Complex Powers of the Laplacian on Affine Nested Fractals as Calder\'on-Zygmund operators

We give the first natural examples of Calder\'on-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with 3 or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1<p<\infty$ and satisfy weak 1-1 bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.


INTRODUCTION
Complex powers of the Laplacian on Euclidean spaces and manifolds and their connection to pseudodifferential operators have been studied intensely (see, for example, [16,17,20,29,3] and the citations within). In this paper we define and study a class of operators built from the Laplace operator ∆ on nested fractals [13], which are a type of post-critically finite self-similar fractal [12,27]. The main focus is to show that the Riesz potentials (−∆) iα and the Bessel potentials (I − ∆) iα , α ∈ R, are Calderón-Zygmund operators in the sense of [21]. These operators are the first explicit examples of Calderón-Zygmund operators on a general class of self-similar fractals. The main result is as follows. Theorem 1.1. Let K be a nested fractal and X be either K or an infinite blowup of K without boundary. Suppose T is an bounded operator on L 2 (µ), where µ is the self-similar measure on X, and there is a kernel K(x, y) such that It is well known (see [11]) that for weights {µ 1 , . . . , µ N } such that 0 < µ i < 1 there is a unique self-similar measure A nested fractal is of the above type, but in addition has a large symmetry group. For K to be a nested fractal one requires that for every pair of points p, q ∈ V 0 the reflection in the Euclidean hyperplane equidistant from p and q maps n cells to n cells. It is also required that any n cell that intersects the hyperplane at a non-boundary point (of the n cell) is mapped to itself by the reflection. For full details see [13].
A key feature of nested fractals is the existence of a regular self-similar Dirichlet energy form E on K with weights 0 < r i < 1, i = 1, . . . , N such that The existence of such forms is non-trivial. In the case that all r i are the same it is due to Lindström [13] via probabilistic methods. Kigami's approach to constructing these as limits of resistance forms may be found in [12,27]. When the r i are not all equal there is no known general solution.
Let u ∈ dom E and f be continuous on X. We say u ∈ dom ∆ with (Neumann) for all v ∈ dom E. We say that u is smooth if ∆ n u is continuous for all n ≥ 1. The operator −∆ is non-negative definite and self-adjoint, with eigenvalues 0 = λ 1 ≤ λ 2 ≤ . . . accumulating only at ∞. We fix an orthonormal basis {ϕ n } for L 2 (µ) where ϕ n has eigenvalue λ n and eigenvalues may be repeated and let D be the set of finite linear combinations of ϕ n .
The effective resistance metric R(x, y) on K is defined by R(x, y) −1 = min{E(u) : u(x) = 0 and u(x) = 1}. It is known that the resistance metric is topologically equivalent, but not metrically equivalent to the Euclidean metric [12,27]).
Examples. An important example of a PCF self-similar set is the unit interval I = [0, 1]. In this case V 0 = {0, 1}. While I is not a fractal, Kigami's construction applies and one recovers the usual energy on the interval and the usual Laplacian ∆u = u ′′ .
The simplest example of a fractal to which the theory applies is the Sierpinski gasket, which has been studied intensively (see, for example, [12,27,6,2,14,24,30]). To describe the Sierpinski gasket, consider a triangle in R 2 with vertices {q 0 , q 1 , q 2 } and consider a set of three mappings F i : R 2 → R 2 , i = 1, 2, 3, defined by The invariant set of this iterated function system is the Sierpinski gasket and Blow-ups. In [23,24] Strichartz defined fractal blow-ups of K. This construction generalizes the relationship between the unit interval and the real line to arbitrary PCF self-similar sets. Let w ∈ {1, . . . , N } ∞ be an infinite word. Then The blow-up depends on the choice of the infinite word w. In general there are an uncountably infinite number of blow-ups which are not homeomorphic. In this paper we assume that the infinite blow-up K ∞ has no boundary. This happens unless all but a finite number of letters in w are the same. One can extend the definition of the energy E and measure µ to K ∞ . The measure µ will be σfinite rather than finite. As before, ∆ is defined by u ∈ dom ∆ with ∆u = f if u ∈ dom E, f is continuous, and It will be important in what follows that if K is a nested fractal and V 0 contains 3 or more points, then the Laplacian on an infinite blow-up without boundary has pure point spectrum [30,15] and the eigenfunctions have compact support. We write {λ n } n∈Z for the eigenvalues of −∆, which are nonnegative and accumulate only at 0 and ∞. As for K, we take a basis {ϕ n } n∈Z of L 2 (µ) consisting of eigenfunctions of −∆ with eigenvalues λ n that may be repeated, and let D be the set of finite linear combinations of ϕ n .
Notation for estimates. We write A(y) B(y) if there is a constant C independent of y, but which might depend on the fractal K, such that A(y) ≤ CB(y) for all y. We write A(y) ∼ B(y) if A(y) B(y) and B(y) A(y). If f (x, y) is a function on X × X, then we write ∆ 1 f to denote the Laplacian of f with respect to the first variable and ∆ 2 f to denote the Laplacian of f with respect to the second variable; repeated subscripts indicate composition, for example Heat kernel on nested fractals. Let K be an affine nested fractal with Dirichlet form as above. Let µ be the unique self-similar probability measure for which µ i = r d i , so that K is of Hausdorff dimension d in the resistance metric, and let X be an infinite blow-up of K without boundary. A fundamental result we require is an estimate for the heat kernel corresponding to the Laplacian. Specifically, the semi-group e t∆ is given by integration with respect to a positive heat kernel h t (x, y) which satisfies where β = d/(d + 1), R(x, y) is the effective resistance metric on X, and γ = γ ′ d+1−γ ′ , where 0 < γ ′ < d+1, is the chemical exponent, a constant depending on the fractal. The same estimates are valid for t k ∂ k ∂t k h t (x, y). These estimates are originally due to Barlow and Perkins [1] for the case of the Sierpinski gasket and have been generalized beyond what is needed here. In particular, (2.1) is a special case of [4, Theorem 1.1 (2) and Remark 3.7 (2)], but see also [7] and [8].
The following lemma implies that the heat kernel is integrable with respect to x and y, and will be important later. The estimate is presumably well-known but we include it for the convenience of the reader.
for all t > 0. Similar estimates hold if we integrate with respect to y.
Proof. Fix y ∈ X and for n ≥ 1 let X n denote the annulus centered at y of inner radius (2 n t) 1 d+1 and outer radius (2 n+1 t) 1 d+1 in the resistance metric. Let X 0 denote the ball radius t 1 d+1 centered at y. Then µ B(y, 2 and on X n the integrand satisfies Thus the integral is bounded below by the contribution from X 0 and above by the series of terms from each X n , which are exponentially decaying and therefore have sum bounded by a constant multiple of the largest term, which is readily computed to have size comparable to t d d+1 .

SINGULAR INTEGRAL AND CALDERÓN-ZYGMUND OPERATORS ON FRACTALS
In this section we define singular integral and Calderón-Zygmund operators on fractals and infinite blow-ups of fractals without boundary. For this we do not need to assume that K is nested, but only that it is a PCF fractal supporting a Laplacian in the sense of Kigami [12]. As usual, X is either K or an infinite blow-up of K without boundary. The following definition can be made for any dense subspace of L 2 , but we consider only the subspace D of finite linear combinations of eigenfunctions. ). An operator T bounded on L 2 (µ) is called a Calderón-Zygmund operator if T is given by integration with respect to a kernel K(x, y), that is T u(x) = X K(x, y)u(y)dµ(y) for u ∈ D and almost all x / ∈ supp u, such that K(x, y) is a function off the diagonal which satisfies the following conditions for some Dini modulus of continuity η and some c > 1. We say, in this case, that K(x, y) is a standard kernel.
The operator T is a singular integral operator if the kernel K(x, y) is singular at x = y.
The next theorem gives conditions which guarantee that (3.2) holds. In the succeeding sections we will show that the purely imaginary Riesz and Bessel potentials satisfy the hypothesis of this theorem. The proof is more involved in our case than the proofs of similar results in the real case due to the lack of a mean value theorem.

Theorem 3.2. Let K be a PCF fractal with regular self-similar Dirichlet form.
Suppose that X is equal to K or an infinite blow-up of K without boundary. If ∆ 2 K(x, y) is continuous off the diagonal of X × X and for all x, y, y ∈ X such that R(x, y) ≥ cR(y, y), for some c > 1 depending only on the scaling values r j of the Dirichlet form.
Suppose that X is an infinite blow-up such that where w = (w n ) is an infinite word. For n ≥ 0 we write ω|n for the finite word w 1 . . . w n and r ω|n := r ω1 · · · · · r ωn . We say that a cell C has size where c 1 is the constant from the estimates in [27, page 110] (see also [27, Lemma 1.6.1 a)]). Then, for a cell of size R we have that µ(C) R d .
for all x ∈ C, where β > 0 is a constant. Then, for all y and y in the interior of C we have Then where h C is the harmonic function on C such that h C | ∂C = f | ∂C , and G C is the (Dirichlet) Green function on C. Therefore where G is the Green function on K 0 . Recall that the Green function is given by the following formula [12, Proposition 3.5.5] or [27, (2.6.15)] and s,s ′ ∈V1\V0 g(s, s ′ ) is a constant depending on the fractal K. By the selfsimilarity of the energy E we have that Inequalities (3.6) and (3.7) imply the conclusion.
Proof of Theorem 3.2. Let r = max i=1,...,n r i and let c > r −3−k0 , where k 0 is such that r k0 < 1/3, and fix x, y, y ∈ X such that R(x, y) ≥ cR(y, y). Then R(x, y) ∼ R(x, y). Let {C n } be a partition of cells of X such that each C n is a cell of size r k0+1 R(x, y), or, equivalently, of size r k0+1 R(x, y). Then there is a cell C of order some m in this family that contains both y and y. We claim that x and y, and x and y, respectively, do not belong to the same or adjacent m − 1 cells. To see this, assume that m < 0, the proof for m > 0 being similar. Suppose that x and y belong to the same or adjacent m − 1 cells.
By the hypotheses, f x (·) has continuous Laplacian on C and satisfies (3.3) and (3.4) with R = R(x, y) and β = d, so Lemma 3.3 implies the conclusion.

PURELY IMAGINARY RIESZ POTENTIALS
Let X be a nested fractal K or an infinite blow-up based on this fractal. We define the class of operators (−∆) iα , with α ∈ R. Recall that for λ > 0 and α ∈ R we have Recall that D is the set of finite linear combinations of eigenfunctions. For ϕ an eigenfunction with eigenvalue λ we define where C α is the constant from (4.1).
We show that these operators are Calderón-Zygmund operators by proving that their kernels satisfy estimates of the form (3.1) and (3.2).
Before doing this we need the following lemma which says that in order to show the kernel coincides with a smooth function off the diagonal in the sense of Theorem 3.2, it suffices to differentiate inside the integral.

Lemma 4.2.
If m ∈ L ∞ ([0, ∞)) and u is a smooth function with compact support on X not intersecting {x} then Proof. Using the Green-Gauss formula (see, for example, [27, Theorem 2.4.1]) we have that Then K iα is the kernel of (−∆) iα , in the sense that for all u ∈ D such that x / ∈ supp u. Moreover, the kernel K iα (x, y) is smooth off the diagonal and satisfies the following estimates Proof. The proof of (4.2) is clear because h t (x, y) is the kernel of the heat operator and u is a linear combination of eigenfunctions. Both the smoothness and the desired estimates rely on the following computation using the estimate (2.1) and with l = j + k where C(m) denotes a constant depending only on k. Since the functions in the integrand are continuous on X × X and the integral converges uniformly on compact sets away from R(x, y) = 0 we conclude that (−∆ 2 ) k (−∆ 1 ) j K iα (x, y) is continuous off the diagonal for each j, k ≥ 0. Proof. Observe that (−∆) iα extends from D to L 2 (µ) by the spectral theorem. By Proposition 4.3, (−∆) iα is given by integration against a kernel K iα that is smooth off the diagonal and satisfies estimates (4.3) and (4.4). Theorem 3.2 implies that (−∆) iα is a Calderón-Zygmund operator.
We believe that the Riesz potentials are singular integral operators, that is, the kernel K iα (x, y) is singular on the diagonal for all α ∈ R \ {0}, but have not succeeded in proving this. Theorem 4.5. For α ∈ R, the operator (−∆) iα defined originally on D extends to a bounded operator on L p (µ) for 1 < p < ∞, and satisfies weak 1-1 estimates.
The kernel K p is smooth off the diagonal and it satisfies the estimates Therefore, for a function p of Laplace transform type the operator p(−∆) is a Calderón-Zygmund operator and it extends to a bounded operator on L q (µ), 1 < q < ∞.
We end this section by describing the dependence of the kernel K iα on α. Proposition 4.9. If x = y, the map α → K iα (x, y) is differentiable.
Proof. Let x, y ∈ X such that x = y and f (t, α) := ∆ 1 h t (x, y)t −iα . We know that | ln t| is integrable and apply a standard theorem (for example, [5, Theorem 2.27]).
As g(t) is continuous on [0, 1] we look only at the integral over [1, ∞). Using that ln t ≤ δ −1 t δ for any δ > 0 we have

BESSEL POTENTIALS
We next study the Bessel potentials on X, where X is a nested fractal K or an infinite blowup, without boundary, of K. Our analysis follows, in large, [19,Chapter 5.3] (see also [25]). In this spirit we consider the strictly positive operator A = 1 − ∆. Then u is an eigenfunction of A if and only if it is an eigenfunction of −∆ and Aϕ n = (1 + λ n )ϕ n . Recall that D is the set of finite linear combinations of the eigenfunctions ϕ n .
(1) If u = a k ϕ k ∈ D then Proof. For the first assertion, recall that by the spectral theorem, if u = a k ϕ k ∈ D is a finite sum then e t∆ a k ϕ k = a k e −tλ k ϕ k .

Therefore we can exchange the sum and the integral in
The second assertion is an immediate consequence of the first.

Based on the above proposition we can extend the definition of
We show next that the operators A α , Re α < 0, defined originally on D, extend to bounded operators on L p (µ) for all 1 ≤ p ≤ ∞. We accomplish this by studying the kernels of the operators. The main tools we use are the heat kernel estimates together with Lemma 2.1. Since estimates of this type will be needed several times, we give the argument for the most general kernel we will encounter. Proposition 5.3. For s ∈ R, m ∈ L ∞ ([0, ∞)), and x = y define In particular, for s > 0, L s,m (x, ·) ∈ L 1 (µ) for all x ∈ X and L s,m (x, ·) 1 ≤ C, with C a constant independent of x. Similarly s > d 2 implies L s,m (x, ·) ∈ L 2 (µ) for all x ∈ X with a uniform bound on the L 2 norm. Moreover if s > d then L s,m (x, y) is uniformly bounded.
Proof. For s ∈ R make the substitution t = uR(x, y) d+1 , from which γ+1 . The intervals of validity of the estimates (5.3) arise naturally in estimating I 1 and I 2 .
To bound I 1 we use that e −tR(x,y) d+1 ≤ 1 on the interval, and make the change of variable t = u −γ to find The exponential decay in the integrand implies this is bounded by a constant multiple c(s) of the integral over the unit length interval [δ −γ , δ −γ +1]. It is easy to see c(s) ≤ 1 for s ≥ d and c(s) ≤ Γ(s) otherwise. We bound the exponential term by e −cδ −γ , and integrate the polynomial term to obtain γ(d+1) For the estimate of I 2 we use that e −ct −γ ≤ 1 on the interval, so that with u = tR(x, y) d+1 we obtain u This integral is the same as in (5.4), except that the power in the integrand is We must therefore have the same estimates for the integral that we did for I 1 , but with the power R( throughout. Multiplying through by the leading R(x, y) d−s factor makes these estimates the same or smaller than the corresponding ones for I 1 , which completes the proof of (5.3).
Observe that the upper bound for s > d is itself uniformly bounded, so L s,m (x, y) is uniformly bounded in this case. Also the bound is integrable for large R(x, y) because it has exponential decay, and the the singularity in the bound (which occurs when s ≤ d) is integrable provided s > 0. Hence L s,m (x, ·) 1 ≤ C with C independent of x. Finally we note that the singularity is in L 2 if s = d, or if s < d and 2(s − d) + d < 0, meaning s > d 2 .
Then K α (·, y) is integrable for all y ∈ X with K α (·, y) 1 ≤ C with C independent of y. The same statements are true for K α (x, ·).
Proof. Let α = a + ib, with a < 0. This is an immediate consequence of the L 1 estimate in Proposition 5.3 with s = −a(d + 1) and m(t) = t ib .
As a consequence of the above result we obtain that the operators A α are bounded operators on L p (µ) for 1 ≤ p ≤ ∞ if Re α < 0. We prove this statement in the following.
Theorem 5.5. Let α be such that Re α < 0. Then A α is given by integration with respect to K α , that is for all f ∈ D. In particular, the operator A α defined originally on D extends to a bounded operator on L p (µ) for all 1 ≤ p ≤ ∞.
Proof. The proof of (5.5) is clear. The second part follows immediately from the estimates of Corollary 5.4 by an argument analogous to the classical proof of the Young's inequality via the generalized Minkowski inequality.
Proposition 5.6. On the compact set K the operator A α is Hilbert-Schmidt when Re α < − d 2(d+1) .
Proof. Let α = a + ib, set s = −a(d + 1) and m(t) = t ib . Then s > d 2 so we can apply Proposition 5.3 to see K α (x, ·) L 2 is uniformly bounded. Integrating with respect to x produces a factor of µ(K) < ∞, so K α (x, y) is in L 2 of the product space.
Proposition 5.8. If Re α < 0 then K α is smooth off the diagonal. If in addition Re α ≤ − d d+1 then K α is continuous and uniformly bounded. Also, the map α → K α (x, y) is analytic on {Re α < 0}, for all x, y ∈ X with x = y.
To obtain smoothness off the diagonal, it suffices by Lemma 4.2 that we differentiate inside the integral. Since h t is the heat kernel, applying the Laplacian is the same as differentiating with respect to t, and we know t k ∂ k ∂t k h t satisfies the same bounds as h t itself. Then applying Proposition 5.3 for m(t) = t ib and s = −(a + j + k)(d + 1) one can see that ∆ k 1 ∆ j 2 K α (x, y) is continuous off the diagonal for all j, k ≥ 1. If Re α ≤ − d d+1 then a second application of the Proposition shows K α (x, y) is uniformly bounded. The second part follows by a standard argument (such as [22,Theorem 5.4]), since the map F (α, t) = h t (x, y)t −α−1 e −t is analytic on α for each t > 0, and continuous in α and t.

Purely imaginary Bessel potentials.
We turn our attention now to the study of the kernel of purely imaginary Bessel potentials, that is, operators of the form (I − ∆) iα , with α ∈ R. We use formula (4.1) as the starting point and, for α ∈ R and u ∈ D, we define For α ∈ R, we define the kernel of (I − ∆) iα via Theorem 5.9. For α ∈ R, G iα (x, y) defined in (5.6) is the kernel of (I − ∆) iα , in the sense that for all u ∈ D such that x / ∈ supp u. Moreover, G iα (x, y) is smooth off the diagonal.
Proof. As u ∈ D is a finite sum, using integration by parts we have that Thus (5.7) holds. From Lemma 4.2 we may apply powers of the Laplacian inside the integral in (5.6) to establish smoothness off the diagonal. Moreover the application of ∆ j 1 ∆ k 2 is equivalent to replacing h t (x, y) with ∂ (j+k) ∂t (j+k) h t , which satisfies the same estimates as t −(j+k) h t . Applying Proposition 5.3 with s = −(j + k)(d + 1), we see that for all j, k ≥ 0, so in particular G iα (x, y) is smooth off the diagonal. Proof. The operator (I − ∆) iα extends to a bounded operator on L 2 (µ) by the spectral theorem. The proof of Theorem 5.9 implies that G iα satisfies the estimates (1.1) and (1.2). Thus (I − ∆) iα is a Calderón-Zygmund operator.
Corollary 5.11. If α ∈ R, then the operator (I − ∆) iα defined originally on D extends to a bounded operator on L p (µ) for 1 < p < ∞ and satisfies weak 1-1 estimates.
We also note that one can easily modify the proof of Proposition 4.9 to obtain the following result.
Proposition 5.13. Let x, y ∈ X with x = y. Then the map α → G iα (x, y) is differentiable.

COMPLEX POWERS ON PRODUCTS OF FRACTALS AND BLOWUPS
In this section we extend our analysis of Calderón-Zygmund operators and the Riesz and Bessel potentials to finite products X N , where X is either a nested fractal K or an infinite blow-up without boundary of K. The study of the energy, the Laplace operator, and the heat kernel estimates on products of PCF fractals was initiated by Strichartz in [26] (see, also, [27, 28]). We begin by reviewing the basic steps in his construction. Consider the product space X N with the product measure µ N . Notice that X N is not, in general, a PCF fractal. Recall from [26] that a measurable function u on X 2 has minimal regularity if and only if for almost x 2 ∈ X, u(·, x 2 ) ∈ dom E, and for almost every x 1 ∈ X, u(x 1 , ·) ∈ dom E. Such a function belongs to the domain of the energy on X 2 , dom E 2 , if and only if exists and is finite [26,27]. This definition can be easily generalized to X N . Then we may define a Laplacian by the weak formulation To avoid confusion, we will henceforth write ∆ ′ for the Laplacian on X. Recall that we fixed orthonormal basis {ϕ n } n for L 2 (µ) such that each ϕ n is an eigenfunction of ∆ ′ . Then if n = (n 1 , n 2 , . . . , n N ) the functions where x = (x 1 , x 2 , . . . , x N ) ∈ X N , form an orthonormal basis for L 2 (µ N ). Let D N be the set of finite linear combinations of ϕ n .
The heat kernel on X N is the product , for x, y ∈ X N ([28, Theorem 6.1]). We extend the metric to X N by Then, if K is an affine nested fractal, the heat kernel estimates become [28, Theorem 2.2] where the symbol ∼ indicates that there are upper and lower bounds by constant multiples, possibly with distinct values of c. The same estimates are satisfied by t k ∂ k ∂t k h N t (x, y) for all t > 0.

Singular integral and Calderón-Zygmund operators on products.
We say that on operator T bounded on L 2 (µ N ) is a Calderón-Zygmund operator on X N if it is given by integration with respect to a kernel K(x, y) which is a function off the diagonal and satisfies for all x = y and N (y, y), for some c > 1, where η is a Dini modulus of continuity. We say that T is a singular integral operator if K(x, y) is singular at x = y. The next theorem, which is the main result of this section, extends Theorem 3.2 to the product setting. Theorem 6.1. Let K be a nested fractal and assume that X is either K or an infinite blow-up of K without boundary. Suppose that T : is given by integration with respect to a kernel K(x, y) which is smooth off the diagonal and satisfies the following estimates where for y = (y 1 , y 2 , . . . , y N ), ∆ ′ y,i K(x, y) is the Laplacian on X with respect to y i . Then T is a Calderón-Zygmund operator. In particular, T extends to a bounded operator on L p (µ N ) for all 1 < p < ∞ and satisfies weak 1-1 estimates.

Purely imaginary Riesz potentials on products.
Having all the ingredients in place, we can define for α ∈ R and u ∈ D N The kernel of (−∆) iα is given by the formula Using, basically, the same computations as in Section 4 we see that, for α ∈ R, K iα (x, y) is smooth away from the diagonal and satisfies the following estimates: . . , N, (6.8) so (−∆) iα is a Calderón-Zygmund operator on X N . We believe that they are singular integral operators but have not succeeded in proving this. Corollary 6.2. For a ∈ R, the operator extends to a bounded operator on L p (µ N ), for all 1 < p < ∞, and satisfies weak 1-1-estimates. Remark 6.3. The boundedness of (−∆) iα on L p (µ N ), 1 < p < ∞, can also be deduced using the multivariable spectral results of [18].
The results about the dependence of the kernels on α extend easily to the product setting. Using essentially the proof of Proposition 4.9 we see that α → K iα (x, y) is differentiable in α for all x, y ∈ X with x = y.

Bessel Potentials on products.
Consider now the strictly positive operator A = I − ∆. As before, for Re α < 0 and u ∈ D N , we define (6.9) A α u = 1 Γ(−α) It is clear that the equivalent of Proposition 5.2 holds so we can define A α = A α−k A k , where k is such that −1 ≤ Re α − 1 < 0, if Re α ≥ 0. Versions of all the statements established in Section 5 remain valid for this class of operators. The crucial ingredients in the proofs there were the heat kernel estimates (2.1) which we have now as (6.2). In particular, we see from (6.2) that all of the integrals we encounter in the product setting differ from (5.2) only in that there is an extra factor of t − (N −1)d d+1 , so Proposition 5.3 is valid if we replace h t (x, y) with h N t (x, y) and replace each occurrence of s − d in (5.3) with s − N d, making the regions for the estimates s < N d, s = N d and s > N d.
The above quickly gives an analogue of Theorem 5.5. Notice that the singularity occurring for s ≤ N d is in L 1 (dµ N ) if s > 0, so that K α (·, y) 1 is bounded by a constant independent of y and similarly for K α (x, ·) 1 . This implies that for Re α < 0, A α extends to be a bounded operator on L p (µ N ) for all 1 ≤ p ≤ ∞.
Our estimates show that the kernel is always smooth away from the diagonal. If Re(α) < − N d d+1 then we have s > N d, from which the kernel is also globally continuous and uniformly bounded. By the same argument as in Proposition 5.8 the map α → K α (x, y) is analytic on {Re α < 0} for all x, y ∈ X N .
For purely imaginary Bessel potentials (I − ∆) iα we have analogues of Theorem 5.9 and its corollaries. Specifically, we define and verify that it represents (I − ∆) iα on those u ∈ D N with support away from x. By the previous reasoning about the analogue of Proposition 5.3 (with d replaced by N d in the conclusions) we see that G iα (x, y) is smooth off the diagonal and satisfies (6.7) and (6.8). Thus (I − ∆) iα is a Calderón-Zygmund operator and it extends to a bounded operator on L p (µ N ) for all 1 < p < ∞ and satisfies weak 1-1 estimates. The map α → G iα (x, y) is also differentiable for all x = y.