Convolution operators on weighted spaces of continuous functions and supremal convolution

The convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.


Introduction
Although necessary and sufficient conditions are known, that make fractional integration a bounded operator between weighted (Lebesgue) spaces, (see [25,28,[31][32][33] and references therein), applications, extensions and generalizations of weighted convolution inequalities continue to attract widespread interest in potential analysis and its applications to partial differential equations [3,6,[13][14][15][16][17]21,27,35,36]. Deviating from the traditional focus on Banach spaces and fractional Riesz integrals, the present article studies weighted convolution algebras of Radon measures and fractional Weyl integrals operating as equicontinuous families of linear endomorphisms on weighted locally convex spaces of continuous functions.
Define the (multiplicative) supremal convolution of two arbitrary weight functions w, v : G → R +∞ as (w · v)(z) := sup{w(x)v(y) : z, y ∈ G, x y = z} (1.3) with the convention 0 · ∞ = ∞ · 0 = 0, and let denote their infimal convolution. Expressing the inequality (1.2) equivalently in terms of infimal convolution w ≤ w · w (1.5) hints at the fundamental role played by supremal and/or infimal convolution for weighted convolution inequalities. In the literature, the additive variant of infimal convolution has been studied as "inf-convolution" or "epi-addition" in the context of convex analysis [24,30]. Given the equivalence (1.1) ⇐⇒ (1.2) ⇐⇒ (1.5), it is natural to investigate its extension to triples of weights (w, v; u) [M], this is shown to be equivalent to local boundedness of w · v. In Theorem 5, the equivalence for some w ∈ W . The brackets · denote upper semicontinuous envelope formation. Assuming χ K ∈ W , V , U for all compact K ⊆ G, each of (1.7a) and (1.7b) is equivalent to being a well-defined linear mapping such that bounded sets are mapped to equicontinuous sets of continuous linear operators Here C v (W ) denotes the set of continuous functions on G with w| f | vanishing at infinity for all w ∈ W and T W is the locally convex topology generated by the weighted supremum norms It is natural to regard (1.7) as a mere consequence of (1.6). But giving a proof of (1.7) requires to drop the positivity assumption on the weights in (1.6). This forces us to deal with convolutes (μ˚f )(x) that can diverge for some or even all x ∈ G, and in turn, to deal with integrals of C ∞ -valued measurable functions that are allowed to diverge. A whole section for notations and results concerning this difficulty has been incorporated to ensure transparency. A byproduct of these preparations is Theorem 2. It guarantees universal measurability of the convolute μ˚f whenever μ is a moderated measure and f a universally measurable C ∞ -valued function.
The paper is organized as follows. Basic notations and conventions are summarized in Sect. 2. This includes a discussion of extended arithmetics. Section 3 treats supremal images as preparation for Sect. 4 where basic properties of supremal convolution on locally compact groups are summarized. Section 5 discusses deconvolution. Section 6 provides definitions and results concerning integration of extended K ∞ -valued functions that are needed for Theorems 5 and 6. Theorem 5 is stated and proved in Sect. 7, where results on images and tensor products of weighted balls of measures, Theorems 3 and 4, are also included. Theorem 6 is stated and proved in Sect. 8. Section 9 applies the general results to fractional Weyl integrals and derivatives as linear endomorphisms on weighted function spaces.

Notations and conventions
Let K be R or C throughout. Define K ∞ := K ∪ {∞}, R + := [0, ∞ + ) and R +∞ := R + ∪ {∞ + }. The symbol ∞ stands for "divergent," "undefined" or "infinite" in a generic unsigned sense. The symbol ∞ + stands for "positive divergent." The extension K ∞ ⊇ K is considered as an Alexandrov compactification, where ∞ is adjoined as the point at infinity. The extension R +∞ ⊇ R + is understood as an ordering theoretic extension, where ∞ + is adjoined as the greatest element.
For any set S, the set of functions S → K is denoted by F (S). To denote functions with values in K ∞ , R + or R +∞ instead of K, we replace F by F ∞ , F + or F +∞ .
Let S be a topological space. The topological space S is called locally compact if every of its points has a compact neighborhood. All locally compact spaces are assumed to be Hausdorff.
is closed for every a ∈ R +∞ . The sets of R +∞ -valued lower, respectively, upper semicontinuous functions are denoted by L +∞ (S), respectively, U +∞ (S). The upper semicontinuous envelope of w ∈ F +∞ (S) is denoted by (2. 2) The set U +∞ (S) is known to be a closure system [7,Definition 2.33] in F +∞ (S), i.e., U +∞ (S) is closed with respect to pointwise formation of suprema of arbitrary subsets.
Restrictions to functions that are locally bounded, uniformly bounded, vanishing at infinity or compactly supported, are denoted using a subscript F lb , F b , F v , respectively, F c . A function f : S → K ∞ is said to "vanish at infinity" if and only if for every ε > 0 there is a compact

Supremal image functions and upper semicontinuity
Let Φ : S → T be a mapping between sets and w : T → R +∞ an arbitrary function. The pullback or inverse image function Φ −1 w of w under Φ can always be defined as Φ −1 w := w • Φ : S → R +∞ using the composition of mappings.
Unless Φ is bijective and thus invertible, there is no general or natural notion for a "pushforward" or an "image function" of a function w ∈ F +∞ (S) under Φ. However, a useful construction resembling the "pushforward" of a function w : S → R +∞ under an arbitrary mapping Φ : S → T arises from considering R +∞ as an ordered set (R +∞ , ≤), where ≤ is the canonical ordering.
Secondly, the mapping Φ : F +∞ (S) → F +∞ (T ) associated with any mapping Φ : S → T by Definition 1 can be seen as the (lower) adjoint to the inverse image operator Φ −1 : F +∞ (T ) → F +∞ (S) in the sense of ordering theory [7,Definition 7.23], where F +∞ (S) and F +∞ (T ) are endowed with the canonical pointwise ordering. This fact is formulated as is a Galois connection (or adjoint pair) in the sense that the two statements The following two results on upper semicontinuity and supremal image functions will be useful. For the remainder of the section, Φ : S → T is a continuous mapping between topological spaces. The brackets · will denote upper semicontinuous envelopes.

Corollary 1 Proposition 1 holds also when F +∞ (T ) is replaced by U +∞ (T ) and Φ is replaced by the assignment w → Φw .
Proof Isotony of Φ and · imply isotony of w → Φw . The equivalence of Φw ≤ v and Φw ≤ v for w ∈ F +∞ (S) and v ∈ U +∞ (T ) concludes the proof.

Lemma 1 The following relations hold:
Proof The relations Φw ≤ Φ w and Φw ≤ Φ w follow from the definitions.

Supremal convolution on locally compact groups
Let G be a locally compact group and let Γ : G × G → G be its continuous multiplication written as x y = Γ (x, y) for x, y ∈ G.
Definition 2 Let w, v ∈ F +∞ (G). The (multiplicative) supremal convolution of w and v, denoted as w · v, is defined by Supremal convolutes w · v are called exact, if the supremum in (4.1) is a maximum for all z ∈ G.
Supremal convolution is an associative binary operation · on F +∞ (G) [24,30] that is Thus, supremal convolution shares many properties with convolution. Supremal convolution can be decomposed as into the supremal tensor product ⊗ and the supremal image Γ under the group multiplication Γ . The supremal tensor product is defined as and S, T are sets. It coincides with the usual tensor product ⊗ of functions whenever the functions are finite valued. The adjective "supremal" refers to the fact that supremal operations (convolution, multiplication or tensor product) are characterized as the unique extensions of the finite-valued case that preserve suprema of arbitrary subsets in each argument. This means that Proof The proposition is an analogue of [24, 4.c. Proposition].
Proof This follows from Proposition 2 and the fact that tensor products of R + -valued upper semicontinuous functions are upper semicontinuous.

Corollary 2
The following inclusions hold: Proof This is merely checking the assumption of Proposition 4.

Remark 1 By means of the exponential transformation
where ∞ − := −∞ + and −∞ − := ∞ + , Proposition 4 is seen to be equivalent to Proposition 1.27 from [30]. Proposition 1.27 from [30] was proved for R d , but the proof in [30] extends to locally compact groups without complications. The special case (4.5a) can also be found in [24,Section 4].
Lemma 2 Let S and T be topological spaces. Upper semicontinuous envelopes · , and tensor products ⊗ are compatible in the following sense: where F + lb denotes locally bounded functions.
Proof Upper semicontinuous envelopes of locally bounded functions are finite valued, and tensor products of finite-valued upper semicontinuous functions are upper semicontinuous.
and v · u are locally bounded. Then, the following associative law holds: Proof Use the definition of · , then the upper semicontinuity of u and then Lemma 2 to get: This, together with Lemma 1 and the definition of · in reverse, gives Now, use the associative law for · and do similar steps in reverse.

Remark 2 Proposition 5 implies that the assignment
defines an associative internal binary operation on the closed subsets of G. Here A denotes the topological closure of A in G.

Supremal deconvolution
Supremal deconvolution is an operation similar to taking the inverse of supremal convolution (see Proposition 6). It is related to infimal convolution · , the dual operation to · , defined as w

Remark 3
Note that \ preserves suprema in its left argument and infima in its right argument.
Left deconvolution can be written as w (a) The following equivalences hold: If u is upper semicontinuous, then w \ u and u / v are upper semicontinuous as well.
Proof Part (a) and (b) of Proposition 6 are analogues of Lemma I-1 and Proposition I-4 in [22] carefully extended to the non-commutative case. Part (c) follows from Proposition 3 by duality.

Integration of K ∞ -valued measurable functions
This section summarizes notations and results for measures and integration on locally compact spaces [4,5]. A slight modification of essential integration is introduced [Eq. (6.2) and (6.4)] to allow integration of arbitrary K ∞ -valued measurable functions with respect to any K-valued Radon measure consistent with the extended arithmetic from Sect. 2. This allows to state and prove that the class of universal measurable K ∞ -valued functions on G is preserved under left convolution with moderated Radon measures on G (Theorem 2). The result is derived from Theorem 1 that concerns integration of universally measurable functions with respect to factors of a product measure.
In the remaining text of the article, the "Radon" in Radon measure is dropped, because solely this kind of measure is used.
Let S be a locally compact space. The set of K-valued continuous functions with compact support is denoted by K(S). The set of K-valued measures on S is denoted by M(S), the set of positive measures by M + (S). The symbols μ, μ and |μ| denote the real part, imaginary part and absolute value of μ ∈ M(S), respectively. The positive/negative part of a realvalued measure μ is denoted by μ ± (See [4, Ch. III, §1, Nos. 3, 5, 6]). The upper integral, respectively, the essential upper integral of f ∈ F +∞ (S) with respect to μ ∈ M + (S) is denoted by The remaining statements can be proved in a similar vein.

Proposition 8 The extended essential integral obeys
Similarly, The standard estimate holds for all f ∈ M ∞ (μ), μ ∈ M(S). When ∞ and ∞ + are identified with each other, one has for all f ∈ M +∞ (S), μ ∈ M + (S).
Proof One uses that μ : I ∞ (μ) → K respects extended addition and K-scalar multiplication, the equivalence in Eq. (6.3) and distinguishes the different cases that appear according to the definition in Eq. (6.4).
Recall the definition of products of measures: For any two measure μ ∈ M(S), ν ∈ M(T ) and f ∈ K(S × T ), let Then μ( f ) and ν( f ) define continuous functions of compact support. The product measure μ ⊗ ν of μ and ν is defined for f ∈ K(S × T ) by [4, Ch. III, §4, No. 1]: , t ∈ T , s ∈ S and this guarantees that μ( f ) and ν( f ) are pointwise well defined in the sense of the conventions for extended essential integration from (6.2) and (6.4). Using the decomposition formula (6.10) yields

Theorem 2 Left convolution (6.14) is well defined as a binary operation
The transposition law holds for f ∈ M ∞ (G) and μ, ν ∈ M σ (G) whenever μ and ν are convolvable and one of the following two expressions is finite: Proof Both statements follow from Theorem 1 and the fact that For a sequence of positive numbers x n ∈ R +∞ , n ∈ N, their series is declared to be As a consequence μ is moderated.
Proof The assumptions on h guarantee that the lower semicontinuous envelope h of h is strictly positive everywhere. The sets H n := { h > 1/n}, n ∈ N are μ-integrable, because they are open and Their union is S. Thus, μ is moderated. By [2, Lemma 1], its support is the union of a sequence of compact subsets of S. Therefore, there exist functions f n ∈ K + (S), n ∈ N such that f · μ = μ, where f := ∞ n=1 f n . Define the measures ν n := f n · μ and the numbers α n := |ν n | • (h) = |μ| • ( f n h) ∈ R + . Then, let μ n := (1/α n ) · ν n if α n = 0 and μ n = 0 otherwise. The numbers α n and the measures μ n fulfill the statements in the theorem. Using which concludes the proof.

Definition 5 Let w ∈ F +∞ (S). The w-weighted ball of K ∞ -valued universally measurable functions is defined as
where the w-supremum norm is The w-weighted ball of w-continuous functions is defined as denote the strict w-weighted ball of functions supported on a single point.
consists of K ∞ -valued universally measurable functions that are K-valued on the set {w > 0}, such that its restrictions to the sets {w ≥ λ}, λ > 0 are continuous and w| f | is vanishing at infinity on the subspace {w > 0}.

Lemma 4 The following equations hold:
The results follow from the definitions by straightforward calculation.
Theorems 3, 4 are now stated and proved in preparation for the main results stated in Theorem 5.

T ). The images Φμ, μ ∈ w[M] exist if and only if
Φw is locally bounded. The following are equivalent: Proof Characterization of existence: If Φw is not locally bounded, then, by local compactness of S, there exists a sequence (s n ) n∈N ⊆ S and a compact subset C ⊆ T such that 2 n ≤ w(s n ) and Φ(s n ) ∈ C for all n ∈ N. Define the positive measure μ by Because w is locally bounded, {s n : n ∈ N} is necessarily a discrete subset of S. This guarantees that μ is a well-defined positive measure, because it renders the sum finite. Then for g ∈ K + (T ). If g = 1 on C, then the expression on the right in (7.6) must be infinite. Thus, the image of μ under Φ does not exist. The reverse implication follows from the estimate (7.8).
The following are equivalent: The converse follows from Remark 5(d) and Eq. (7.5b).

Lemma 5 Let
. Then the function f := ∞ k=1 α k f k , that is defined pointwise by this series (in the sense of Sect. Proof First note that f ∈ M ∞ (S) because of Remark 4 and because M ∞ (S) is closed under pointwise formation of arbitrary series of sequences. One has and thus, f ∈ M ∞ [w]. For each fixed n ∈ N, choose a sequence g n,m ∈ K(S)∩M ∞ [w] such that g n,m → f n with respect to · w when m → ∞. Clearly, n k=1 α k g k,m ∈ K(S)∩M ∞ [w] for each n ∈ N and the limit for m → ∞ coincides with n k=1 α k f k on each point s ∈ S where w(s) > 0. (This follows from the fact that α n , f n (s) ∈ K for all n ∈ N and s ∈ S with w(s) > 0.) Therefore, n k=1 α k f k ∈ C ∼ v [w]. As (7.9) implies the convergence of with respect to · w for n → ∞ and C ∼ v [w] is, by definition, a closed subset of M ∞ [w] with respect to · w one can finally conclude f ∈ C ∼ v [w].
are convolvable if and only if w · v is locally bounded. The following statements are equivalent: In addition, the following statements are equivalent to (a)-(c) if u is upper semicontinuous: holds. Proof Characterization of existence: Assume that w · v is not locally bounded. One finds a sequence (z n ) n∈N in G and a compact subset C of G such that (w · v)(z n ) ≥ 5 n and such that C is a neighborhood of the set {z n : n ∈ N}. One finds sequences (x n ) n∈N and (y n ) n∈N in G such that w(x n )v(y n ) ≥ 4 n and x n y n = z n for all n ∈ N. Passing to a subsequence and using symmetry, if necessary, one may assume that w(x n ) ≥ 2 n . Then, the set {x n : n ∈ N} is discrete because w is locally bounded, and therefore, one may assume that the sets x −1 n C, n ∈ N are disjoint (again, by passing to a subsequence if necessary). By construction, x −1 n C is a neighborhood of y n for all n ∈ N. Thus, the set {y n : n ∈ N} is discrete as well. On the other hand, the estimate holds for any g ∈ K + (G). Choosing g = 1 on C, one obtains an infinite expression, and thus, the image of μ ⊗ ν under Γ does not exist. The reverse implication follows from Theorems 3 The first statement follows from Remark 5(c) and Theorem 2. For the second statement, let y ∈ G and set ν : "(c) ⇒ (a)": This follows from (7.5d), Remark 6(c) and Proposition 6. "(a) ⇒ (c')": The estimate (7.11) implies (7.12) where M c (G) denotes the set of all measures on G with compact support. Equation (7.11) implies that the inclusion in Eq. (7.12) holds also after the right factor on the left-hand side is replaced by its closure in M ∞ (G) with respect to · u and likewise for the term on the right-hand side: By Lemma 3, every element μ ∈ q w[M] can be represented by a vaguely convergent series for n ∈ N and ∞ n=1 α n = 1. It will be shown in the following paragraph that Then Eq. (7.13) implies that μ n * f ∈ C ∼ v [v] for all n ∈ N. Thus, the inclusion in (c') follows from Lemma 5.
Finally, to prove (7.14), let . Using (6.6) and the premise w · v ≤ u reformulated as one obtains, for n ∈ N, that which proves (7.14) because ∞ k=n+1 α k → 0 for n → ∞. The last inequality uses Lemma 3, Eq. (7.2). "(c') ⇒ (a)": Let f ∈ K + (G) with f ≤ 1/u and x ∈ G. By assumption, This statement is equivalent to q w(x)L x −1 v ≤ 1/ f and, in turn, to w(x)L x v ≤ 1/ f . Taking the infimum over f , one obtains w(x)L x v ≤ u by the upper semicontinuity of u. Taking the supremum over x ∈ G, one obtains w · v ≤ u.
With the results of Sect. 7, the main objective of this work can now be achieved. First some notations are introduced, and some basic facts on the weighted spaces of measures or continuous functions are summarized. Throughout the section, S is a locally compact space and G is a locally compact group. The elements of U + (S) will be called weights on S.

Definition 6
Let W be a set of weights on S. The linear space of W -vanishing-at-infinity continuous functions [26,Section 22] is The space of W -vanishing-at-infinity W -continuous functions is introduced as Introducing the set C ∼ v (W ) allows a more elegant formulation of Theorem 6. The relation is not even linear, but the natural Hausdorff quotients of (C v (W ), T W ) and (C ∼ v (W ), T ∼ W ) are isomorphic as locally convex spaces.

Definition 7
Those sets W of weights on S such that for all u ∈ U + (S), w, v ∈ W , λ ∈ R + are called cone ideals of weights on S (or cone ideals on S for short). The cone ideal generated by a set V of weights on S is denoted by V and given by For any set W of weights on S, the cone ideal W is the largest set V of weights on S such that T V = T W . It follows that the assignment W → (C v (W ), T W ) defines a bijection between cone ideals W and the locally convex spaces of W -vanishing continuous functions (C v (W ), T W ). Working with cone ideals instead of general Nachbin families allows a convenient formulation of Theorem 6. (8.3a) The bornology K W is defined as It was shown in [29, p. 152] (see also [34,Theorem 3.1]) that W (M) is the topological dual of (C v (W ), T W ) where the pairing ·, · W : C v (W ) × W (M) → K is given by the integration is the polar set of w [M]. Therefore, K W is the equicontinuous compactology on W (M) associated with T W [12,18] (see also [34,Section 4] (a) The following inclusion holds: The following two conditions hold: is well defined. Here L μ f = μ * f denotes the left convolution of f with μ as given in Eq. (6.14). Images of bounded subsets of ( q W (M), K W ) under L are equicontinuous sets of continuous mappings The statement in part (c) means that and that for all ε > 0, w ∈ q W , v ∈ V there exist δ > 0, u ∈ U such that for all f , g ∈ C ∼ v (U ) the following implication holds The condition (8.8e) for this replacement is fulfilled if (8.8c) holds. Statements (8.8c) and (8.8f) in turn are equivalent to the more concise formula This statement, in turn, implies (8.8c). Thus, the statement in part (c) is equivalent the statement (8.8g). By Theorem 5 the statements (8.8a), (8.8b) and (8.8g) are equivalent which concludes the proof.
The sets C ∼ v (U ) and C ∼ v (V ) in statement (c) of Theorem 6 can be replaced by the sets C v (U ) and C v (V ), if the assumption χ K ∈ V for all compact K ⊆ G is added. Then the mapping L with domain q W (M) is a linear mapping with continuous linear operators between locally convex spaces as co-domain.
The implication "(b) ⇒ (a)" in Theorem 6 fails if the condition (8.6b) is dropped. The reason is that the assignment W → K W , with W a cone ideal, is injective, but the assignment This shows that (8.6a) by itself does not imply (8.5) contradicts (8.5), but convolution * is known to be a well-defined internal operation on M f (G) in agreement with (8.6a).

Application to fractional Weyl integrals
Following [23], the space E := f ∈ C ∞ (R) ∀ m, n ∈ N : t n f (m) (t) → 0 for t → ∞ + denotes complex valued smooth functions that vanish rapidly at ∞ + . Such functions are called "good" in [23]. The generalized fractional Weyl integral of order α ∈ C for a function f ∈ E is defined as [11,20,23] ( where m ∈ N 0 is such that α + m ∈ H with H := {z ∈ C : z > 0}. It is known that (9.1) is well defined and the I α are linear endomorphisms, even automorphisms, on E. Further, the index law I α • I β = I α+β for all α, β ∈ C, (9.2) holds. An alternative proof for these facts based on Theorem 6 is given below. Using our theorem, continuity of the fractional integrals with respect to a suitable weighted topology is obtained in addition. Furthermore, sets of linear combinations of generalized fractional integrals are seen to be equicontinuous whenever coefficients and orders are bounded.
Let P denote the set of upper semicontinuous functions u : R → R + with the property that u(t) ≤ λ(1 + |t| p ), t ∈ R for some λ ∈ R + . Let U + + (R) denote the set of upper semicontinuous functions u : R → R + such that supp u ⊆ [t, +∞) for some t ∈ R. Then define P + := P ∩ U + + (R). The sets U + + (R), P and P + are cone ideals. Define the weights w +, p , p ≥ 0 as It is calculated that w +, p · w +,q = w +, p+q (9.4) for all p, q ≥ 0. The elements of P + are characterized as being less or equal than a translate of w +, p for some p ≥ 0 large enough. Together with Proposition 4 and (9.4), this implies that the inclusion P + · P + = P + · P + ⊆ P + (9.5) holds and that P + is invariant under translations. Observe that C v (P + ) is a continuous variant for the space E.
For α ∈ H, let μ α denote the measure on R with Lebesgue density λ α (t) := t α−1 /Γ (α) for t > 0, 0 f o rt ≤ 0, (9.6) and let μ 0 = δ 0 be the Dirac measure at the origin. Note that μ α˚μβ = μ α+β for all α, β ∈ H ∪ {0}, (9.7) corresponding to the index law for the operators I α , α ∈ H∪{0}. Using (1+t) p ≥ max{1, t p }, the following uniform estimate can be proved d|μ α |(t) w +, p (t + 1) ≤ C β,φ,ε (9.8) for all α ∈ H with α ≤ β and |arg α| ≤ φ where β < p, φ < π/2 and ε > 0 are fixed and C β,φ,ε < ∞. This translates to the fact that the set {q μ α : α ∈ H ∩ {0}, α ≤ β, |arg α| ≤ φ} is contained in a scalar multiple of a translate of the weighted ball q w +, p [M] for p > β. Using Theorem 6 and the above remarks, one arrives at the following conclusion. For C < ∞, φ < π/2 and d ∈ N, the set {λ 1 I α 1 + · · · + λ d I α d : α i ∈ H ∪ {0}, λ i ∈ C, |λ i |, |α i | ≤ C, |arg α i | ≤ φ} (9.9) is an equicontinuous set of continuous linear endomorphisms of C v (P + ). For d = 1, this is referred to as I α , α ∈ H ∪ {0} being an equicontinuously parameterized family of continuous linear convolution endomorphisms of C v (P + ). The result is readily extended to general orders from C using the following construction. For any set of weights W on R, the space of smooth functions with W -vanishing-at-infinity derivatives is defined as (9.10) and endowed with the weighted topology T ∞ W that is generated by the seminorms f → max{ f (k) w : m ≥ k ∈ N 0 } with w ∈ W , m ∈ N 0 . Clearly, classical derivatives define continuous linear endomorphisms of C ∞ v (W ). The linear space C ∞ v (P + ) coincides with the space E. Note that I α , α ∈ C is defined as the composition of a classical derivative of order m, a multiplication with the constant (−1) m and a convolution with the measure μ α+m . The index law for α, β ∈ C is obtained from the case α, β ∈ H ∪ {0} by interchanging derivatives and integration. This is permitted due to the properties of functions in E. Therefore, the results above carry over to the case α ∈ C. For C < ∞ and d ∈ N, the set {λ 1 I α 1 + · · · + λ d I α d : α i , λ i ∈ C, |λ i |, |α i | ≤ C} (9.11) is an equicontinuous set of continuous linear endomorphisms of C ∞ v (P + ). Using the index law for β = −α ∈ C and the fact that I 0 is the identity operator, it follows that the operators I α , α ∈ C are also automorphisms.
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