On extremal domains and codomains for convolution of distributions and fractional calculus

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz potentials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.


Introduction
A recurring problem in applications of distribution theory is to find an optimal domain for a given convolution operator or semigroups of such [1,20,34,56]. Here, "optimal" means as large as admissible for convolution of distributions [45,50,52,55]. In other Communicated by Adrian Constantin. B R. Hilfer hilfer@icp.uni-stuttgart.de 1 Fakultät für Mathematik und Physik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Deutschland words, optimal spaces are defined as the convolution duals (originally called "c-duals" [24,58]) of a given set of distributions.
The main goal of this work is to develop a systematic method to construct optimal domains and codomains of distributions. Our method applies to general sets or semigroups of convolution operators.
Different from most investigations on this topic we study sets of function spaces endowed with a composition of spaces that is naturally induced by convolution. The emerging algebraic and order theoretic structures are investigated in detail, and they reveal interesting links to quantale theory [47], [17, p. 114-116]. Subsequently, the distribution spaces are endowed with topologies and bornologies in a uniform way, which ensures continuity and boundedness properties of convolutions between these spaces. The application of our general theory is exemplified for convolution operators used in fractional calculus and for the Hilbert transform.
To explain the crosslinks with quantale theory recall first the general order theoretic concept of Galois connections on power sets (also called "polarities") [3, p. 122], [17, p. 116-120]. According to [3,Thm. 19 & Cor.,p. 123] any symmetric binary relation R ⊆ X × X on a set X induces a "symmetric Galois connection" G R : P(X ) → P(X ), Y → G R (Y ) := {z ∈ X ; ∀y ∈ Y : y Rz} (1.1) and an associated closure operator H R := G R • G R . Taking the set X := D of distributions and the relation y Rz := "y and z are convolvable" furnishes convolution duals G R = (−) * D and perfections H R = (−) * * D [58, p. 20]. The elements of (P(D )) * * D = {U ⊆ D ; (U ) * * D = U }, the closure system associated to (−) * * D , are called convolution perfect spaces. These spaces are linear by the definition of convolvability and (P(D )) * * D constitutes a complete lattice [3]. The link between convolution perfect spaces and quantales emerges when studying extremality of inclusions U * V ⊆ W , where U , V , W ⊆ D are convolution perfect. This becomes clear from our extremality Theorem 1.
Theorem 1 Let U , V and W be convolution perfect spaces such that convolution of distributions is well-defined as a bilinear mapping * : U × V → W . Then, there exists a largest convolution perfect space U containing U and a smallest convolution perfect space W contained in W such that * : U × V → W and * : U × V → W are well-defined bilinear mappings in the same sense.
The second part of our extremality theorem is immediate from linearity properties of convolution and the fact that convolution perfect spaces constitute a closure system. Let * denote the partially defined composition The space (U ) * D serves as the largest joint domain for convolution operators with kernels from U and (U ) * * D is the smallest convolution perfect space containing the kernels. Note, that when all convolution operators K with kernel u ∈ U are represented by K (v) = u * v = B(u, v) for all v ∈ (U ) * D with some fixed bilinear convolution mapping B between convolution perfect spaces, then U * V is the smallest possible codomain for this mapping B.
The partially defined operation (1.2) furnishes an algebraic and order theoretic structure (P(D )) * * D , ⊆, * . This triple is identified as a "commutative quantale with partially defined operation", as explained in the following.
Up to formalities a commutative quantale [47,48] is a triple (Q, ≤, •) with (Q, ≤) a complete lattice and (Q, •) a commutative semigroup such that Now, consider for (Q, ≤) the set of convolution perfect distribution spaces, ordered by inclusion, enriched with the null vector space {0} and an artificially adjoined largest element "∞". Define • as the extension of * to Q by {0} • U = {0} for U ∈ Q, by U • V = ∞ for non-convolvable U , V ∈ Q \ {∞} and by U • ∞ = ∞ for U ∈ Q \ {{0}}. It will be proved that this defines a quantale (Q, ≤, •) and we will call it the quantale associated to (P(D )) * * D , ⊆, * . Commutative quantales (Q, ≤, •) possess residuals (using the nomenclature from [48, p. 922]) [47, p. 15]. The residual called "c by b" is defined as (1.5) The property (1.4) and the definition (1.5) result in the equivalence (1.6) see [47, p.15], [17,Def. 3]. Because the quantale associated to (P(D )) * * D , ⊆, * possesses residuals it follows that the residual "W by V ", given by the set is a well-defined convolution perfect space if V ⊆ W and is empty otherwise. Here V ⊆ W is equivalent to E * V ⊆ W , because E is identified as the neutral element for * . By construction, the space defined by (1.7) is the unique solution for U in Theorem 1. Wrapping up, Theorem 1 reflects the closure system property of convolution perfect spaces and the quantale property (1.4) of the composition * (more precisely, of its extension • from above).
Let us now outline the paper section by section, thereby introducing notations and highlighting some important results. Beginning in Sect. 2 we introduce and study the so called Φ-absolute value where B(D) := {B ⊆ D ; B bounded}. Note, that |−| Φ is absolutely homogeneous and subadditive, just like a seminorm. In Theorems 2 and 3, these functionals are used to characterize mapping properties of convolution of distributions via convolution on The first part of Sect. 3 studies the quantale structure induced by convolution on the set system I T , which consist of the cone ideals, i.e. non-empty additively closed downsets I ⊆ I + , that are moderated, i.e. Here (1.11) The regularization-solid closure of U is defined as the set V ⊆ D adjoint to the cone ideal I = |U | B with respect to the mapping |−| B , in other words (1.12) Proposition 6 establishes a bijection between regularization-solid spaces and moderated cone ideals I ⊆ I + lb := { f ∈ I + ; f locally bounded}, that is interpreted via order theoretic adjoints. Further, Theorem 5 states that any convolution inclusion of regularization-solid spaces is equivalent to the convolution inclusion of the corresponding moderated cone ideals. Section 4 is devoted to algebraic and order theoretic properties of convolution duals by studying the quantale (I T , ⊆, q * ). Proposition 9 yields a correspondence between the convolution dual (−) * D and the duality operation where / * T denotes residuals (1.5) formed in (I T , ⊆, q * ). The correspondence is induced by the bijection from Proposition 6, mentioned above. Defining the convolution perfect distribution spaces are found to map bijectively to the nondegenerate J ∈ I * * T (this means J ∈ I * * T with {0} = J ⊆ I + lb ). With the composition I * J := (I * J ) * * T for I , J ∈ I * * T this gives rise to a Girard-quantale (I * * T , ⊆, * ) with dualizing element I + lb [47,Sec. 6]. Identifying the space I + with ∞, it follows that (I * * T , ⊆, * ) is isomorphic to the quantale associated to (P(D )) * * D , ⊆, * , defined below Equation (1.4). These results are applied to maximal domains for composites of convolution operators in Theorem 6.
The construction (1.2) by itself does not yield optimal modules and algebras associated to semigroups of convolution operators, as has been observed for the Hilbert transform in [20, p. 301]. Therefore, Sect. 5 studies universal constructions of maximal modules and associated algebras. Let A ⊆ D be totally convolvable, that is, every p-tuple from A is convolvable. The space . . , a p ∈ A, p ∈ N : a 1 * · · · * a p * m exists} (1.15) and related constructions are studied in Sect. 5. This construction was introduced and applied to the mathematical modeling of fractional relaxation as linear translation invariant systems involving fractional derivatives on distributional domains in [31]. Theorem 7 establishes, that (A) * M D defines a convolution module over In Sect. 6 topological structures are introduced and investigated. Inspired by Köthes' "normale Topologie" on perfect sequence spaces [33, §30] convolution perfect spaces U will be endowed systematically with a weighted L 1 -type topology T * (U ) and a bornology B * (U ). For these definitions we obtain two functional analytic results. Theorem 8: Convolution between convolution perfect spaces is hypocontinuous whenever well-defined in the algebraic sense. Theorem 9: For given convolution perfect spaces V ⊆ W , the quantale theoretic residual "W by V " from Equation (1.7) is equal to the largest regularization-solid space contained in the space of convolutors Section 7 treats applications to causal fractional integrals and derivatives considered as translation invariant convolution operators. Their domains are significantly extended and the index laws are generalized to larger classes of distributions. Special cases of these results were recently discussed in [31] and in [22] with reference to desiderata for fractional integrals and derivatives. The negative fractional Laplacian (−Δ) α/2 , α > 0 is discussed in Sect. 8. We prove that the operator (−Δ) α/2 , α > 0 defines a continuous  linear endomorphism of D α by applying the maximal domain operator. This result was also stated in the recent "Handbook of Fractional Calculus with Applications" [34], but the characterization of the distributional domain, which is provided therein, is not accurate due to a subtle error in its functional analytic construction. Some remarks on notation: We always treat distributions on R d and suppress the attribute (R d ) with the exception of Sect. 7 and the space of causal distributions D + = D + (R). The reader's familiarity with notations for function spaces from [51], such as E , D, B etc. is assumed. The compact and bounded subsets of a locally convex space E are denoted as K(E) and B(E). The abbreviation K := K(R d ) is used and we write D K := {φ ∈ D ; supp φ ⊆ K } for K ⊆ R d . Translation, reflection and the support of a distribution u are denoted as T x u,ǔ and supp u respectively. The absolute convex closure of a subset B of a linear space is denoted by acx B. Dual pairings are denoted by −, − . The shorthand { f = g} stands for {x ∈ R d ; f (x) = g(x)}. Because many uncommon or new notations are used in this work, we included Table 1 for convenient reference.

Generalized absolute values and convolution of distributions
This Section summarizes properties of generalized absolute values and translation shells and then describes their application to convolution.

Basic properties of generalized absolute values and translation shells
Definition 1 Let u ∈ D and Φ ∈ B(D). Define the Φ-absolute value of u as Let f ∈ I + and K ⊆ R d . The K -translation shell of f is defined as The Φ-absolute value |u| Φ of a distribution u is a locally Lipschitz continuous function R d → R + . In particular |u| Φ ∈ I + lb and |u| Φ is a regular distribution. Note, that the mapping D u → |u| Φ ∈ I + is absolutely homogeneous and subadditive, just like a seminorm. Similar to the definition of the spaces D L p the definition of generalized absolute values depends on the Lie group structure of R d . The fact that makes generalized absolute values more convenient to use in connection with translation shell operators as compared to absolute values of regularizations. The latter could be called "weak generalized absolute values". Formation of K -translation shells equals supremal convolution with the indicator function 1 K . Supremal convolution arises in the context of convolution operators of measures on weighted spaces of continuous functions [30]. By virtue of [30,Prop. 3] T K I + ⊆ I + and T K I + lb ⊆ I + lb for all K ∈ K. Let us summarize some readily verified relations. Generalized absolute values and compact translation shells are connected by the relations Moreover, for all K ∈ K with non-empty interior one finds φ ∈ D K such that Generalized absolute values preserve supports up to compact sets, that is For 1 ≤ p ≤ +∞ and u ∈ D one readily derives from [51, Thm. XXV] that (2.6) Proposition 1 Let (θ n ) be an approximate unit and Φ ∈ B(D). Then and there exists Ψ ∈ B(D) such that Proof Let x ∈ R d and u ∈ D . Then the functions from the set (1 − θ n )T xΦ converge to zero uniformly within D for n → ∞ and

Proposition 2 The following three sets of functions
generate the same cone ideal of functions D → I + .
Proof It is known that every finite Φ ⊆ D is contained in acx(Ψ * Ψ ) for some other finite Ψ ⊆ D [14]. An examination of the proof in [14] reveals, that every Φ ∈ B(D) is contained in acx(Ψ * Θ) for some Ψ ∈ B(D) and some finite Θ ⊆ D (see also [12], where this is further generalized). Using this, one estimates for some C < ∞ and K ∈ K with a finite sum on the right-hand side. For the converse, note that C · T K θ ∈ B(D) and apply the relation (2.4a). (2.10) Proof Let u ∈ D . Let L ∈ K and ψ ∈ D L with ψ(x) dx = 1. Using (2.4c) and (2.4a) one obtains Define K and λ as in (2.4b). By virtue of (2.4b) and (2.4a) one obtains ThusΦ := T L Θ,Ψ := {ψ} andΘ := λ · T K Φ satisfy (2.10).

Characterization of convolution in terms of generalized absolute values
In the following we recall some properties of convolution * on the set I + of lower semicontinuous functions R d → R + . For definitions and properties of convolution of distributions we refer to [13,27,39,42,43,46,52,59,60]. In particular, for convolution of p-tuples see [27,43,46,52,60]. The most common definition for convolvability of p-tuples seems to be condition (a) in Theorem 2 below. The convolution f * g of f , g ∈ I + is defined by the formula via an upper integral [5]. By virtue of Fubini's Theorem for lower semicontinuous functions [19, p. 55 (a)], one obtains that (I + , * ) is a semigroup. Further, * is homogeneous with respect to R + -scalar multiplication, additive, isotone, reflection invariant and commutes argumentwise with translations. Conveniently, convolution on I + does not require a convolvability condition for well behavedness. Isotony implies the inequality where T K f := sup{T x f ; x ∈ K } as in Definition 1. The inequality (2.12) with compact K , L is fundamental for the quantale structures studied in Sect. 3.
The following are equivalent: Proof It suffices to give a proof for p = 2, the generalization to p ∈ N is straightforward. Denote u = u 1 and v = u 2 . Due to (2.6) condition (a) implies (2.13) Using Equations (2.5a) and (2.5b), and the fact that θ Δ has uniformly bounded derivatives for θ ∈ D, this is found to be equivalent to (2.14) According to [54,Theorem 51.7], Φ ∈ B(D(R 2d )) can be replaced by Ψ ⊗ Ψ with Ψ ∈ B(D) in (2.14). Using |u ⊗ v| Ψ ⊗Ψ = |u| Ψ ⊗ |v| Ψ it follows that is equivalent to (2.14). The integral in (2.15) is rewritten as and therefore (2.15) is equivalent to |u| Ψ * |v| Ψ ∈ L 1 loc for all Ψ ∈ B(D). Due to Equations (2.4a) and (2.12), this entails In the remaining part of this section we demonstrate a possible utilization of generalized absolute values by giving another proof for the associativity properties of convolution that were obtained in [27,52,60].
for all n ∈ N. Moreover, uniformly on compact sets.
Proof Assume p = 2. Let Ψ ∈ B(D) such that (2.17) holds and set Clearly, Θ ∈ B(D). Using Equation (2.17) one estimates Lebesgue's Theorem of dominated convergence and Proposition 1 yield (2.19b). The generalization to general p ∈ N is straightforward.
Corollary 1 Under the assumptions of Lemma 1 the sequence φ n u 1 * · · · * φ n u p converges to u 1 * · · · * u p for n → ∞ with respect to the strong topology of D .

Corollary 2 Convolvable p-tuples of non-zero distributions can be arbitrarily rewritten by introducing parentheses without changing the result.
Proof Use "(a) ⇔ (c)" from Theorem 2, f * I + lb ⊆ I + lb for all f ∈ I + lb with compact support and Corollary 2.

Regularization-solid spaces and moderated cone ideals
This section first describes the convolution quantale structure on the set system of moderated cone ideals I T . Then regularization-solid distribution spaces are introduced using generalized absolute values. Convolution inclusions between such spaces are then characterized by the convolution inclusions of the corresponding moderated cone ideals via Proposition 6 and Theorem 5.

The convolution quantale of moderated cone ideals
2) The set system of (moderated) cone ideals and the corresponding closure operator are denoted by I (I T ) and (−) ((−) T ). One defines the binary operation The residual operator of (I, ⊆, q * ) and (I T , ⊆, q * ) is denoted by / * I and / * T , respectively. Equations (3.6) below extend / * I and / * T to arbitrary subsets of I + . Ideals Remark 1 1. Cone ideals are precisely the non-empty additively closed down sets of I + . The set systems I and I T are closure systems over I + . The set system I T is a complete sublattice of I. 2. The closure operators (−) and (−) T can be described explicitly as Here "sup" denotes the pointwise supremum of sets of functions in I + . Note, that T K (sup F) = sup{T K f : f ∈ F} for F ⊆ I + and K ∈ K, because suprema over two independent variables commute. 3. Equation (3.2) is a typical mild assumption for weight function systems to obtain well behaved weighted (ultra-)distibution spaces and relates to the condition [wM] for weight function systems from [11].

Proposition 4
The triple (I, ⊆, q * ) is a commutative quantale. The triple (I T , ⊆, q * ) is a subquantale of (I, ⊆, q * ) in the sense of [47,Def. 3.1.3]. The residuals formed in (I, ⊆, q * ) can be described by the formula Residuals formed in (I T , ⊆, q * ) can be described by the formula The quantale (I T , ⊆, q * ) is unitary with unit I + c , that is Further, it holds Proof Using isotony and additivity of convolution one obtains the inclusion Equation (3.9) and the fact that * is associative and respects unions as an operation on P(I + ) [ .7), because q * is monotone and because I + J is equal to the supremum of I and J formed in I T .

Definition 3
The moderated cone ideal associated to U ⊆ D is defined as (3.11a) and the regularization-solid set of distributions associated to I ⊆ I + is The regularization-solid closure is the composite operator for every cone ideal I ⊆ I + that satisfies 1 K * I ⊆ I for all K ∈ K, which holds for all I ∈ I T in particular. This yields another proof for Equation (2.6). By an adjoint pair we will refer to a tuple ( f , g) of isotone mappings f : X → Y and g : Y → X between to ordered sets X and Y such that f (x) ≤ y if and only if x ≤ g(y) for all x ∈ X and y ∈ Y (this is called "Galois connection" in [10, 7.23], [17]). The range of f is the associated kernel system (also called "interior system") in Y and the range of g is the associated closure system in X . Kernel systems are the closure systems with respect to the reversed order. The associated kernel system in I is equal to I T ∩ P(I + lb ). Proof Clearly, the criteria (Gal1) and (Gal2) from [10, 7.26] are satisfied, implying adjointness. Equation (2.4a) implies |U | B ∈ I T for all U ⊆ D and it remains to prove I = |(I ) D | B for I ∈ I T with I ⊆ I + lb . Here, adjointness implies "⊆" and the reverse inclusion follows from (2.4b).

Corollary 4 The assignment U → (U ) ‚
D defines a closure operator on P(D ) and U → |U | B defines an order isomorphism from the set system of regularization-solid distribution spaces to the set system I T ∩ P(I + lb ), which constitutes a kernel system in I with kernel operator I → |(I ) D | B .
Proof These are standard order theoretic conclusions found in [10, 7.27].
Corollary 5 It holds |I | B = I for all I ∈ I T with I ⊆ I + lb . Lemma 2 Let p ∈ N. Convolution, regularization-solid closures and associated moderated cone ideals satisfy the compatibility relations |I 1 * · · · * I p | B = I 1 q * · · · q * I p for all I k ∈ I T ∩ P(I + lb ),

Convolution perfect moderated cone ideals as a Girard quantale
Definition 4 Let I ⊆ I + . Define the operators Proof Let f ∈ I , h ∈ (K ) * T and g ∈ J . It suffices to prove ( f * h ) * g ∈ I + lb because (K ) * T ∈ I T . Associativity and commutativity imply ( f * h ) * g = ( f * g) * h . By assumption f * g ∈ K and thus ( f * g) * h ∈ I + lb .

Corollary 6
For moderated cone ideals I , J , K such that I * J ⊆ K it follows Proof First note, that K / * T J = (K / * T J ) * * T holds due to Equation (4.4b). The proposition then follows by calculating, for all I ∈ I T , that Here it was used that (−) * * T is a closure operator, Equations (1.6) and (4.4a) and that (−) * T is a Galois connection.

Lemma 3 The convolution dual satisfies
The space (U ) * D is regularization-solid for all U ⊆ D and

Equation (4.7) implies that (U ) *
D is regularization-solid for any U ⊆ D . Using this fact and Equations (4.8) yields (4.11) Proof Let I , J , K be the moderated cone ideals corresponding to U , V , W . By Theorem 5, the left-hand side of (4.11) is equivalent to the left-hand side of (4.3) and the right-hand side of (4.3) follows from Proposition 7. It holds J = {0} and thus (J ) * T ⊆ I + lb . Theorem 5, Theorem 2 and Proposition 9 imply the right-hand side of (4.11).

Theorem 6 Let U , V ⊆ D be regularization-solid and non-zero with
(4.12a) Moreover, for u ∈ U , v ∈ V and w ∈ (U * V ) * D , one has the associative law  (4.13b) and the functions of μ-power growth P μ := P μ;0 . As sets, the associated distribution spaces and their convolution duals coincide with the spaces 14) that were studied in [1,20,44,57]. For μ ∈ (−∞, ∞] we define further When d = 1 the notation will be used. All the defined sets belong to I T . It can be calculated that P μ;k q * P ν;l = P μ+ν+d;k+l for all μ, ν > −d, k, l ∈ N 0 , (4.17a) (4.17c) whenever μ + ν < −d and P μ;k * P ν;l = ({+∞}) otherwise. The relations (4.17) hold for all μ, ν ∈ R with d = 1 when P μ;k is replaced by P μ;k + . As many of the relations (4.17) were proved previously in [4,8,41]     The convolution perfect algebra generated by A is defined as

Proposition 12 Let A ⊆ D be totally convolvable. Then:
Proof Let B := (A) * ‚a D and 0 = m ∈ D . Using the inclusion |B| B * · · · * |B| B ⊆ B and Theorem 2 the statement "(b 1 , . . . , b p , m) is convolvable for all b 1 , . . . , b p ∈ B, p ∈ N" is seen to be equivalent to "(b, m) is convolvable for all b ∈ B". This . The spaces S + and (S + ) * D arise naturally in causal fractional calculus on the real line, see Sect. 7. It is well known that S + is a convolution algebra of distributions [55]. The space (S + ) * D consists of distributions vanishing rapidly for t → −∞.
Example 8 Continuing Example 6 consider an arbitrary sum (supremum of ideals) P formed from a subset P of {P ν;l + ; ν ∈ R, l ∈ N 0 }. Equations (4.17) imply that of such sums precisely the sets P −μ , P + , Q + and R + , where μ ∈ (1, +∞) and k ∈ N 0 , are closed with respect to convolution. Similar statements are true for the sets P μ,k . Due to Proposition 11, the corresponding regularization-solid distribution spaces are regularization-solid convolution algebras.

Hypocontinuity with respect to a weighted L 1 -type topology
Topologies and bornologies are now introduced on every convolution perfect distribution space using generalized absolute values and convolution. Hypocontinuity [26,Ch. 4, §7] and boundedness [25, 1:2] of convolution are established in Theorem 8 for these topologies and bornologies. A residual formed in (I * * T , ⊆, * ) is characterized as the largest regularization-solid space contained in the corresponding space of convolutors in Theorem 9. The Hilbert transform on D L p is a convolutor not contained in this subspace, see Example 9.
Recall, that the space of convolutors O C (V , W ) from V to W consists of the distributions u with the property that the mapping D φ → u * φ extends to a continuous linear mapping V → W , see [2,Def. 12]. Here V and W are normal distribution spaces in the sense of [26, p. 319].

Definition 8
Let U be a convolution perfect distribution space. The locally convex topology T * (U ) is generated by the seminorms The bornology B * (U ) is defined as the set of subsets B ⊆ U such that

Remark 6
The topology T * (D ) coincides with the strong topology on D due to I + c = |(D ) * D | B and Proposition 2.

Remark 7
Any convolution perfect space U ⊆ D defines a normal space of distributions U = (U , T * (U )): Clearly, the space E K is continuously included in U for any compact K ⊆ R d , and thus, E is continuously included in U by [26, Thm. 1, p. 321]. Proposition 1 and Lebesgue's theorem of dominated convergence yield that E , and thus D, is dense in U . Remarks 5 and 6 yield that U is continuously included in D .

Remark 8
The seminorm in (6.1a) is equal to sup K ∈K K |u| Φ (x)|v| Φ (−x) dx and therefore Remark 6 implies that any T * (U )-neighborhood is D -closed. By virtue of [28,Thm. 3.2.4] it follows, that any T * (U )-Cauchy-filter that has a D -limit, has the same T * (U )-limit.
In the following, when a particular bornology [25, 1:1] is specified on the spaces V , W then O C (V , W ) is endowed with the topology of uniform convergence with respect to the bornology on V [28,Sec. 8.4]. The bornology on O C (V , W ) is defined as the sets of mappings L ⊆ O C (V , W ) such that L(B) is bounded in W for any bounded B ⊆ V . In the following we will occassionally write U instead of (U , T * (U ), B * (U )), when the meaning is clear.
Proof Using (2.4c), (2.12) and (2.4a) one obtains Lemma 5 Let V , W be convolution perfect distribution spaces with V ⊆ W and let u ∈ O C (V , W ), where V , W are endowed with the topologies from (6.1a). The continuous extension C u : V → W of the mapping D φ → u * φ ∈ W is given by is continuous as a mapping E → D by Remarks 5, 6 and 7. Because D is dense in E and convolution of distributions is hypocontinuous as a mapping Now, let (φ n ) be an approximate unit and v ∈ V . Proposition 1 and Lebesgue's theorem of dominated convergence imply that (φ n v) converges in V . Thus, C u (φ n v) = u * (φ n v) is a T * (W )-Cauchy-filter that converges in D by continuity, completeness and Remarks 5 and 6. Then [39,Thm. 7.1] implies that (u, v) is convolvable and that lim n→∞ C u (φ n v) = u * v within D . According to Remark 8,this implies C u  . Lemma 4 yieldsũ ∈ |{u}| B ,ṽ ∈ V and Ψ ∈ B(D) with Φ ⊆ Ψ and |ũ * ṽ| Ψ ≥ |u| Φ * |v| Φ . By Lemma 5 (ũ,ṽ) is convolvable andũ * ṽ ∈ W . Thus, Theorem 2 yields Finally, Equation (6.7) and Proposition 8 imply

Example 9
The Hilbert transform H : [29]. One calculates that

Distributional causal fractional calculus on the real line
The machinery developed so far is now applied to the causal fractional integration and differentiation operators. We determine their largest natural domains, generalize the index laws and determine spaces on which semigroups of these operators operate continuously and linearly. Following Schwartz [51] the fractional integrals I α + and derivatives D α + with α ∈ C are defined as convolution operators [21,Sec.9] for α > −m and m ∈ N 0 . The distributions {Y α ; α ∈ C} form a convolution group, more precisely 3) The operators I α + and D α + are continuous and linear on the space of causal distributions D + . Equation (7.3) entails the index law I α u for all u ∈ D + and α, β ∈ C. Enlarged domains to be obtained now are described using the spaces P μ;k + from Example 6 on page 18.

Largest distributional domains and index laws
Calculating the generalized absolute values of the distributions Y α one obtains Using the convolution dual operator (−) * D we specify the domain of I α + as Using Remark 2 the domain of I α + with α ∈ C \ −N 0 can also be written as  This includes the special case The proof is an application of Theorem 6. Equation (7.6a) follows using U := (P α−1 according to equations (7.4), (4.17a) and Theorem 5. The proofs for the remaining equations are similar.

Remark 10
The conditions on u in Theorem 10 have a similar form as the index laws for fractional powers of generators of semigroups [16,Thm. 5.32], [32,Prop. 5.2]. See [7,21,38] for more references on this topic. Using the theory developed in the present work has the advantage that one obtains simple and explicit formulas for the domains.

Remark 11
Interpreting I α + as an improper integral leads to different sufficient conditions for the index law, see Theorem 1.3 and Remark 1.4 in [37].

Largest distributional domains for endomorphic operation
In order to realize the operators I α + as convolution endomorphisms of distribution spaces we apply the operator (−) * M D . Using Equations (4.17) one obtains One observes, that Dom D α Applying the operator (−) * A D induces natural semigroups of operators D α + as described in the following: Let I denote the closure operator associated to the closure system Using Equations (4.17) we calculate that for all A ⊆ C. This means that the closure operator I on C generates those subsemigroups of {Y α ; α ∈ C}, that are maximal with respect to total convolvability on some joint domain space from the class of convolution perfect distribution spaces.

Theorem 11
Let p > 0. The convolution semigroup X operates continuously and linearly on the distribution space Y by convolution of distributions for The convolution group X operates bijectively and continuously on Y for

In all cases compact sets of indices α map to equicontinuous sets of operators.
Proof This is proved similar to Theorem 10 using (7.8), Theorems 7 and 8 .

Remark 12
The above results can be placed into the context of the desiderata for fractional calculus proposed in [23].

Remark 16
The Riesz potentials R α with α ∈ H \ (d + 2N 0 ) can be turned into a group with α ∈ C, if they are defined as operators on the dual space Φ of the Lizorkin space [49, (25.16)] where F denotes the Fourier transform on the Schwartz space. However, the elements of Φ can not be interpreted as distributions because Φ ∩ D = {0}. This follows from the Wiener-Paley Theorem and the Taylor formula for multivariate power series.
Remarks 15 and 16 combined with D α ⊆ S lead us to conjecture that there exist random variables w ∈ D α and a measure on D α such that for all g ∈ S the dual pairing (−Δ) −α/2 w, g is a centered Gaussian with variance g 2 L 2 (R d ) . If true, the conjecture would render the traditional construction of multidimensional fractional Brownian fields more direct.
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.