A list of six
desiderata was recently proposed in [
1] for calling families of operators
with family index
from some index set
fractional derivatives (
) and fractional integrals (
) of order
. Distributional domains for
seem to require a minor modification of these
desiderata.
Multiplication of distributions is ill-defined so that for distributions desideratum (f) (Leibniz rule) requires generalization. A slightly modified list of desiderata might read as follows:
- (a)
Integrals and derivatives of fractional order should be linear operators on linear spaces.
- (b)
On some subset
,
,
the index law (semigroup property)
holds true for
and
, where
denotes the domain of
.
- (c)
Restricted to a suitable subset
of the domain of
the fractional derivatives
of order
operate as left inverses
for all
with
, where ∘ denotes composition of operators, and
is the identity on
.
- (d)
Each of the two limits
should exist in some sense on some set
,
,
. Moreover, the limiting maps
and
should be linear.
- (e)
is the identity on , i.e., in Equation (3b).
- (f)
Endowed with a suitable multiplication
the limiting map
obeys the Leibniz rule
for all
with
,
. If
consist of numerical functions, then ⊙ is pointwise multiplication and
.
Given these modified
desiderata, the objective in this short note is to introduce fractional calculi for distributions. Let us stress that the distributional domains
,
given in Theorem 1 below are maximal in a precise mathematical sense. One cannot enlarge them without violating either the
desiderata or the interpretation of fractional derivatives and integrals as convolution operators. Recall that numerous other mathematical interpretations exist [
2], that may have different maximal domains. In this paper fractional operators are interpreted as convolutions with power law kernels (cf. [
2], Equation (28)). A comprehensive analysis of convolutions with power law kernels on weighted spaces of continuous functions was recently given in [
3].
Define the spaces of continuously differentiable functions, test functions, and smooth functions with bounded derivates
in the usual way [
4]. The spaces
are endowed with the norm
. The topology on
is induced by the seminorms
with
and
, where
is the space of continuous functions vanishing at infinity.
The space of distributions is the topological dual of . The dual space is the space of integrable distributions. The pairing is denoted , the pairing as .
Definition 1. Two distributions are called convolvable iff for all , where . Their convolution is defined by requiring thatholds for all . Let denote the space of causal distributions defined as elements whose support is bounded on the left.
Definition 2. Fractional integrals and derivatives are defined for all and all distributions as convolution operatorswith kernels The operators
and
are linear and continuous on
. The kernels
form a convolution group
for all
. This entails the index law
for all
and
. Clearly, all
desiderata are fulfilled for
with
and
.
The domain
of causal distributions will now be enlarged using certain sets of lower semicontinuous functions as convolution weights. A function
, where
, is called lower semicontinuous, if the set
is closed for every
. The set of all lower semicontinuous functions is denoted
, the set of lower semicontinuous functions whose support is bounded on the left is denoted
. For
let
be the set of lower semicontinuous functions of power-logarithmic growth of order
. Then
are the sets of interest.
Definition 3. Let and let denote the set of all bounded subsets of . Thendenotes the set of all distributions convolvable with the given set U. A locally convex topology on is defined by the family of seminormswith and . Here, the V-modulus of an element is defined asfor all . Theorem 1. The convolution group , resp. , can be extended from to operate linearly, bijectively, and continuously on the space with , resp. , in such a way that compact sets of indices α map to equicontinuous sets of operators.
Corollary 1. The desiderata (a)–(e) are fulfilled for withand for with as in (15a) andIn both cases it is possible to choose . The proof of Theorem 1 and its corollary will be published elsewhere, because it is lengthy and giving it here would distract attention from the main message. The domains , , are maximal with respect to convolvability in both cases. The second case yields a (purely imaginary) “fractional calculus of order zero” in the sense that for all operators in that subset.