Codomains for the Cauchy-Riemann and Laplace operators in R

A codomain for a nonzero constant-coefficient linear partial differential operator P (∂) with fundamental solution E is a space of distributions T for which it is possible to define the convolution E ∗ T and thus solving the equation P (∂)S = T . We identify codomains for the Cauchy-Riemann operator in R and Laplace operator in R . The convolution is understood in the sense of the S ′ -convolution.

Suppose we wish to solve where P (∂) is a nonzero constant-coefficient linear partial differential operator, T is a known distribution and S is an unknown distribution. Let E denote a fundamental solution of the operator P (∂). If T has compact support, then S := E * T solves (1), where the convolution is interpreted in the classical sense as defined by L. Schwartz [13]. The convolution operator T → E * T becomes a two-sided inverse for the differential operator P (∂) in the space E of distributions with compact support. That S := E * T solves (1) is good news. Unfortunately, the condition that T must have compact support is very restrictive. There is need for more general solvability conditions for (1). Given a specific nonzero constant-coefficient linear partial differential operator P (∂), we can weaken the condition that T has compact support. We achieve this by employing a certain type of convolution called S -convolution. A codomain for the operator P (∂) is a space of distributions T for which it is possible to define the convolution E * T and thus solving the equation P (∂)S = T . It is important to point out that finding a codomain depends on having an explicit formula for a fundamental solution E of the operator P (∂) and on how we define the convolution of E with distributions T that may not have compact support.
There are different types of convolutions in the literature, each designed to achieve a specific purpose. It is expected that any genuine convolution operation should be commutative, should commute with the operation of differentiation and satisfy, in some sense, the Fourier exchange formula: F (S * T ) = F (S)F (T ), where S and T are tempered distributions. We will be working with a certain type of convolution called S -convolution defined on S , the space of tempered distributions. If S and T are tempered distributions, we have no reason to expect that S * T exists in the classical sense or if it does exist, we cannot expect it to be a tempered distribution and as such we cannot expect the Fourier exchange formula to be valid. To ensure that the classical convolution is a tempered distribution, L. Schwartz [13] introduced the space O C of rapidly decreasing distributions. He proved that any tempered distribution is convolvable with any distribution in O C . Moreover, he proved that the convolution satisfies the Fourier exchange formula. The definition of the S -convolution was developed by Y. Hirata and H. Ogata [7] with the purpose of extending the validity of the Fourier exchange formula F (S * T ) = F (S)F (T ), which was originally proved by L. Schwartz ([13], p. 268) for pairs of distributions in the Cartesian product O C × S . Y. Hirata and H. Ogata showed that the S -convolution satisfies the three conditions given above that are expected of a genuine convolution. Later R. Shiraishi [15] introduced an equivalent definition which is the definition we are going to use.
The goal of our work is to identify codomains in the context of the Sconvolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 and Laplace operator in R 2 . Once a codomain for the operator P (∂) is obtained, it leads immediately to existence results for the equation P (∂)S = T when the distribution T belongs to the appropriate codomain. Moreover, a solution S is given explicitly by the formula S = E * T .
In [14], L. Schwartz found a class of weighted distributions whose elements are S -convolvable with the distribution pv 1 x . A weighted distribution space was also used by J. Horváth [9] and N. Ortner [12] to identify those tempered distributions that are S -convolvable with the n-dimensional hypersingular kernels of the form k x |x| |x| −λ , where λ ∈ C and k is a bounded function on the unit sphere. J. Alvarez and L.E.S. Moyo [3] used weighted integrable distribution spaces to identify the maximal codomains in the context of the S -convolution as defined by R. Shiraishi [15] for the Laplace operator Δ := n j=1 ∂ 2 ∂x 2 j associated with the Euclidean space R n , n = 2 and the product Laplace operator Δ Our work is organized in the following way. In Section 2, we collect definitions and results that will be used in the proving of our main results in Sections 3 and 4. In Section 3 and Section 4, we identify codomains in the context of the S -convolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 and Laplace operator in R 2 , respectively. Finally, in Section 5, we conjecture that it is impossible to identify maximal codomains in the context of the S -convolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 and Laplace operator in R 2 .
The notation that will be used is standard. The symbols indicate the usual spaces of functions or distributions defined on R n with values in C. The Euclidean norm on R n is denoted by |·|, whereas ||·|| p denotes the L p -norm. The conjugate exponent of p will be denoted by p . For a multi-index α = (α 1 , . . . , α n ) ∈ N n 0 and a point x = (x 1 , . . . , x n ) ∈ R n , we will use the following notations: appropriate test function φ, will be denoted by ∨ T . The Fourier transform of a function g will be denoted by F (g) or g . The letter C will indicate a positive constant which may be different at different instances. Subindices appended to C will indicate quantities on which the constant C depends. We will introduce other notation at the appropriate time.

Preliminary definitions and results
In this section, we collect definitions and results that will be used in the proving of our main results in Section 3 and Section 4. We begin with the following lemmas.
Next, we define the space D L p , 1 ≤ p ≤ ∞, as introduced by L. Schwartz in [13] and the weighted space w λ D L 1 . Before we define these spaces, we need the following definitions.
The space • B is a closed subspace of B . We equip D L p with the topology generated by the countable family of norms The space B c is the space B endowed with β -convergence. Following W. Kierat and U. Sztaba [10], pp. 75-76, a sequence {φ k } , φ k ∈ B is said to be β -convergent to zero in B if it satisfies the following two conditions: (1) For each α ∈ N n 0 , there exists a positive real number C α such that and in B c (see, for example, W. Kierat and U. Sztaba [10], pp. 75-76) and in D L p , for p ∈ [1, ∞). We state a characterization theorem, due to L. Schwartz [13], for the distribution space D L p . Theorem 4 ([4], p. 173). Given T ∈ D and p ∈ [1, ∞] fixed, the following statements are equivalent: Proof. See J. Barros-Neto [4], pp. 173-175.
As a consequence of Theorem 4, the space D L p is contained in the space S .
Our next theorem says that the space D L p is closed with respect to multiplication by functions in B . The theorem will play a pivotal role in the proofs of the main results in Sections 3 and 4. Proof. Let T ∈ D L p and g ∈ B . Since T belongs to D L p , by Theorem 4, we have where each f α belongs to L p , m ∈ N 0 and α ∈ N n 0 . Therefore We wish to show that gT belongs to D L p . In other words, by Theorem 4, we wish to show that gT = |β|≤q ∂ β h β , where each h β belongs to L p , q ∈ N 0 , and β ∈ N n 0 . Now, for all φ in D , we have By Leibniz's formula, we have Hence, in the sense of distributions, we have It remains to show that f α ∂ α g belongs to L p . We consider two cases.
Since g belongs to B , given any γ in N n 0 , there exists a positive constant This shows that f α ∂ α g belongs to L p .

Case 2. p = ∞.
Since g belongs to B , given any γ in N n 0 , there exists a positive constant C γ such that Also, since each f α belongs to L ∞ , there exists a positive constant C α such that for almost all x in R n . Now, by using (3) and (4), we have for almost all x in R n . This shows that f α ∂ α g belongs to L ∞ . Up to this point, we have shown that where f α ∂ α (g) belongs to L p . Hence, by Theorem 4, we conclude that g∂ α f α belongs to D L p . Finally, by (2), we conclude that gT belongs to D L p .
Next, we define the space w λ D L 1 .
Definition 6 ([2], p. 153). For n ≥ 1 , set w(x) = 1 + |x| 2 , x ∈ R n . Then for fixed λ ∈ R, we define with the topology on w λ D L 1 induced by the map By examining the definition of w λ D L 1 in (5), we have w −λ T ∈ D L 1 so that by invoking Theorem 4, we have w −λ T = |α|≤m ∂ α f α , where f α belongs to L 1 . This gives T = |α|≤m w λ ∂ α f α . Thus, we may also define w λ D L 1 as follows: We may also view w λ D L 1 as the topological dual of w −λ B c and w −λ • B . It is clear from Definition 6 that the space w λ D L 1 consists of tempered distributions. The space w λ D L 1 is referred to as the space of weighted integrable distributions. The space w λ D L 1 will be used in Section 3 to identify a codomain in the context of the S -convolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 . The same space w λ D L 1 , with a modified weight function, will be used in Section 4 to identify a codomain in the context of the S -convolution as defined by R. Shiraishi [15] for the Laplace operator in R 2 . For more information on the space w λ D L 1 , consult J. Alvarez and L.E.S. Moyo [3].
Finally, it is time to define S -convolution, a type of convolution that we will be working with. Here is the original definition of S -convolution, due to Y. Hirata and H. Ogata [7].

Definition 7 ([7], p. 148). Two distributions S, T
In [15], R. Shiraishi proved that Definition 7 is equivalent to the following definition which is the definition that we are going to use.

Definition 8 ([15], p. 26; [5], p. 193). Two distributions S, T ∈ S are
S -convolvable if (φ * ∨ S)T ∈ D L 1 , for all φ ∈ S . In this case, the Sconvolution S * T is defined by R. Shiraishi [15] proved that if S and T are S -convolvable, where S and T belong to S , then the map defines a tempered distribution, denoted by S * T . It turns out that S and T are S -convolvable if and only if T and S are S -convolvable. That is, For additional equivalent definitions of the S -convolution, see P. Dierolf and J. Voigt [5], p. 193 or R. Shiraishi [15], p. 26.

Codomain for the Cauchy-Riemann operator in R 2
In this section, we identify a codomain in the context of the Sconvolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 . We recall that E(x 1 , x 2 ) = Proof. Assume that T belongs to wD L 1 R 2 . We wish to prove that T is S -convolvable with h. That is, we wish to show that for every φ in S(R 2 ). But then for any φ in S( By the hypothesis, we know that w −1 T belongs to D L 1 (R 2 ).
So by Theorem 5, to prove (6), it suffices to prove that By Lemma 2, we have where and We will prove that w(φ * ∨ h) is bounded in R 2 by proving that both I 1 and I 2 are bounded in R 2 . We first prove that I 1 is bounded in R 2 .
so that Since φ belongs to S R 2 , we conclude that the function (1 + |x| 2 ) 1 2 φ is bounded in R 2 . Hence, from (7), we have By switching to polar coordinates, we obtain Therefore, from (8), we have I 1 ≤ C . This completes the proof that I 1 is bounded in R 2 . Next, we prove that I 2 is bounded in R 2 .
Since φ belongs to S(R 2 ), we conclude that the function (1 + |x| 2 ) Consequently, from (9), we have I 2 ≤ C . This shows that I 2 is bounded in R 2 . Up to this point, we have proved that I 1 and I 2 are bounded in We claim that each derivative of the C ∞ (R 2 )-function p is also bounded in R 2 . In fact, for each α in N 2 0 , by Leibniz's formula, we have We know that if φ belongs to S(R 2 ), then ∂ γ φ belongs to S(R 2 ), for all γ in N 2 0 . Therefore, ∂ α−α φ belongs to S(R 2 ). But then we have already shown above that . Now, let us put things together. By the hypothesis, we know that w −1 T belongs to D L 1 (R 2 ). Also, we have just shown that w(φ * ∨ h) belongs to B(R 2 ). Therefore, by Theorem 5, we conclude that

Codomain for the Laplace operator in R 2
In this section we identify a codomain in the context of the S -convolution as defined by R. Shiraishi [15] for the Laplace operator in R 2 . We recall that E(x) = 1 2π ln |x| is a fundamental solution of the Laplace operator in R 2 .
Proof. Assume that T belongs to 1 wa D L 1 (R 2 ). We wish to prove that T is S -convolvable with h. That is, we wish to show that for every φ in S(R 2 ). But then for any φ in S(R 2 ), we have Let us examine the right hand side of (11). By the hypothesis, w a T belongs to D L 1 (R 2 ). So by Theorem 5, to prove (10), it suffices to prove that where |φ(x − y) ln |y|| dy, and |φ(x − y) ln |y|| dy.
We prove that g is bounded in R 2 by proving that both I 1 and I 2 are bounded in R 2 . We begin by proving that I 1 is bounded in R 2 . Since φ belongs to S(R 2 ), we conclude that φ is bounded in R 2 . So, from |φ(x − y) ln |y|| dy, we have |ln |y|| dy.
By switching to polar coordinates, we obtain Therefore, This shows I 1 is bounded in R 2 . Next, we prove that I 2 is bounded in R 2 . But then Since φ belongs to S(R 2 ), we conclude that the function yφ is integrable in R 2 . Hence, |yφ(x − y)| dy is finite. Therefore, from (12), we have Up to this point, we have shown that both I 1 and I 2 are bounded in R 2 . Therefore for all x ∈ R 2 . Next, we claim that each derivative of g is also bounded in R 2 . For each α in N 2 0 , by Leibniz's formula, we have We show that w a ∂ α ( 1 wa ) and 1 wa (∂ α−α φ * ∨ h) are bounded in R 2 . We begin by showing that 1 wa (∂ α−α φ * ∨ h) is bounded in R 2 . From (13), we know that g(x) := 1 wa (φ * ∨ h)(x) is bounded in R 2 for any φ in S(R 2 ). But then, we know that if φ belongs to S(R 2 ) then ∂ γ φ also belongs to S(R 2 ) for any γ ∈ N 2 0 . Therefore ∂ α−α φ belongs to S(R 2 ). Hence, 1 wa (∂ α−α φ * ∨ h) is bounded in R 2 . Also, in the spirit of Lemma 1, it can be shown that w a ∂ α ( 1 wa ) is bounded in R 2 for any α in N 2 0 . Therefore ∂ α g is bounded in R 2 for each α in N 2 0 . This proves our claim. We therefore conclude that g(x) := 1 wa (φ * ∨ h)(x) belongs to B(R 2 ). Let us put things together. By the hypothesis, we know that w a T belongs to D L 1 (R 2 ). Also, we have just shown that g(x) := 1 wa (φ * ∨ h)(x) belongs to B(R 2 ). Therefore, by Theorem 5, we conclude that w a gT belongs to D L 1 (R 2 ). But w a (x)gT = (φ * ∨ h)T .

Conjecture
J. Alvarez and L.E.S. Moyo [3] used weighted integrable distribution spaces to identify the maximal codomains in the context of the Sconvolution as defined by R. Shiraishi [15] for the Laplace operator Δ := n j=1 ∂ 2 ∂x 2 j associated with the Euclidean space R n , n = 2 and the product Laplace operator Δ x Δ y := m j=1 ∂ 2 ∂x 2 j n k=1 ∂ 2 ∂y 2 k associated with the Cartesian product space R m × R n , m = 2, n = 2 . We conjecture that it is impossible to identify maximal codomains in the context of the Sconvolution as defined by R. Shiraishi [15] for the Cauchy-Riemann operator in R 2 and Laplace operator in R 2 .