Optimal codomains for the Laplace operator and the product Laplace operator

An optimal codomain for an operator P (∂) with fundamental solution E , is a maximal space of distributions T for which it is possible to define the convolution E ∗ T and thus to solve the equation P (∂)S = T . We identify optimal codomains for the Laplace operator in the Euclidean case and for the product Laplace operator in the product domain case. The convolution is understood in the sense of the S′ -convolution.


Introduction
Given a linear partial differential operator P (∂) with constant coefficients, the equation P (∂) S = T has always a solution for any distribution T with compact support.Indeed, a solution S is given as the convolution of the distribution T with a fundamental solution E of the differential operator.This convolution operator T → E * T becomes a twosided inverse for the operator P (∂) in the space E of compact supported distributions.An optimal codomain for the operator P (∂) is a maximal space of distributions T for which it is possible to define the convolution E * T and thus to solve the equation P (∂) S = T .Given an specific operator, the condition that T have compact support can be weakened greatly.Finding the maximal space depends on having an explicit formula for the distribution E and on how we define the convolution for distributions T that might not have compact support.
Regardless of the definition, any bonafide convolution is expected to be commutative, to commute with derivatives, and it should satisfy, in some sense, the Fourier exchange formula F (S * T ) = F (S)F (T ), if S and T are tempered distributions.Several definitions, (see, for instance, [5], [14], [12], [13], [4], [7], [8], [9], [11]), have been introduced and studied in great detail to extend the convolution to appropriate distributions without any compactness assumption.In particular, the S -convolution was developed by Y. Hirata and H. Ogata ( [5]) and R. Shiraishi ([14]) with the purpose of extending to appropriate pairs of tempered distributions the classical definition of convolution given, for instance, by L. Schwartz ([13]).Y. Hirata and H. Ogata showed that their definition satisfies all the expected properties.R. Shiraishi ([14]) introduced an equivalent definition, which is the definition we will use.
In this article we identify the optimal codomain in the context of the S -convolution as defined by R. Shiraishi ([14]), for the Laplace operator associated with the product domain R n1 ×R n2 .Both operators have tempered fundamental solutions given by explicit formulas.Each optimal codomain is a weighted space of tempered distributions.Already L. Schwartz had observed without proof (see [13]) the existence of the Newtonian potential of distributions in an specific weighted space.The weighted space relevant to the Euclidean case was also used by J. Horváth [7] and N. Ortner [9] to identify those tempered distributions that are Sconvolvable with n-dimensional hypersingular kernels.Their results apply, in particular, to appropriate tempered fundamental solutions of the Laplace operator.The weighted space relevant to the product domain case is an extension of a space of distributions considered in [1].
The proofs of our results use simple characterizations of the spaces of distributions involved in each case, extending results proved in [2].As we will explain later, the fact that we consider tempered distributions in the definition of the convolution does not restrict the identification of maximal spaces of distributions.
Once an optimal codomain for an operator P (∂) is obtained, it immediately leads to existence results for the equation P (∂) S = T when the distribution T belongs to the appropriate maximal space.Moreover, a solution S is given explicitly by the formula E * T .For specific operators, the fundamental solution might have smoothness properties that will give, by known results (see for instance [15] p. 73, [6] p. 109), smoothness properties of the solutions of the equation P (∂)S = T .For instance, in the case of the Laplace operator, ellipticity implies that any fundamental solution will be locally integrable, and C ∞ outside of zero.On the other hand, as it is expected in a product domain structure, fundamental solutions for the operator Δ x Δ y are not C ∞ outside of zero, which implies that the convolution operator T → E * T is not pseudo-local, that is to say, smoothness properties of T not necessarily translate into smoothness properties of E * T .
Our article is organized as follows: In Section 2 we present a brief account of the S -convolution and the relevant spaces of test functions and distributions, with their main properties.In Section 3 we obtain simple characterizations of two weighted spaces of tempered distributions.Section 4 and Section 5 carry our main results for the operators Δ and Δ x Δ y , respectively.Finally, in Section 6 we discuss the unlikely possibility of identifying an optimal codomain for a general operator P (∂).We prove, however, that is always possible to extend the convolution operator T → E * T to an appropriate weighted space of distributions.The paper ends with a list of references.
The notation used in this article is standard.The symbols C ∞ 0 , S , C ∞ , L p , L p loc , D , S , E , etc., indicate the usual spaces of distributions or functions defined on R n , with complex values.With |•| we denote the Euclidean norm on R n , while • p will denote the norm in the space L p .When we need to emphasize that we are working on a particular setting, we will write Other notation will be introduced at the appropriate time.

Preliminary definitions and results
We will first summarize the definitions of those spaces of test functions and distributions that are relevant to the notion of S -convolution.For more details, we refer to [13] p. 199.
With B we denote the space of C ∞ functions ϕ : R n → C such that ∂ α ϕ is bounded in R n for each multi-index α .We define in B the topology of the uniform convergence in R n of each derivative.With Ḃ we indicate the closed subspace of B consisting of those The space D L 1 of integrable distributions is, by definition, the topological dual of the space Ḃ , with the strong dual topology.
Since C ∞ 0 is dense in Ḃ , the space D L 1 is contained in D .Furthermore, every distribution T with compact support belongs to D L 1 .According to [13] p. 201, each distribution T ∈ D L 1 can be written as where f α ∈ L 1 .This explains why the distributions in D L 1 are called integrable distributions.Moreover, it shows that they are tempered distributions.
Straightforward calculations show that D L 1 is closed under differentiation as well as under multiplication by functions in B .
We will need to consider a second topology in the space B .This topology is the finest locally convex topology that induces the usual topology of C ∞ on the subsets of B that are bounded with respect to the topology previously defined on B .We will denote with B c the space B endowed with this topology.A sequence {ϕ j } converges to ϕ in B c when, for each multi-index α , sup j ∂ α ϕ j ∞ < ∞ and the sequence {∂ α ϕ j } converges to ∂ α ϕ uniformly on compact sets of R n .The spaces C ∞ 0 , and Ḃ , are dense in B c ([13] p. 203).Moreover, the space D L 1 is the topological dual of B c ([13] p. 203).Now we are ready to take up the notion of S -convolution.The first definition was proposed by Y. Hirata and H. Ogata [5].Their purpose was to extend the validity of the Fourier exchange formula F (T * S) = F (T )F (S), originally proved by L. Schwartz for pairs of distributions in the Cartesian product O C × S ([13] p. 268).Here, O C denotes the space of rapidly decreasing distributions ( [13]).Later on, R. Shiraishi introduced in [14] an equivalent definition which is the definition we state now and we will use in this article.Definition 1 ([14]).Given two tempered distributions U and V , it is is linear and continuous.Thus, it defines a tempered distribution which will be denoted by U * V .
In this definition, U ( ∨ V * ϕ) is the multiplicative product of the tempered distribution U with the function given by the regularization x → ( It can be shown that this regularization is a C ∞ function and each derivative has at most polynomial growth at infinity.R. Shiraishi proved in [14] that the operation introduced in Definition 1 is commutative, as expected of a bonafide convolution.Moreover ( [5]), the multiplicative product U V can be defined so that the Fourier exchange formula, (U * V ) ∧ = U V holds.We point out that the S -convolution coincides with the classical convolution defined by L. Schwartz, in all the cases in which both exist.
Particularly, when the distributions U and V are integrable functions, we can write where U * V indicates the classical convolution of two integrable functions.
If two tempered distributions U and V are S -convolvable, by definition, Thus, it can be written as where f α ∈ L 1 .Thus, we go back to the classical pairing L 1 , L ∞ .L. Schwartz studied in [12] the S -convolution with the one-dimensional Hilbert kernel H = pv 1 x .To this purpose he introduced a weighted version of the space D L 1 (R).Namely, with the topology induced by the map This definition can be extended as follows.
with the topology induced by the map with the topology induced by the map It is immediate from Definition 3 that the space w μ D L 1 is the topological dual of the spaces w −μ Ḃ and w −μ B c .
Likewise, the space It is clear from Definitions 3 and 4 that w μ D L 1 and w μ1 ), we observe that The assumptions on μ 1 , μ 2 , μ imply that the function w −μ w μ1 1 w μ2 2 belongs to B .Thus, we have the inclusion.To prove that the inclusion is strict we will consider the tempered distribution δ x ⊗ w λ 2 for an appropriate value of λ, where δ x denotes the Dirac measure supported at 0 ∈ R n1 .
We first show that where the integral is finite.Now we will prove that and θ 2 (y) = θ (|y|) .We claim that the sequence {θ 1 (x/j) θ 2 (y/j)} j converges to 1 in B c (R n1+n2 ).In fact, On the other hand, if we fix a compact "box" K 1 × K 2 in R n1+n2 we can find j K1,K2 so that θ 1 (x/j) = 1 on K 1 and θ 2 (y/j) = 1 on K 2 , for j ≥ j K1,K2 .Thus, for j ≥ j K1,K2 .If we fix multi-indexes α, β with |α| + |β| > 0, we have should have a limit as j → ∞.Let us see that this is not the case.Indeed, and this last integral diverges as j → ∞.
Since we can choose λ so that ).This completes the proof of Lemma 5.
As we will prove in Section 4, the space w μ D L 1 , for an appropriate positive value of μ, is the maximal weighted space of distributions associated with the operator Δ.Likewise, we will prove in Section 5 that w μ1 1 w μ2 2 D L 1 (R n1+n2 ) , for appropriate positive values of μ 1 and μ 2 , is the maximal weighted space of distributions associated with the operator Δ x Δ y .Let us point out that these appropriate values of μ 1 , μ 2 and μ satisfy the assumptions of Lemma 5.
Fundamental solutions for the operators Δ and Δ x Δ y do not satisfy the same homogeneity and smoothness properties, so, as convolution kernels, they define operators with different pseudo-local properties.These differences are reflected in the fact that the spaces of distributions that will be S -convolvable with each of them are not the same, as shown above.

The characterization of w µ D L 1 and the characterization of w
The following results provide simple representation formulas for the distributions in w μ D L 1 and w μ1 1 w μ2 2 D L 1 (R n1+n2 ) , extending results proved in [2].These representation formulas will be used in Sections 4 and 5 to identify optimal codomains for the operators Δ and Δ x Δ y .Proposition 6.Given T ∈ D and μ ∈ R, the following statements are equivalent.
(1) T ∈ w μ D L 1 . (2) Proof.Let us first assume that 1) holds.Consider a cut-off function ψ ∈ C ∞ 0 such that 0 ≤ ψ ≤ 1, ψ(x) = 1 for |x| < 1 and ψ(x) = 0 for |x| > 2. Given T ∈ w μ D L 1 we can write It is clear that the distribution ψT has compact support.Since (1 − ψ) w μ |x| μ belongs to B , we conclude that the distribution 1−ψ |x| μ T belongs to D L 1 .Furthermore, it is zero in a neighborhood of zero.Thus, we have obtained the representation stated in 2).
Let us now assume 2).Since E ⊂ w μ D L 1 and w −μ |x| μ ∈ B where T 2 is not zero, we conclude that T 1 and |x| μ T 2 belong to w μ D L 1 .Thus, This completes the proof of Proposition 6.
Remark 7. The distributions T 1 and T 2 in Proposition 6 are not unique.Indeed, if U is a distribution with compact support that is zero near the origin in R n , the distributions T 1 − |x| μ U and T 2 − |x| μ U provide another representation.As we mentioned before, the representation given in Proposition 6 will be central to our characterization of an optimal codomain for the Laplace operator.However, it is not the only possible representation.In fact, if T ∈ w μ D L 1 , then w −μ T can be written as finite ∂ α f α for f α ∈ L 1 .So we obtain the representation Alternatively, we can also write where the functions g α belong to the weighted space L 1 (w −μ ).
(1) Proof.In order to simplify the notation, we will not indicate in what follows the underlying Euclidean space, which will always be R n1+n2 .If we assume that 2) holds, we claim that each of the terms in the representation of T belongs to w μ1 1 w μ2 2 D L 1 .Indeed, since T 0 ∈ E the claim is true for the first term.We observe that w −μ1 1 w −μ2 2 |x| μ1 belongs to B where T 1 is not zero, thus the claim is also true for the second term.The other terms are treated similarly.
An appropriate version of Remark 7 applies to the representation obtained in Proposition 8.

The optimal codomain for the Laplace operator Δ
For brevity we will assume in what follows that the dimension n of R n is ≥ 3. The case n = 1 is similarly treated and the case n = 2 uses a modification of the weighted space of distributions by considering a logarithmic weight.
It is well known (see, for instance, [15] p. 73), that the locally integrable function E n defined as where Γ indicates the Euler gamma function, is a tempered fundamental solution for the Laplace operator Δ when n ≥ 3 .Moreover, the function E n is real analytic and thus C ∞ , outside the origin in R n , reflecting the fact that the Laplace operator is elliptic.As a consequence, the convolution operator T → E n * T is pseudo-local, which means that E n * T will coincide with a C ∞ function in any open set of R n where T coincides with a C ∞ function.This implies that solutions of the equation ΔS = T will have the same smoothness properties as T .
Here is the main result of this section.
Theorem 9. Let T ∈ S and let n ≥ 3 .Then, the following statements are equivalent Theorem 9 is an immediate consequence of the following result.
Theorem 10.Let T ∈ S and let μ < n.Then, the following statements are equivalent.
Proof.We will prove the theorem under the assumption 0 < μ < n.The proof of the case μ ≤ 0 uses the same ideas.We first prove that 1) implies 2).According to Definition 1, we need to show that the distribution T (|x| −μ * ϕ) belongs to D L 1 for each ϕ ∈ S .We can write ( 6) Thus, it is enough to show that w μ (|x| −μ * ϕ) ∈ B .Since we can write it suffices to prove that w μ (|x| −μ * ϕ) ∈ L ∞ , which is a straightforward calculation that we will omit.
To prove that 2) implies 1), we will show that the distribution T can be written as in Proposition 6.We start by writing T = ψT +(1 − ψ) T , where ψ is the cut-off function introduced in Proposition 6.Clearly, ψT ∈ E and (1 − ψ) T is zero in a neighborhood of zero.
We claim that we can choose in (6) a test function ϕ in C ∞ 0 so that In fact, let us consider a function ϕ in For this particular test function we can write

By hypothesis, (|x|
This completes the proof of Theorem 10.

Remark 11.
Let us observe that in the proof above we were able to choose an appropriate test function in the space C ∞ 0 .This turns out to be important, because it is possible to modify Definition 1 stating that Clearly S -convolvability implies convolvability in this sense.However, it is known ( [4], [10]), that there are pairs of tempered distributions for which the convolution in this new sense is not a tempered distribution.Thus, these notions of convolution are in general different.That we only use test functions in C ∞ 0 in the proof of 2) implies 1) in Theorem 10, means that w μ D L 1 is optimal for both definitions of convolution.Thus w μ D L 1 is the optimal space of distributions that are convolvable with |x| −μ .Remark 12.By definition of the S -convolution, it is clear that the convolution operator T → |x| −μ * T maps w μ D L 1 into S .Continuity results for the S -convolution (see [11]) can be used to refine this observation.

The optimal codomain for the product Laplace operator Δ x Δ y
We assume for simplicity that n 1 , n 2 ≥ 3 , so a fundamental solution E n1,n2 for the operator Δ x Δ y can be written as The function E n1,n2 is locally integrable in R n1+n2 and it defines a tempered distribution.As it is typically the case in a product domain, E n1,n2 has singularities outside the origin in R n1+n2 .This reflects the fact that the operator Δ x Δ y is not elliptic in R n1+n2 .As a consequence, the convolution operator T → E n1,n2 * T is not pseudo-local, that is to say, smoothness properties of the distribution T do not necessarily translate into smoothness properties of the distribution E n1,n2 * T .Theorem 13.Let T ∈ S .Then, the following statements are equivalent: Theorem 13 is an immediate consequence of the following result.Theorem 14.Let T ∈ S and let μ 1 < n 1 , μ 2 < n 2 .Then, the following statements are equivalent: Proof.We will assume 0 < μ 1 < n 1 , 0 < μ 2 < n 2 .The other cases can be proved using the same ideas with minor modifications.First we show that 1) implies 2).According to Definition 1, we need to show that the distribution T (|x| −μ1 |y| −μ2 * ϕ) belongs to D L 1 for each ϕ ∈ S .We can write (7) T (|x| Thus, it is enough to show that w μ1 1 w μ2 2 (|x| −μ1 |y| −μ2 * ϕ) ∈ B .Since we can write which is a straightforward calculation that we will omit.
Let us now assume that 2) holds.We will prove that ) by showing that the distribution T can be represented as in Proposition 8.In fact, with the notation introduced in the proof of Proposition 8, we can write It is clear that the distribution θ 1 θ 2 T belongs to E .To handle the second term in (8) we formally write We will show that the function is well defined and belongs to B(R n1+n2 ) for an appropriate test function ϕ.Indeed, we claim that there exist functions To prove (9) we consider a function ϕ 2 .Thus, we have proved (9).To prove (10), let us consider a function ϕ 2 ∈ C ∞ 0 (R n2 ) such that ϕ 2 (y) > 0 for 3 < |y| < 4 , it is zero otherwise, and R n 2 ϕ 2 dy = 1.
To deal with the third term in (8), we write where we fix this time ϕ = ϕ 1 ⊗ ϕ 2 , with ϕ 1 satisfying an estimate similar to (10) in R n1 and ϕ 2 satisfying an estimate similar to (9) in R n2 .With this choice of ϕ the function ).Finally, we write the last term in (8) as where the test function ϕ is the tensor product ϕ 1 ⊗ϕ 2 of two test functions ϕ 1 and ϕ 2 , both of them satisfying estimates similar to (9), ϕ 1 in R n1 and ϕ 2 in R n2 .Thus, ).This completes the proof of Theorem 14.
Remarks 11 and 12 apply to the product domain setting with obvious modifications.

Closing remarks
Although every non-zero operator with constant coefficients P (∂) has a tempered fundamental solution, it is an open problem to find an explicit formula.Besides, considering the many possible behaviors of this operator, it is quite unlikely that an optimal codomain can be identified.However, it is always possible to identify a codomain much larger than the space E of compactly supported distributions or even the space O C ([13]) of rapidly decreasing distributions.Before stating and proving such result, we state a known auxiliary lemma (see [3] for a proof).Proposition 16.Given U ∈ S there exists μ ∈ R such that U is S -convolvable with every distribution in the space w μ D L 1 .
Proof.If U is a tempered distribution, it is known (see [13]) that there exists a continuous function F of slow growth and a multi-index α such that U = ∂ α F in the sense of S .That F is a function of slow growth means that Remark 17.As observed in [11], O C = ∩ μ∈R w μ D L 1 with strict inclusion in each space w μ D L 1 .

2 xj
associated with the Euclidean space R n and the product Laplace operator Δ x Δ y =
etc. Partial derivatives will be denoted as∂ α or ∂ α ∂x α , where α is a multi-index (α 1 , ..., α n ) ∈ N n .We will use the abbreviations |α| = α 1 + ... + α n , x α = x α1The letter C will denote a positive constant, probably different at different occurrences.When necessary, we will append relevant parameters to the constant C , as in C n,p , etc.