A STRUCTURE THEOREM FOR FUNDAMENTAL SOLUTIONS OF ANALYTIC MULTIPLIERS IN R N

. Using a version of Hironaka’s resolution of singularities for real-analytic functions, we show that any elliptic multiplier p , real-analytic near its zero-set, has a fundamental solution µ 0 , and give an integral representation of µ 0 in terms of the resolutions supplied by Hironaka’s theorem. This µ 0 can be weakly approximated in a Sobolev space by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute a fundamental solution for an example non-polynomial symbol.

1. Introduction.Let S ′ (R n ) denote the space of tempered distributions on R n .A fundamental solution of Op(p) = F −1 pF is a µ 0 ∈ S ′ (R n ) such that Op(p)µ 0 = δ 0 in S ′ (R n ), where δ 0 is the unit measure at 0, F is the Fourier transform, and p is the symbol.The study of these is classical, and most results are recorded in standard texts [7,8].The Hörmander-Lojasiewicz theorem [6,9] ensures existence when p is a polynomial, and provides a way to construct a µ 0 , at least in principle, explicitly from the symbol.But the situation becomes nebulous when p is not a polynomial or globally smooth.We address this problem when p −1 (0) is compact and p is real-analytic near p −1 (0), and obtain an integral representation that clearly shows the structure of µ 0 .
In order to do so, we use a variant of Hironaka's resolution of singularities [1].Generally, the local charts that are supplied by Hironaka's theorem are unknowable, but they allow us to build an integral representation out of the geometry of p −1 (0).Occasionally, a diffeomorphism that brings p into a resolved form can replace them, and such a diffeomorphism can often be constructed when there is global symmetry.Important examples with this property include any sum of powers of the Laplacian, or any sum of powers of certain elliptic self-adjoint second-order differential operators.However, the novel and most interesting case for us here is when p is not a polynomial, and p is not necessarily globally smooth, but with dimension n > 1 and order d > 0.
We give examples showing the utility of this approach.
A lot of research has been devoted to the construction of explicit representations.See e.g.Ortner and Wagner [10] and Camus [3,4] for a broad class of operators.Usually, it is very difficult to find explicit representations of fundamental solutions, and the study is often focused on a particular operator of fixed order and dimension.In the case of general homogeneous elliptic and some types of non-elliptic operators, Camus [3,4] obtained explicit representations valid for any number of dimensions.Apart from the base practical value of constructing general solutions via convolution, an explicit form may find application in proofs of mapping properties of its operator.See e.g.Rabier [11], where the solution obtained in [3], implicit in [7], is used.

Notation
It must be smooth outside an open ball B(0, R) centered at 0 of some radius R > 0.
and the ellipticity constraint inf Definition 2.1 (The Paley-Wiener spaces).Let K ⊂ R n be compact and convex.Define PW d K (R n ) to be the space of entire functions u satisfying If {K j } ∞ j=1 is an exhaustion of R n by compact convex sets, we put These spaces are related to E ′ (R n ), the compactly supported distributions on R n .We write H t loc (R n ) for the Frechet space of distributions locally belonging to H t (R n ), and H −t comp (R n ) for its dual space of compactly supported distributions in H −t (R n ).Finally, S(R n ) is the Schwartz space, u ∈ S(R n ) decays faster than any polynomial, and we put F u(ξ) = R n e −ix•ξ u(x) dx and F −1 u(x) = 1 (2π) n F u(−x) for x, ξ ∈ R n .3. Solution Operator.Our main tool is Hironaka's resolution of singularities.Often it is stated abstractly [2], but we need a local embedded version.Theorem 3.1 (Local embedded version of Hironaka's theorem.From Atiyah [1]).Let U ⊂ R n be an open neighbourhood of 0, and let f be a function 0 ≡ f ∈ C ω (U ).Then there is an open 0 ∈ V ⊂ U , a real-analytic manifold M , and a map

It has the following properties:
1. Ψ : M → V is proper and real-analytic.

Ψ :
As p is real-analytic near its zero-set, it is compact with Lebesgue measure zero.The resolution theorem implies that p can be written locally in normal crossing form.Fix an open cover x αj for all x ∈ U j for some α j ∈ N n 0 , where each c j is a complex-valued, but nowhere zero, real-analytic function on U j .Also, we put m = max{α j } N j=1 .
Theorem 3.2.Let {χ j } N j=1 be any partition of unity subordinate to {V j } N j=1 .There is a fundamental solution µ 0 , smooth away from x = 0, of the form where χ = 1 − N j=1 χ j , and the I j are given a.e. by In this case, all other fundamental solutions differ from µ 0 by such a v .
The first step is to prove a lemma about principal value integrals with log kernel.It is used here in a way similar to Björk [2, Chapter 6, Theorem 1.5].
Lemma 3.3.Let ψ ∈ C ∞ (R) be either rapidly decaying or compactly supported.Then, for any k ∈ N, we have Proof.The proof of this is a routine exercise in repeated integration by parts.Observe that we can write ψ(r) = r −k r k ψ(r), and where all boundary terms at 0 in the final integration vanish, because and boundary terms at ±∞ vanish by the hypothesis on ψ.
Then P = Op( χ p ) + Q satisfies both Proof.The proof is an application of Lemma 3.3 and the Fubini-Tonelli theorem.Let ψ ∈ S(R n ).Using Lemma 3.3 coordinate-wise, we compute and we then get Note that the properties of Ψ j ensure that all the above integrals are well-defined.The determinant of dΨ j on each component of U j \ (p • Ψ j ) −1 (0) never becomes zero.This completes the proof.Lemma 3.5.P : Using the Leibniz rule, we get for any α ∈ N n 0 some C ′ α , C α > 0 such that which by the continuity of F : S(R n ) → S(R n ) implies the lemma.
Proof.Because each map Ψ j is proper, each (χ j F v) • Ψ j is compactly supported.By the well-known [7, Paley-Wiener-Schwartz Theorem 7.
where K is a compact and convex set so large that and so Qv is entire with Qv ∈ PW m (R n ).

Define the reflection map
The transpose of Q is AQA.Using this fact and Lemma 3.5, we extend Q, hence P , by duality: Applying Lemma 3.8, we get the fundamental solution µ 0 = P δ 0 for the operator.Using e.g.[12, Theorems 5.2 and 7.1], or similar in [5], it is smooth in x = 0, and Lemma 3.9.µ 0 is weakly approximated in and so where u j,k ∈ E ′ (U 0 j ) and Ψ j are precisely as stated in Theorem 3.2.Proof.Observe that pF v = 0 implies supp F v ⊂ p −1 (0) so that F v is compact.Again by [7,Theorem 7 and since c j is never zero, by [7, Theorem 2.3.5],we must have where u j,k ∈ E ′ (R n−1 ) are some distributions supported inside the z n = 0 slice of U j .It follows that Deformation of a star-convex zero-set onto a circle.
The main Theorem 3.2 is finally obtained by combining the above partial results.Unfortunately, it is impossible to obtain Ψ j explicitly for any given multiplier symbol.But if, for example, p −1 (0) is the real-analytic boundary of some star-convex domain, we can replace the charts by a single deformation Ψ of the boundary onto a sphere.Given p, we look for Ψ so that Ψ * p factorizes.Our main theorem gives where χ appropriately suppresses a region surrounding p −1 (0) on which Ψ is defined.The symbol p is taken into a polynomial form by the map Pulling back, we find that where c is a polynomial with no root in [0, ∞), and r j > 0 are the positive real roots.Let C j,k be the unique coefficients in ) is independent of ω ∈ S n−1 , and equal to 1 in a neighbourhood of ∪ m j=1 {r j } × S n−1 , all of the spheres of radius r j .If the multiplicities satisfy m j < n, we have where J n 2 −1 is the cylindrical Bessel function of order n 2 − 1, and 3.2.A perturbation of ∆ g .Let arg ξ be the multi-valued argument of ξ ∈ R 2 .Consider instead the multiplier symbol where n ∈ N and a < 1 2 .It is certainly real-analytic near its star-shaped zero-set.This p is taken into normal crossing form by the map Ψ : (− 1 2 , 1 2 ) × (0, 2π) → R 2 : (r, θ) → r + 1 + a cos(nθ) It is clear that Ψ is a diffeomorphism onto its image, not covering the whole zero-set.But a representation using only Ψ is still possible, because it misses just a single point.
Pick χ such that 1 − χ • Ψ ∈ C ∞ ((− 1 2 , 1 2 ) × (0, 2π)) does not depend on θ ∈ (0, 2π), and is compactly supported in (− 1 4 , 1 4 ) and equal to 1 in a neighbourhood of r = 0. We tacitly extend χ by one to all of R 2 .In that case, we have 8π 2 where u ∈ D ′ (R/2πZ) is a distribution on the space of 2π-periodic smooth functions.We could replace |ξ| 2 g in p by any integer power of |ξ| 2 g and still get a similar result, provided that we adjust the fractional power 1  2 in Ψ in accordance with this change.A similar technique can be applied to sums of powers of such multipliers too.

3. 1 .
Sums of powers of ∆ g .be a positive-definite symmetric matrix.Consider for d ∈ N and {c j } d j=0 ⊂ C with c d = 1 and c 0 = 0