrizable toological sace X equied with continuous inective self-mas F so that there is a continuous surection from the sace of infinite words over {1,, J} N to X, as in Chater 1 of [9] The reader may refer, however, to think of the more intuitive situation in which the F are Lischitz contractions on a finite dimensional Euclidean sace and X is their fixed oint in the sense of Hutchinson [5] In either case, the boundary of X is the ost-critical set V 0, which is a finite set with the roerty that F X) F k X) F V 0 ) F k V 0 ) for k We assume that X \ V 0 is connected The best known examle of a set of this tye is the Sierinski Gasket, for which the full analytic theory we will use is exounded in [19] Note that the Sierinski Caret is not of this tye, as it is not ost-critically finite If w is a finite word on the letters {1,, J} then w is its length The set V m is w =m F w V 0 ) and V = m=0 V m Points in V \ V 0 are called unction oints For a word w = w 1 w m we write F w = F w1 F w F wm The set of finite words is W, and the set of infinite words is Σ If σ = σ 1 σ is an infinite word we use [σ] m to denote the subword σ 1 σ m For a finite word w we use the notation ww resectively wσ) for finite resectively infinite) words that begin with w

3 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 3 We assume that we have a regular self-similar harmonic structure on X which rovides an irreducible self-similar Dirichlet form E with domain dome) and scalings 0 < r < 1 so that Eu) = J 1 r Eu F ) Details of the construction and roerties of such forms may be found in [9] We also fix a robability measure µ satisfying 31) µa) = J 1 µ µf A)) We will require that this measure is related to the Dirichlet form by µ = r S for the unique S so that r S = 1 This measure is known to be the natural one for many asects of the analysis on X; it arises in studying the analogue of Weyl-tye asymtotics of Lalacian eigenvalues [1], and in determining heat kernel estimates [9, 13, 3, 4] It is the Hausdorff measure with Hausdorff dimension S ) for the resistance metric, which is the metric in which the distance between oints x and y is Rx, y) given by Rx, y) = min { Eu) : ux) = 0, uy) = 1 } Remark 1 In most of the theory of analysis on fractals one can take any Bernoulli measure defined by 31) for some 0 < µ < 1 with µ = 1 In articular, this is all that is required for the construction of the Lalacian resolvent in [6] The author would be curious to know whether it suffices for Theorem 97 or even Theorem 73 regarding the resolvent on blowus The Dirichlet form and measure give rise to a weak Lalacian by defining u dom ) with u = f if there is a continuous f such that Eu, v) = f v dµ for all v dom 0 E), the functions in dome) that vanish on V 0 The Lalacian is selfadoint and has comact resolvent, with negative eigenvalues λ of finite multilicity that accumulate only at The asymtotic distribution of the eigenvalues was determined by Kigami and Laidus [1], who roved a more general version of the following result Proosition 31 [1]) Let Nx) = #{λ : λ x} Then 0 < lim inf x Nx) lim su xs/s +1) x Nx) < xs/s +1) We also recall from [9] that there is an exlicit formula for a continuous Green kernel that is ositive on the interior of X, zero on V 0, and inverts with Dirichlet boundary conditions Kigami s construction was generalized in [6], from which we will need the following results Proosition 3 If V 0 and z C is such that none of the values r w µ w z is a Dirichlet eigenvalue of then there is a function η z) with measure-valued Lalacian satisfying zi )η z) = 0 on X \ V 0 η z) = δ q for q V 0 where δ q is the Kronecker delta We call this a iecewise z-eigenfunction

4 4 LUKE G ROGERS Under the same assumtions on z we define functions that form a natural basis for the iecewise z-eigenfunctions on 1-cells For V 1 \ V 0, let 3) ψ z) = η r µ z) F F where by convention the sum ranges only over those terms that are well-defined, in this case those so F V 0 ) Then ψ z) has a measure-valued Lalacian, with zi )ψ z) = 0 on all 1-cells and Dirac masses at the oints of V 1 \ V 0 The main result of [6] is that for suitable values of z the resolvent G z) x, y) of the Lalacian with Dirichlet boundary conditions may be written in terms of these iecewise eigenfunctions Proosition 33 For each z C such that none of the values r w µ w z is a Dirichlet eigenvalue of, let 33) G z) x, y) = r w Ψ rwµwz) Fw x, Fw y) w W in which 34) Ψ z) x, y) = G z) qψ z) x)ψ z),q V 1 \V 0 q y) where G z) q is the inverse of the symmetric matrix B z) q with entries 35) B z) q = F X) n ψ z) q) :F V 0 ) q Then G z) x, y) is continuous and vanishes on V 0, and it inverts zi ) in that if f L 1 µ) then zi ) G z) x, y) f y) dµy) = f x) X In what follows, we will suose that z = λ > 0 and study the decay of η λ) and its deendence on λ, so as to estimate the decay of the terms in G λ) This will then be used to study the deendence of η z) and G z) for those z C that are not on the negative real axis As usual these estimates will include constants deending on the structure of the fractal, for which reason we define a shorthand notation as follows Notation 34 We write a b or b a if there is a constant c > 0 that deends only on the fractal, the harmonic structure or the measure, and for which a cb If a b and b a then we write a b The imlicit constant, as well as constants exlicitly named, may vary from line to line in a comutation 4 Resistance Partitions and Chemical Paths The short-time behavior of diffusion on fractal sets may be analyzed by artitioning the fractal such that all ieces have a rescribed resistance u to a constant factor) and studying the lengths of minimal aths in this decomosition see, for examle, [4]), which are often called chemical aths We should therefore exect that the behavior of the resolvent G λ) x, y) for large λ > 0 may be determined in the same manner In this section we record some known estimates for chemical aths that will be useful later Definition 41 A artition of X is a finite set Θ W with the roerties 1) If θ, θ Θ then θσ θ Σ only if θ = θ, ) θ Θ θσ = Σ

5 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 5 We will refer to the cells F θ X), θ Θ, as the cells of Θ The artitions we use are Θ k = { θ 1 θ m : r θ1 r θm e k < r θ1 r θm }, and we write VΘ k ) for the unction oints corresonding to words in Θ k VΘ k ) = F θ V 0 ) θ Θ k We will frequently use that when θ Θ k, 41) r θ e k, µ θ e ks, r θ µ θ e ks +1) There is a grah structure corresonding to the artition Θ k Let Γ k be the grah with vertices VΘ k ) and edges between every x and y for which there is θ Θ k such that x, y F θ V 0 ) Points oined by an edge are said to be neighbors As a consequence of the definition of Θ k we have see Lemma 3 of [4] for a roof) 4) Rx, y) e k for any x, y VΘ k ) that are neighbors in Γ k A ath oining x and y in Γ k is a finite sequence {x i } i=0 I of vertices such that adacent vertices are connected by an edge and x 0 = x, x I = y The length of the ath is I and the grah distance d k x, y), also called the chemical distance, is the length of the shortest ath oining x and y in Γ k It is generally difficult to obtain good estimates relating the grah distance d k x, y) and the resistance Rx, y); the interested reader is directed to [11] for a detailed analysis of this question and its connection to heat kernel estimates We will satisfy ourselves with some elementary but crude results For examle, it is well known that Rx, y) does not exceed the resistance along a ath from x to y, so that for x, y VΘ k ) we must have Rx, y) e k d k x, y) In articular if x, y V 0 then d k x, y) e k Conversely, in Lemma 33 of [4] it is shown that for any x, y X, the bound d k x, y) e S +1)k/ holds We will need these estimates in a form that comares d k+k x, y) to d k x, y) Lemma 4 e k d k+k x, y) e S +1)k / d k x, y) Proof Observe that the artition by words { θ = θθ : θ Θ k, θ Θ k } has r θ e k+k ), so is a subartition of Θ k+k and consequently has longer aths Since a ath in this artition describes a ath Γ k in which any air of vertices is oined by a ath no longer than the maximal ath between oints of V 0 in Γ k, which we know is at most e S +1)k /, we conclude that d k+k x, y) e S +1)k / d k x, y) For the lower bound we note that there is a constant c deending only on the harmonic structure and such that the artition by words { θ = θθ : θ Θ k, θ Θ k c} contains Θ k+k and therefore has shorter aths Again each such ath restricts to a ath in Γ k, and now every air of Γ k vertices is searated by a ath of length at least the minimal distance between oints of V 0 in Γ k c, which is e k c e k

6 6 LUKE G ROGERS For the secial classes of nested and affine nested fractals there are quite recise results about chemical distances in [13, 3] In articular the authors construct a geodesic metric on affine nested fractals that is comarable to a ower of the resistance see Proosition 36 and Remark 37) of [3]), from which the following is easily obtained Proosition 43 [3]) For an affine nested fractal there is γ 0, S + 1) which may be obtained by solving an exlicit otimization roblem, such that for all sufficiently large k, d k x, y) e kγ Rx, y) γ Sketch of Proof Knowing that Rx, y) γ is comarable to a geodesic metric we see that edges of Γ k have geodesic length like e kγ, so the geodesic distance from x to y is comarable to d k x, y)e kγ if k is large This must be comarable to Rx, y) γ, so we have d k x, y) e kγ Rx, y) γ The bounds on γ are evident from the receding discussion The roof in [3] is a generalization of an earlier argument, Proosition 35 of [13], for the case of nested fractals We do not wish to give the definitions of nested or affine nested fractals here, but we recall that they are subsets of R n, are generated by iterated function systems consisting of Euclidean similarities that have a high degree of symmetry Full details may be found in [13, 3] and the references therein Definition 44 For later use we extend the definition of d k x, y) from Γ k to all of X in the obvious fashion For x in the interior of a cell F θ X) of Θ k and y Γ k let d k x, y) = min{d k z, y) : z F θ V 0 )} For x in the interior of F θ X) and y in the interior of F θ X) set d k x, y) = min{d k z, z ) : z F θ V 0 ), z F θ V 0 )} When we refer to a d k geodesic between oints x and y that are not in Θ k we mean a geodesic oining the cells containing them 5 Estimates for iecewise eigenfunctions with ositive eigenvalue In this section we develo decay estimates for the iecewise λ-eigenfunctions η λ) in the case where λ is a ositive real number We summarize the results in the following theorem Theorem 51 There is kλ) with e kλ) 1 + λ) 1/S +1) and a constant c > 0 deending only on the fractal and harmonic structure such that for λ 0, ) and V 0, each iecewise λ-eigenfunction η λ) satisfies the following bounds For all q V 0, q, and all x X 51) 5) 53) ex cd kλ), x) ) η λ) x) ex d kλ), x) ), n η λ) ) 1 + λ) 1/S +1), + λ) 1/S +1) ex d kλ), q) ) n η λ) q) + λ) 1/S +1) ex cd kλ), q) ), excet that the lower bound of 51) is not valid on cells F θ X), θ Θ kλ) such that F θ V 0 ) contains a oint of V 0 \ {} Proof The number kλ) is introduced in Definition 51, where it is shown it is comarable to 1 + λ) 1/S +1) The uer bound of 51) for sufficiently large λ is Corollary 515, and continuity ensures we may take a suitably large constant multile to make it true for all λ > 0 Both 5) and the lower bound of 53) are in Lemma 518, while the lower bound of 51) is Corollary 51 and the uer bound of 53) is Corollary 5 Remark We note in assing that the estimate 51) is sufficient to comlete Strichartz s roof of the smoothness of finite-energy harmonic functions on roducts of fractals [18, Theorem 114] Strichartz roves this modulo an assumtion on the behavior of the normal derivatives of the heat kernel [18, Equation 83)], but the key estimate in his roof is actually the decay estimate [18, Equation 85)], which is a consequence of the above in the

7 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 7 case of nested fractals by Proosition 43 More generally, for roducts based on fractals and harmonic structures of the tye discussed in this aer, the estimate 51) is sufficient to imly the convergence in [18, Equation 819)], and thus the argument roving [18, Theorem 114] The author hoes that is a first ste toward roving hyoelliticity of solutions of ellitic PDE on roducts of cfss fractal sets [18], Section 11 and [16] Section 8) The remainder of the section is devoted to the roof of this theorem As the working is at times technical it may hel the reader to have a concrete examle in mind The most elementary choice is to let X be the unit interval with Lebesgue measure and the mas F be the contractions onto the left and right halves of X In this case we obtain the usual Dirichlet energy, the Lalacian is the second derivative, and the normal derivative is the outwarddirected first derivative at the endoints [19] Eigenfunctions and iecewise eigenfunctions are exonentials, and a quick comutation shows that η λ) 1 = sinh λx/ sinh λ Our goal is to show that the basic features of this function, as listed in the theorem, are also true for the functions η λ) on any cfss set An observation we use throughout this section is 54), which says that we may decomose these functions into ieces corresonding to cells of a artition Θ k and obtain a linear combination of ieces; moreover if Θ k is suitably chosen then each iece looks almost like a coy of a fixed function In the case of the unit interval the ieces are literally translates and reflections of a fixed hyerbolic sine, and the decomosition 54) can be obtained by iteration of the double angle formula see Section of [6]) The main idea of this section is that in the decomosition each iece has a larger normal derivative near its eak than at the other boundary oints, and in order for such ieces to oin together in a smooth fashion it is necessary that the eak of each is smaller than that of its neighbor by a constant factor Thus the exonential decay may be derived from the shae of a collection of basic functions and counting the cells in aths on Θ k In essence, our estimates on the relative sizes of the normal derivatives come from the fact that η λ) is subharmonic, but there is some unavoidable technical work to obtain the correct quantative bounds It should be emhasized that most of the work in this section deends on the fact that λ is ositive Assumtion 5 In this section we require that λ > 0 We will extensively use the decomosition of η λ) as a iecewise λ-eigenfunction on the cells of a artition Θ k as defined in Section 4 above Using the Lalacian scaling η r wµ w λ) F w ) = rw µ w ) η r wµ w λ) ) F w = λη r wµ w λ) F w we see that η λ) may be written as a linear combination of the functions η r θµ θ λ) y for θ Θ k and y VΘ k ) More recisely, 54) η λ) x) = θ Θ k η λ) F θ q) η r θµ θ λ q q V 0 F θ x)) Before roceeding we require several rearatory lemmas about the function η λ) The first is a maximum rincile for smooth subharmonic functions It is well known but does not seem to aear in the literature, excet for that art which is in Proosition 11 of [15] Lemma 53 Suose u dom ) and u 0 If u attains its global maximum at an interior oint then u is constant Moreover u 0 at any local maximum oint

8 8 LUKE G ROGERS Proof The roof uses the fact that on any cell F w X) there is a Green kernel g w that is non-negative on F w X) and strictly ositive on F w X) \ F w V 0 ), and such that 55) ux) = h w x) g w x, y) uy) ) dµ F w X) where h w x) is the harmonic function on F w X) with h w x) = ux) for all x F w V 0 ) Let h m be the iecewise harmonic function at scale m with h m x) = ux) for all x V m Since u 0 we find from 55) that u h m This imlies that n ux) n h m x) for all x V m However the sum of the normal derivatives of u at any oint of V m \V 0 must vanish because u dom ), so the sum of the normal derivatives of h m must be non-ositive This gives that the m-scale grah Lalacian of h m is non-negative, so if h m achieves its maximum at a oint x V m \V 0 then this maximum is also attained at all neighbors of x in the m-scale grah Connectivity then imlies h m is constant on V m If u attains its global maximum at a oint x X \ V 0 then u h m and h m harmonic imlies that h m achieves the same value at a oint on the boundary of any cell containing x For all sufficiently large m, this oint which could be x) is not in V 0, so h m attains its maximum value ux) at a oint of V m \ V 0, and is therefore constant by our revious reasoning Alying this for all large m imlies u = ux) on the dense set V, and since u is continuous it must be constant Suose in order to obtain a contradiction that u has a local maximum at x X \ V 0 and ux) > 0 The easy case is when x V, because it is then easily seen that h m has a local maximum at x for all sufficiently large m This contradicts the above reasoning showing the grah Lalacian of h m to be non-negative if there is a neighborhood on which u 0 The alternative is that x V Then there is an infinite word σ Σ so m F [σ]m X) = {x} and these sets form a neighborhood base of x From 55) we see that h [σ]m x) > ux) for all m large enough that u > 0 on F [σ]m X) The maximum rincile for harmonic functions then gives max { uy) : y F w V 0 ) } = max { h [σ]m y) : y F w V 0 ) } > ux), so that every neighborhood of x contains a oint at which u exceeds ux), in contradiction to ux) being a local maximum Definition 54 Let ζ be the function harmonic on X with ζ q) = 0 for q V 0, q and ζ ) = 1 Lemma 55 0 η λ) ζ Proof If the first inequality fails then the fact that η λ) has a strictly negative minimum at some interior oint x However by Lemma 53 the Lalacian 0 on V 0 imlies that η λ) must be non-negative at a minimum oint, in contradiction to η λ) x) = λη λ) x) < 0 Having established the first inequality, it follows that η λ) is subharmonic and thus is bounded above by the harmonic function with the same boundary values, which is recisely ζ Corollary 56 On any connected oen set in X the maximum of η λ) boundary Proof We have η λ) Lemma 57 If λ > λ then η λ) η λ ) is attained at the = λη λ) 0 and so may aly Lemma 53

9 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 9 Proof Suosing the contrary we find that η λ ) η λ) has a negative local minimum at a oint x However using that η λ ) 0 and the hyothesis, η λ ) η λ) ) x) = λ η λ ) λη λ) ) x) λη λ ) η λ) ) x) < 0 which cannot occur at a minimum oint by Lemma 53 Corollary 58 For any, q V 0 with q, the values n η λ) ) are non-negative and increasing in λ, while the values n η λ) q) are non-ositive and increasing in λ In articular for q the values n η λ) q) are bounded below by min,q V0 n ζ q) < 0 Proof Positivity of n η λ) ) is evident from Lemma 55 because η λ) Then the fact that η λ) η λ ) when λ > λ from Lemma 57 imlies n η λ) Similarly n η λ) q) is negative because η λ) imlies n η λ) q) n η λ ) ζ < 1 on X \ {} ) n η λ ) ) 0 = η λ) q) from Lemma 55, and η λ) η λ ) q) The lower bound comes from Lemma 55, because η λ) ζ for all λ > 0 imlies n η λ) q) n ζ q) for all, q V 0 With these basic observations in hand we look more closely at the decomosition 54) Lemma 59 For all V 0, X η λ) dµ 1 + λ) S S +1 Proof For λ 1 the result is clear from the ositivity of η λ) and the monotonicity shown in Lemma 57; in fact we have rather than an inequality For λ 1, fix k such that e ks +1) λ e, 1], so that for θ Θ k we have r θ µ θ λ 1 The monotonicity of Lemma 57 then imlies η r θµ θ λ) dµ 1 X All terms in the decomosition 54) are ositive by Lemma 55, so writing θ for a word in Θ k such that F θ X), we have X η λ) dµ η r θ µ θ λ) Fθ dµ F θ X) = µ θ η r θ µ θ λ) dµ X However µ θ λ S/S +1) for θ Θ k and our choice of k, so the roof is comlete Lemma 510 For any word w W, x F w V 0 ) n η r w µ w λ) q F w ) x) λ rw µ w ) + λ ) S S +1

10 10 LUKE G ROGERS Proof Alying the Gauss-Green formula to η r wµ w λ) q yields x F w V 0 ) n η r w µ w λ) q where we used Lemma 59 and µ w = r S w Fw ) x) = = λ F w X) F w X) = λµ w X F w and the constant function 1 η r wµ w λ) q η r wµ w λ) q η r wµ w λ) q dµ λµ w 1 + rw µ w λ ) S S +1 = λ rw µ w ) + λ ) S S +1 Fw ) dµ F w dµ On the first reading of the following lemma one should think of the case Θ = Θ k It will later be used for Θ = Θ k \ {θ : F θ X) } and more comlicated sets This lemma is the main argument in this section of the aer, in that it uses smoothness of the oin between ieces of the decomosition 54) to show that η λ) must decay Lemma 511 For Θ Θ k and Y = θ Θ F θ X) we have η λ) y) C e ks +1) λ + e k λ /S +1)) η λ) z) y VΘ )\ Y z Y Proof At any oint x VΘ k ) the sum of the normal derivatives of η λ) over the cells meeting at x must be zero because η λ) dom ) If we sum this cancelation over all VΘ k ) oints that are interior to Y we may use 54) and rearrange to obtain 56) θ Θ q V 0 η λ) F θ q)) x F θ V 0 )\ Y n η r θ µ θ λ) q F θ ) ) x) = 0 We estimate the innermost sum in 56) using Lemma 510 Suose first that θ and q are such that F θ q is an interior oint of Y Then Corollary 58 tells us that the terms n η r θ µ θ λ) q Fθ ) x) for x Y are negative Thus the lower bound of Lemma 510 is still valid with these oints removed Substituting into 56) we find η λ) F θ q)) n η r θ µ θ λ) q Fθ ) ) 57) x) θ Θ {q V 0 :F θ q) Y} = θ Θ {q V 0 :F θ q) Y} θ Θ {q V 0 :F θ q) Y} λ ce ks +1) + λ ) S S +1 x F θ V 0 )\ Y η λ) F θ q)) η λ) x F θ V 0 )\ Y n η r θ µ θ λ) q λ F θ q)) rθ µ θ ) + λ ) S S +1 η λ) y) y VΘ )\ Y Fθ ) ) x) Now in the terms n η r θ µ θ λ) q Fθ ) x) = r θ nη r θµ θ λ) q F θ x)) on the left in 57), the oints q and Fθ x) cannot coincide, because x Y and F θq Y Again aealing to Corollary 58 we see that such normal derivatives are negative,

11 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 11 increasing in λ, and bounded below by c = min,q V0 n ζ q), which deends only on the harmonic structure of X Putting this and the estimate r θ e k into 57) gives a ositive constant C deending on c and the degree of vertices so that Ce k z Y η λ) z) θ Θ {q V 0 :F θ q) Y} λ ce ks +1) + λ ) S S +1 η λ) F θ q)) y VΘ )\ Y x F θ V 0 )\ Y η λ) y) ce k ) We also use that ce ks +1) + λ ) S S +1 e ks + λ S S +1 We may obtain decay estimates of η λ) choice of k and sets Θ by iterative use of Lemma 511 for an aroriate Definition 51 Given λ > 0 we let kλ) be the larger of 0 and the greatest integer such that kλ) 1 log λ log C S + 1 where C is the constant in Lemma 511 Note that e kλ) 1 + λ) 1/S +1) The following result is an immediate consequence of the definition of kλ) Corollary 513 If λ is large enough that kλ) 1, then for Θ Θ kλ) η λ) y) 1 η λ) z) e y VΘ )\ Y Lemma 514 For λ as in Corollary 513 and each V 0, let X kλ), i) = { x VΘ kλ) ) : d kλ), x) i } Then η λ) y) e i y X kλ),i) Proof We induct over i The base case i = 0 is simly the fact that η λ) ) = 1 Observe that oints y X kλ), i) for i satisfy d λ), y) i 1 or are in V 0 \ {} The latter may be ignored because η λ) is zero on V 0 \ {}, so substituting the inductive estimate into Lemma 513 gives the result If we restrict η λ) to a cell, Corollary 56 imlies the maximum is at the boundary If a oint x X has d kλ), x) = i then the boundary oints of the cell of Θ kλ) that contains x are in X kλ), i) Thus Lemma 514 imlies z Y Corollary 515 For any x X and λ as in Corollary 513, η λ) x) ex d kλ), x) ) Lemma 514 also allows us to show that Lemma 59 gives the correct value for the integral as λ Corollary 516 For all V 0, X η λ) dµ 1 + λ) S S +1

12 1 LUKE G ROGERS Proof For λ 1 the result was observed in Lemma 59 If λ > 1 then for θ Θ kλ) we reason as in the roof of Lemma 59 to find η r θµ θ λ) q dµ µ θ λ S S +1 F θ X) Integrating the decomosition 54) then yields η λ) dµ λ S S +1 X = λ S S +1 λ S S +1 θ Θkλ)) i i η λ) F θ q) q V 0 η λ) x) x X kλ),i) where the last inequality is from Lemma 514 The reverse inequality is Lemma 59 Using the result of Corollary 516 in the roof of Lemma 510 imroves it as well Corollary 517 For any word w W, n η r w µ w λ) q Fw ) λ x) rw µ w ) + λ ) S S +1 x F w V 0 ) From here it is not difficult to obtain estimates of the normal derivatives of η λ) at oints of V 0 Recall from Section 4 that d k, q) is e k, so ex d kλ), q) ) tends to be small when λ is large The following lemma therefore tells us that the normal derivatives n η λ) q) are much smaller when q than when q = Lemma 518 For, q V 0, e i n η λ) ) 1 + λ) 1 S +1, + λ) 1 S +1 ex dkλ), q) ) n η λ) q) 0 Proof Suose q Let θ 1,, θ n be those words from Θ kλ) for which F θ iv 0 ) q r Using 54) and the lower bound for the normal derivatives n η θ i µ θ i λ) ) x y) for x, y V0 with y x from Corollary 58, as well as r e kλ) 1 + λ) 1/S +1), θ i n 0 n η λ) q) = η λ) r F θ i x) n η θ i µ θ i λ) x F ) θ q) i i=1 x V 0 n = η λ) F θ i x)r r θ i n η θ i µ θ i λ) ) x F θ q) ) i x V 0 i=1 1 + λ) 1 S +1 C1 + λ) 1 S +1 max nζ x y) ) n x,y V 0,x y i=1 η λ) z) z X kλ) q,1) + λ) 1 S +1 ex 1 dkλ), q) ) x V 0 η λ) F θ i x) where in the final ste we alied Lemma 514 and used that oints in z X kλ) q, 1) have d kλ), x) d kλ), q) 1 This gives the desired estimate for q

13 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 13 As a secial case of the above estimate we have 0 n η λ) q) λ 1 S +1 Combining this with Corollary 517 for the emty word w gives the desired result for n η λ) ) when λ 1 For λ 1 the conclusion is clear from Corollary 58, because n η λ) ) is increasing, so is bounded below by n ζ ) and above by n η 1) ) To comlete our icture of the behavior of η λ) we need a lower estimate on its decay and a corresonding uer estimate for the normal derivative n η λ) q) when q After a reliminary lemma, these may be obtained by somewhat simler reasoning than that used earlier Lemma 519 1) The function η λ) is non-zero all oints in V \ V 0, ) For q we have n η λ) q) < 0, 3) For w Θ kλ) and q we have n η r wµ w λ) q) Proof The first ste is to rove a weaker version of the second statement From the Gauss- Green formula we have n η λ) q) n ζ q) = n η λ) q) n ζ q ) = n η λ) x)ζ q x) η λ) x) n ζ q x) x V 0 = η λ)) ζq dµ X = λ η λ) ζ q dµ X so that n η λ) q) n ζ q) as λ 0 With q we have n ζ q) < 0, hence we may find some c < 0 and λ such that n η λ) q) c for all λ λ Now at x V \ V 0 the fact that η λ) dom ) requires that the normal derivatives sum to zero Take k so large that x VΘ k ) and for all θ Θ k we have r θ µ θ λ 0 λ Suose that η λ) x) = 0 Using 54) we can write this sum of normal derivatives as 0 = η λ) F θ ) n η r θ µ θ λ) F θ x)) {θ Θ k :F θ V 0 ) x} V 0 = η λ) F θ ) r n η r θµ θ λ) ) F x)) 58) c {θ Θ k :F θ V 0 ) x} { V 0 :F θ ) x} {θ Θ k :F θ V 0 ) x} { V 0 :F θ ) x} θ rθ ηλ) F θ ) where the first ste uses the scaling of the normal derivative and the fact that η λ) x) = 0, while the second uses that n η r θµ θ λ) F θ x)) c because r θµ θ λ λ and Fθ x) However c < 0 and all rθ ηλ) F θ ) 0, so the only way 58) can be true is if these vanishes at the boundary oints F θ V 0 ) of each cell meeting at x Reeating the argument inductively we see after finitely many stes that η λ) must vanish at all oints in VΘ k ), which is imossible because VΘ k ) and η λ) ) = 1 values are all zero, meaning that η λ) This roves the first statement of the lemma θ

14 14 LUKE G ROGERS The second assertion of the lemma now follows fairly easily from the first Using the artition Θ k as above and the decomosition 58) for the normal derivative at x = q we see n η λ 0) q) = η λ) F θ ) r n η r θµ θ λ) ) F q)) 59) c {θ Θ k :F θ V 0 ) q} { V 0 :F θ ) q} {θ Θ k :F θ V 0 ) q} rθ V 0 ηλ) F θ ) and we have already shown that the values rθ ηλ) F θ ) > 0 when F θ V 0 Finally the third statement follows from the second because n η λ) q) is continuous, non-ositive and increasing in λ, and r w µ w λ is bounded above when w Θ kλ), with all of these deending only on the fractal and harmonic structure Lemma 50 There is a constant a > 0 such that for any oint x VΘ kλ) ) \ V 0, η λ) x) a max { η λ) y) : θ Θ k with F θ V 0 ) {x, y} } These oints y are the neighbors of x in VΘ kλ) ) Proof Smoothness of η λ) requires that the first ste of 58) holds, where we take k = kλ) As we are not assuming η λ) x) = 0 we obtain instead of the second ste of 58) η λ) x) = {θ Θ kλ) :F θ V 0 ) x} rθ n η r θ µ θ λ) ) F x) θ x)) F θ {θ Θ kλ) :F θ V 0 ) x} {q V 0 :F θ q) x} {θ Θ kλ) :F θ V 0 ) x} {q V 0 :F θ q) x} η λ) F θ q))r rθ ηλ) F θ q)) θ θ n η r θ µ θ λ) ) F x)) where we used that n η r θ µ θ λ) ) q F θ x)) 1 from the third art of Lemma 519 On the left of this inequality each of the normal derivatives n η r θ µ θ λ) ) F F θ x) θ x)) is bounded above by a constant deending only on the harmonic structure, as shown in the first art of Lemma 518 Since the values rθ are comarable on both sides of the equation, the result follows q θ θ Corollary 51 Let X kλ), be the subset of X obtained by deleting those cells F θ X), θ Θ kλ) that intersect V 0 at oints other than There is c > 0 such that for all x X kλ), η λ) x) ex cd kλ), x) ) Proof If x VΘ kλ) ) \ V 0 and d kλ), x) = i then there is some y VΘ kλ) ) with d kλ) x, y) = i 1 By the revious result, η λ) x) aη λ) y) and by induction η λ) x) a i η λ) ) = a i The result for oints of VΘ kλ) ) \ V 0 follows by setting c = log a, and we note that the estimate η λ) x) C ex d kλ), x) ) from Corollary 515 ensures c > 0 To obtain the bound for a general oint x X kλ),, let θ Θ kλ) be such that F θ X) x The value d kλ), x) is the distance from to the nearest oint of F θ V 0 ); all other oints of this form are at most d kλ), x) + 1 from, and none is in V 0 \ {}, so the result for VΘ kλ) ) \ V 0 alies to them It therefore suffices to know that the restriction of η λ) to F θ X) is bounded below by a multile of its boundary values Observe that this function is a linear combination of the functions η r θµ θ λ) q Fθ with the boundary values as coefficients Hence it is enough to know that each η r θµ θ λ) q y) has a ositive lower bound on those cells F X) that

15 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 15 do not contain a oint of V 0 \ {q} This follows from the third art of Lemma 519, continuity of η λ) and the fact that the values r θ µ θ λ lie in a bounded interval, and we see that the constant deends only on the harmonic structure of the fractal Corollary 5 For the constant c of Corollary 51 n η λ 0) q) + λ) 1/S +1) ex cd kλ), x) ) Proof If we rewrite 59) with k = kλ) we have instead of c a constant as determined in the third art of Lemma 519, so that n η λ 0) q) rθ ηλ) F θ ) {θ Θ k :F θ V 0 ) q} { V 0 :F θ ) q} however on the right the values rθ 1+λ) 1/S +1) and there is at least one oint F θ from VΘ kλ) )\V 0 for which d kλ), F θ ) = d kλ), q), so that η λ) F θ ) C ex cd kλ)+c, x) ) as seen in Corollary 51 Since all terms on the right have the same sign, this term gives an uer bound 6 Estimates of the resolvent kernel on the ositive real axis In this section of the aer we use the estimates of Section 5 and the series exression 33) for the resolvent G λ) x, y) of the Lalacian to obtain estimates in the case λ 0, ) The main result is as follows Theorem 61 There are constants κ 1 and κ deending only on the fractal, harmonic structure and measure, and such that if λ > 0, 61) 1 + λ) /S +1) ex κ 1 d kλ) x, y )) G λ) x, y) 1 + λ) /S +1) ex κ d kλ) x, y )) excet that the lower bound is not valid if x or y is in a cell F θ X), θ Θ kλ) such that F θ V 0 ) contains a oint of V 0 \ {} For these latter cells, the aroriate estimate is instead one on the normal derivative Secifically, if V 0, y is not in one of the above cells F θ X) and the normal derivative n is taken with resect to the first variable, then 6) ex κ 1 d kλ), y )) n G λ), y) ex κ d kλ), y )) A symmetrical result holds for the normal derivative n with resect to the second variable There are also bounds aroriate to the case where both oints are near the boundary If and q are oints in V 0 then 63) 1 + λ) 1/S +1) ex κ 1 d kλ), q )) n ng λ), q) 1 + λ) 1/S +1) ex κ d kλ), q )) It is easy to see these estimates are equivalent to the following global bounds Let RV 0, x) denote the resistance distance from x to V 0 Then ex κ 1 d kλ), y )) λ /S +1) Rx, V 0 ) +λ 1/S +1)) Ry, V 0 ) +λ 1/S +1)) G λ) x, y) ex κ d kλ), y )) With this in hand will be relatively easy to obtain a similar result for the Neumann resolvent G λ) N x, y), which has vanishing normal derivatives rather than zero values at oints of V 0 Corollary 6 Let λ > 0, G N x, y) be the Neumann resolvent kernel, and 1x) denote the function identically equal to 1 on X Then G N x, y) 1 λ1x)1y) satisfies the uer and lower bounds of 61) everywhere on X

16 16 LUKE G ROGERS The roofs of these results occuy the rest of this section We begin by exanding the exression 33) for the resolvent of the Lalacian and using 34) to obtain 64) G λ) x, y) = r w G rwµwλ) q ψ r wµ w λ) Fw x) ψ r wµ w λ) q Fw y),,q V 1 \V 0 w W where we recall that G λ) q is the inverse of the matrix B λ) q defined in 35) In the next few lemmas we aly the estimates from Section 5 to the terms in this series, for which urose we require the following assumtion Assumtion 63 For the remainder of this section we require that λ > 0 Lemma 64 Let D λ) be the diagonal matrix with entries D λ) = 1 + λ) 1/S +1) B λ) ) and E λ) = D λ)) 1 + λ) /S +1) B λ) Then for all λ and we have D λ) 1, E λ) = 0, and for q ex c 1 cd kλ), q) ) E λ) q ex c d kλ), q) ), where c 1 and c deend only on the fractal and its harmonic structure and c is the constant in Theorem 51 Proof Comaring 35) and 3) we have B λ) q = = F X) n η r µ λ) F :F V 0 ) q :F V 0 ) q r F q) n η r µ λ) F q) ) F However Theorem 51 then shows that for = q we have B λ) 1 + λ) 1/S +1), so D λ) 1 Also from Theorem 51 we have for q + λ) 1/S +1) ex d kr µ λ), q) ) r n η r µ λ) F q) ) F + λ) 1/S +1) ex cd kr µ λ), q) ), and the observation that d kr µ λ), q) d kλ), q) lets us choose aroriate constants c 1 and c Lemma 65 For any word w W and all λ > 0, ex c 1 cd krw µ w λ), q) ) r w µ w ) + λ ) 1 S +1 r w G r wµ w λ) q ex c d krw µ w λ), q) ) where c 1, c and c are as in Lemma 64 Proof Recall that d kλ), q) e kλ) 1 + λ) 1/S +1), so that ex d kλ), q) ) can be made arbitrarily small by taking λ large enough Lemma 64 then imlies B λ) is close to diagonal, so we may find G λ) = B λ)) via the Neumann series Secifically, 1 + λ) 1/S +1) G λ) = I D λ) E λ)) D λ) = D λ) + D λ) E λ))k D λ) rovided λ C 1 where C 1 is chosen large enough that the series converges Observe that this C 1 deends only on the structure of the fractal, because D λ) contains only values 1 and the values in E λ) satisfy the estimate in Lemma 64 Notice also that all values in D λ) and E λ) are ositive, hence the same is true of all terms in the series Making the obvious uer and lower estimates of the sum of the series we conclude that for all and q ex c 1 cd kλ), q) ) 1 + λ) 1/S +1) G λ) q ex c d kλ), q) ) k=1

17 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 17 Substituting r w µ w λ) in lace of λ and using r w µ w = r S +1 w gives ex c 1 cd krw µ w λ), q) ) r w µ w ) + λ ) 1 S +1 r w G r wµ w λ) q ex c d krw µ w λ), q) ), which roves the estimate for λ C 1 r w µ w ) However if 0 λ C 1 r w µ w ) then it is immediate that r w µ w ) + λ ) 1 S +1 r w is bounded above and below, and we see G r wµ w λ) q is bounded above and below by continuity of its deendence on r w µ w λ At the same time d krw µ w λ), q) is bounded because it is the distance on a cellular artition of bounded scale All constants deend only on the fractal and harmonic structure, so the result follows Lemma 66 There are ositive constants c 3 and c 4 deending only on the harmonic structure, such that if c is the constant from Theorem 51 and V 1 \V 0, then for x in the suort of ψ r wµ w λ) Fw x), which is the union of all cells F w X) that contain F w ), 65) ex c 3 cd kλ) F w, x) ) ψ r wµ w λ) Fw x) ex c 4 d kλ) F w, x) ), excet that the lower bound does not hold on the cells F w F θ X) for those θ Θ krw µ w λ) such that F θ X) intersects V 0 \ {} For these excetional cells the correct estimate is that if x = F w q) for some q V 0 then 66) ex c 3 cd kλ) F w, F w q) ) r w µ w ) + λ ) /S +1) n ψ r w µ w λ) ex c 4 d kλ) F w, F w q) ) Fw ) x) Proof From 3) we see that ψ r wµ w λ) Fw x) is a iecewise λ-eigenfunction on F w X) with value 1 at F w ) and zero at the other oints of F w V 1 ) It is non-zero recisely on the the cells F w X) such that F V 0 ) On each such cell it is equal to η r µ r w µ w λ) F F w According to Theorem 51 we have 67) ex cd krw µ w λ)f, Fw x))) η r w µ w λ) F w x)) ex d krw µ w λ)f F, F w x))), excet that the lower bound does not hold on the cells excluded in the statement of the lemma From the artition Θ kr µ r w µ w λ) of X, form the artition r w Θ kr µ r w µ w λ) of F w X) and observe that the scale is comarable to that of Θ kλ) restricted to F w X) It follows that there are ositive c 3 and c 4 deending on the harmonic structure so that for y, z X, 68) c 3 d kλ) F w y, F w z) d krw µ w λ)y, z) c 4 d kλ) F w y, F w z) rovided we round aroriately Substituting 68) into 67) and eliminating the rounding by taking a suitably large multile of the exonential gives the desired estimate for ψ r wµ w λ) Fw x) because it is a finite sum of such terms For the normal derivative we note that if x = F w q for q V 0 then x = F w q for some q V 0 \ {} Using Theorem 51 at q we obtain ex cd krw µ w λ)f, q ) ) + r w µ w λ) /S +1) n η r w µ w λ) F q ) 69) ex d krw µ w λ)f, q ) ) Recalling that we see that n η r w µ w λ) F Fw ) = r w n η r w µ w λ) ) F F r w µ w ) + λ ) /S +1) n η r w µ w λ) F w F w ) x)

18 18 LUKE G ROGERS also satisfies the estimate in 69) As n ψ r wµ w λ) Fw is a finite sum of such terms, the bound 66) may be obtained by substituting 68) into this estimate We divide the roof of Theorem 61 into two arts: the roof of the uer bounds and the roof of the lower bounds Proof of Theorem 61: Uer bounds Fix x, y in X If x y let w W be the longest word such that x, y F w X), and otherwise let w be an infinite word such that F w X) = {x} = {y} in this latter case the exression 610) below may need an additional sum over the ossible choices of w, but we suress this because it does not otherwise affect the working) Then the series 64) terminates at scale w which is + if x = y) and may be written 610) G λ) x, y) = w i=0,q V 1 \V 0 r [w]i G r[w]i µ[w]i λ) q ψ r [w] i µ [w]i λ) F [w] i x) ψ r [w] i µ [w]i λ) q F[w] i y) It will be convenient to divide the sum into three ieces according to the size of r w µ w λ Define min { i : r [w]i µ [w]i λ 1 } if r w µ w λ 1 611) i 0 = + if r w µ w λ > 1, so that if i 0 < then r [w]i0 λ /S +1) The three ieces of the sum are as follows, where we note that it is ossible for a iece to be emty I 1 = I = I 3 = w i 0,q V 1 \V 0 r [w]i G r[w]i µ[w]i λ) min{i 0, w } 0 min{i 0, w } 0 q ψ r [w] i µ [w]i λ) F q V 1 \V 0 r [w]i G r[w]i µ[w]i λ) V 1 \V 0 r [w]i G r[w]i µ[w]i λ) [w] i x) ψ r [w] i µ [w]i λ) q q ψ r [w] i µ [w]i λ) F ψ r [w] i µ [w]i λ) F For both I 1 and I we use the trivial estimate F[w] i y) [w] i x) ψ r [w] i µ [w]i λ) q [w] i x) ψ r [w] i µ [w]i λ) F[w] i y) F[w] i y) 0 ψ r [w] i µ [w]i λ) F[w] i x) ψ r [w] i µ [w]i λ) q F[w] i y) 1 for the ψ factors Then for I 1 we have r [w]i µ [w]i λ 1, from which the values r G [w]i µ [w]i λ) q 1, and so 61) 0 I 1 w i 0 r [w]i r [w]i0 1 + λ) /S +1) if r w µ w λ 1 I 1 = 0 if r w µ w λ > 1 where we used the geometric decay of the r [w]i and the definition of i 0 Note that we have the uer bound 1+λ) /S +1) rather than ust λ /S +1) because when λ < 1 we have i 0 = 0 and r [w]i0 = 1 corresonding to the emty word

19 RESOLVENT ESTIMATES AND FRACTAL BLOWUPS 19 For I we instead estimate the factors r [w]i G r [w] i µ [w]i λ) q 613) min{i 0, w } I 0 min{i 0, w } 0 r [w]i G r[w]i µ[w]i λ) q q V 1 \V 0 using Lemma 65 to obtain r[w]i µ [w]i ) + λ ) /S +1) ex c d kr[w]i µ [w]i λ), q) ) 1 + λ) /S +1) if r w µ w λ λ) /S +1) ex c d krw µ w λ), q) ) if r w µ w λ > 1 because d k, q) is at least exonentially increasing in k, so the sum is dominated by a constant multile of the largest term, which is the one with the maximal value of i Moreover the constant multile deends only on the r and µ values and the way ath lengths on Θ k grow with k, all of which are roerties only of the fractal, harmonic structure and measure In the case that the maximal value of i is i 0 we have r w µ w λ 1 so d krw µ w λ), q) 1 and the exonential term is trivial The factors 1 + λ) /S +1) can be used rather than the more obvious choice λ /S +1) because if λ < 1 the sum is emty For the I 3 term we must take a different aroach, because in this case i < i 0 imlies we can aly Lemma 65 to obtain r [w]i G r [w] i µ [w]i λ) q r [w]i µ [w]i ) + λ ) /S +1) D r [w] i µ [w]i λ) q r [w]i µ [w]i ) + λ ) /S +1) and therefore the only estimate we have is that from Lemma 66, which gives 614) min{i 0, w } I λ) /S +1) i=0 min{i 0, w } 1 + λ) /S +1) i=0 ψ r[w]i µ[w]i λ) V 1 \V 0 F [w] i x) ψ r [w] i µ [w]i λ) F[w] i y) ex c4 d kλ) F[w]i, x ) c 4 d kλ) F[w]i, y ) ) V 1 \V 0 where we again have relaced the obvious λ /S +1) factor with 1 + λ) /S +1) because the sum is emty if λ < 1 The above indicates that we need a lower bound on d kλ) F[w]i, x ) + d kλ) F[w]i, y ) for the oints F [w]i, V 1 \ V 0 and a mechanism for counting how many of them there are Let L l be the set of such oints that are in the boundary of the l-cell containing x but not in the boundary of the l + 1)-cell containing x, then the number of oints in L l is smaller than the number of oints in V 1 \ V 0, which is a constant deending only on the structure of the fractal Moreover any oint z L l is searated from x by an l + 1)-cell By the same reasoning as in Lemma 66, the d kλ) diameter of any such l + 1) cell F [w]l X) is bounded below by a constant multile of the d kr[w]l µ [w]l λ) diameter of X, rounded down to the nearest integer Writing the diameter of X with resect to d k as diam k X) we obtain: cdkλ) F[w]i, x ) ) ex c diamkr[w]l µ [w]l λ) X) ) F [w]i L l ex We use this estimate only for l w Any oints F [w]i which occur in the sum but are not in any of the L l, l w must be in F w V 1 ), so the number of these is bounded by a constant deending only on the structure of the fractal For these oints we use the triangle inequality to estimate the corresonding terms of 614) ex c 4 d kλ) F[w]i, x ) c 4 d kλ) F[w]i, y ) ) ex c4 d kλ) x, y ))

20 0 LUKE G ROGERS Combining these estimates and substituting into 614) we have shown 615) I λ) /S +1) ex min{i )) 0, w } c 4 d kλ) x, y + ex c diam kr[w]l µ [w]l λ)x) ) ) 1 + λ) /S +1) if r w µ w λ λ) /S +1) ex )) c 4 d kλ) x, y if r w µ w λ > 1 where we used the same reasoning about the exonential decay as was used for the I term, along with the fact that d kλ) x, y ) diamkrw µ w λ)x) because x and y have resistance searation r w Comaring 61), 613) and 615) we see that we always have a bound by 1 + λ) /S +1) if r w µ w λ 1, and in this case the resistance searation of x and y is at most a constant multile of r w = r w µ w ) 1/S +1) λ /S +1), so that d kλ) x, y ) 1 Setting κ = min{c, c 4 } we therefore obtain G λ) x, y) 1 + λ) /S +1) 1 + λ) /S +1) ex κ d kλ) x, y )) For the case r w µ w λ > 1 we have I 1 = 0, and the estimate for I is dominated by a multile of that for I 3, so again G λ) x, y) 1 + λ) /S +1) ex κ d kλ) x, y )) and we have established the uer bound of 61) stated in the theorem We now turn to the uer bounds for normal derivatives Recall that n denotes the normal derivative with resect to the first variable If x V 0 and y x then the series for G λ) x, y) is finite and we can comute the normal derivative term by term The uer bound on ng λ) x, y) may then be obtained in almost the same way as the uer bound on G λ) x, y) The reasoning for both I and I 3 is unchanged excet that instead of bounding r [w]i G r [w] i µ [w]i λ) q by λ /S +1) we use the full uer bound r [w]i µ [w]i ) + λ ) /S +1) from Lemma 65 The uer bound for nψ r [w] i µ [w]i λ) l=0 F [w] i x) from 66) cancels this factor and leaves recisely the exonential decay terms seen in 613) and 614), so the only change to these estimates is that the λ /S +1) factor is no longer resent The I 1 term requires slightly more changes Again we use Lemma 65 and 66) to see that 616) r [w]i G r [w] i µ [w]i λ) q n ψ r [w] i µ [w]i λ) is bounded, but then use it to conclude that ni 1 w i 0 F [w] i x) ) ex c 4 d kλ) F [w]i, x) ) ex c4 d kλ) F [w]i, x) ) ψ r [w] i µ [w]i λ) q,q V 1 \V 0 ex c 4 d kλ) F [w]i, y) c 4 d kλ) F [w]i, x) ), F[w] i y) where the last ste is from 65) This sum can be bounded by the same argument as was used for I 3 in assing from 614) to 615), so we may sum the terms from I 1, I and I 3 to obtain the desired bound ng λ) x, y) ex κ d kλ) x, y) ) rovided y x This verifies the uer bound in 6) An easy argument then shows that for oints z within the Θ kλ) cell containing x, the eak size of G λ) z, ) is comarable to Rx, z) rather than 1 + λ) /S +1) ; we will later need the immediate consequence 617) G λ) z, ) L λ) /S +1) Rx, z) as z x V 0