Multiplicative Sonine Pairs

• Multiplicative Sonine pairs are inverse kernel pairs defined via convolution relationships in settings like Volterra, Laplace, and Mellin transforms.
• They emerge through logarithmic rectification and weighted Weyl-Sonine operators, converting additive structures into multiplicative (Mellin-type) ratios.
• In higher-rank frameworks such as symmetric cones and Dunkl theory, critical thresholds and discrete admissibility conditions govern measure positivity and index shifts.

Multiplicative Sonine pairs are not governed by a single universal definition in the current literature. Instead, closely related constructions appear in several technically distinct settings: as inverse pairs for Volterra convolution, as reciprocal multiplier pairs in Laplace or Fourier variables, as Mellin-type kernels obtained after the logarithmic rectification $y=\ln t$, and as index-shift kernels for Bessel, Dunkl, and symmetric-cone special functions. In each case, the common Sonine feature is an exact or generalized inverse relation, while the multiplicative feature arises either from transform-domain factorization or from a geometry in which additive differences in a rectified variable become multiplicative ratios in the original variable (Zheng, 2024, Dorrego, 8 Jan 2026, Rösler et al., 2018).

1. Foundational definitions and the algebraic core

Across several frameworks, a Sonine pair is first defined additively. On a finite interval $(0,b)$, the Volterra convolution is

$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$

and the classical Sonine condition is

$\int_0^t K(t-s)k(s)\,ds = 1$

for almost all $t\in(0,b)$. A kernel $k\in L^1(0,b)$ satisfying this condition with some $K\in L^1(0,b)$ is called a Sonine kernel, and $K$ is its associate kernel. The generalized Sonine condition replaces the constant $1$ by a differentiable $g$ with $(0,b)$0 and $(0,b)$1, and Theorem 3.5 proves that this generalized condition is equivalent to the original Sonine condition (Zheng, 2024).

On $(0,b)$2, the same idea is written as

$(0,b)$3

In this formulation, $(0,b)$4 are called a Sonine pair. The associated Laplace transforms satisfy

$(0,b)$5

and uniqueness of the Sonine partner follows because the Laplace transform determines an $(0,b)$6-function uniquely (Mishura et al., 2020).

For general fractional calculus with Sonine kernels, the same normalization is written

$(0,b)$7

and, when Laplace transforms exist,

$(0,b)$8

In the Mikusiński field of convolution quotients, this turns Sonine partnership into an explicit multiplicative inverse relation up to the integration kernel (Luchko, 2021).

A Weyl-history version replaces the constant function by the Heaviside distribution. In the tempered-distribution setting, an admissible pair of order $(0,b)$9 satisfies

$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$0

so for $(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$1,

$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$2

This is the global Weyl-Sonine normalization on $(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$3 (Dorrego, 8 Jan 2026, Dorrego, 25 May 2026).

Framework	Sonine relation	Multiplicative reading
Volterra kernels	$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$4	inverse under additive convolution
Laplace/Mikusiński calculus	$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$5	reciprocal transform factors
Weyl-Sonine distributions	$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$6	global inverse on history space
Cone and Dunkl formulas	integral representation of higher index through lower index	multiplicative or index-shift kernel

The term “multiplicative Sonine pair” therefore names a structural pattern rather than a single formalism. In some papers the multiplicativity is explicit, while in others it emerges only after a change of variables or passage to transform space.

2. Logarithmic rectification and Mellin-type Sonine structure

A direct route from additive Sonine pairs to multiplicative ones is provided by weighted Weyl-Sonine operators. Given a scale function $(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$7 and weight $(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$8, the weighted Weyl-Sonine integral is

$(K*k)(t):=\int_0^t K(t-s)k(s)\,ds,$9

while the paired derivative is

$\int_0^t K(t-s)k(s)\,ds = 1$0

These operators are obtained by conjugating additive convolution in the rectified variable $\int_0^t K(t-s)k(s)\,ds = 1$1 (Dorrego, 8 Jan 2026).

The multiplicative case appears when $\int_0^t K(t-s)k(s)\,ds = 1$2 and $\int_0^t K(t-s)k(s)\,ds = 1$3. Then

$\int_0^t K(t-s)k(s)\,ds = 1$4

and the corresponding integral becomes

$\int_0^t K(t-s)k(s)\,ds = 1$5

This is explicitly described as multiplicative or Mellin-type memory dependence: the kernel depends on the ratio $\int_0^t K(t-s)k(s)\,ds = 1$6, and the measure is $\int_0^t K(t-s)k(s)\,ds = 1$7 (Dorrego, 8 Jan 2026).

The spectral side makes the multiplicative pairing even sharper. If $\int_0^t K(t-s)k(s)\,ds = 1$8, then the weighted Fourier transform diagonalizes the pair: $\int_0^t K(t-s)k(s)\,ds = 1$9

$t\in(0,b)$0

The multipliers are reciprocal, and for $t\in(0,b)$1 the weighted Fourier transform is essentially a Mellin transform in disguise (Dorrego, 8 Jan 2026).

A semigroup-based reformulation reaches the same conclusion from another angle. It defines

$t\in(0,b)$2

shows that the corresponding integral operator is

$t\in(0,b)$3

and proves the inversion identity

$t\in(0,b)$4

The paper states that it does not explicitly study multiplicative Sonine pairs in the Mellin-convolution sense, but it also shows that after $t\in(0,b)$5, additive kernels in $t\in(0,b)$6 become ratio kernels in $t\in(0,b)$7 with measure $t\in(0,b)$8 (Dorrego, 25 May 2026).

The generalized Sonine condition of Zheng provides a complementary criterion. Since $t\in(0,b)$9 with $k\in L^1(0,b)$0 and $k\in L^1(0,b)$1 already implies the exact Sonine identity, a plausible implication is that in logarithmic or Mellin-type settings the decisive normalization should be imposed at the multiplicative identity $k\in L^1(0,b)$2, which corresponds to the additive starting point $k\in L^1(0,b)$3 under $k\in L^1(0,b)$4 (Zheng, 2024).

3. Symmetric cones and genuinely multiplicative Sonine kernels

The most explicit non-additive realization of multiplicative Sonine pairs appears for Bessel functions on symmetric cones. Let $k\in L^1(0,b)$5 be a simple Euclidean Jordan algebra with symmetric cone $k\in L^1(0,b)$6, rank $k\in L^1(0,b)$7, dimension $k\in L^1(0,b)$8, and Peirce constant $k\in L^1(0,b)$9. The critical index is

$K\in L^1(0,b)$0

For $K\in L^1(0,b)$1, the cone beta measure is

$K\in L^1(0,b)$2

where $K\in L^1(0,b)$3 (Rösler et al., 2018).

The classical Sonine formula is

$K\in L^1(0,b)$4

Here $K\in L^1(0,b)$5 is the quadratic representation; for matrix cones,

$K\in L^1(0,b)$6

This is multiplicative in the Jordan-theoretic sense: the lower-index Bessel function is evaluated after a cone multiplication or congruence action by the integration variable (Rösler et al., 2018).

The paper also proves a composition law for the beta kernels,

$K\in L^1(0,b)$7

where the composition is defined through the pushforward under

$K\in L^1(0,b)$8

This gives an index-shift semigroup structure. In this setting, multiplicative Sonine pairs are not merely reciprocal kernels; they compose coherently under the cone multiplication (Rösler et al., 2018).

A central higher-rank phenomenon is the appearance of gaps. The beta measures admit analytic continuation to compactly supported distributions below the classical range, and for $K\in L^1(0,b)$9 the paper characterizes exactly when these continued beta distributions are actual measures. Positive Sonine measures exist precisely for

$K$0

while below $K$1 only the discrete Wallach values survive. Thus, unlike the one-dimensional case $K$2, higher-rank multiplicative Sonine pairs are constrained by a critical index and a discrete exceptional set (Rösler et al., 2018).

4. Dunkl theory, Sonine formulas, and higher-rank obstructions

In Dunkl theory, Sonine formulas appear as positive integral representations between kernels or Bessel functions with different multiplicities. For a root system $K$3 and multiplicities $K$4, the relative intertwining operator is

$K$5

and it intertwines the Dunkl operators at multiplicities $K$6 and $K$7. Positivity of $K$8 is equivalent to the existence, for each $K$9, of a probability measure $1$0 such that

$1$1

and, after averaging over the reflection group, to the corresponding Bessel-function formula

$1$2

In this literature, a Sonine formula is precisely such an integral representation of the higher-multiplicity object through the lower-multiplicity one (Rösler et al., 2019).

For root system $1$3, with multiplicities $1$4 and one-parameter shift

$1$5

the paper derives an explicit positive Sonine formula for

$1$6

where

$1$7

This realizes $1$8 as a positive Sonine pair in an index-shift sense (Rösler et al., 2019).

The same paper establishes sharp obstructions. If a bounded complex Radon measure formula exists, then either

$1$9

or

$g$0

For positive measures, $g$1 must also be real. Writing

$g$2

the paper shows that if $g$3, then no positive Sonine formula exists and $g$4 is not positive. This disproves the conjecture that $g$5 always implies positivity of the relative intertwiner (Rösler et al., 2019).

The rank-one case is exceptional. There the classical Sonine formula

$g$6

and Xu’s positive formula for $g$7 show that every shift $g$8 is admissible. A common misconception is therefore corrected by the higher-rank theory: monotone increase of multiplicity does not in general produce a positive Sonine pair (Rösler et al., 2019).

5. Fractional calculus, operational inverses, and generalized Sonine kernels

General fractional calculus with Sonine kernels treats the pair $g$9 as the structural basis of generalized integration and differentiation. The general fractional integral is

$(0,b)$00

and the operational role of the Sonine partner $(0,b)$01 is fixed by

$(0,b)$02

The theory proves the fundamental inversion laws

$(0,b)$03

together with their $(0,b)$04-fold analogues (Luchko, 2021).

The transform-domain multiplicative structure is explicit: $(0,b)$05 In the Mikusiński field $(0,b)$06 of convolution quotients, the algebraic inverse of $(0,b)$07 is

$(0,b)$08

and the generalized derivative becomes multiplication by $(0,b)$09 modulo initial-data terms. This is one of the clearest exact meanings of a multiplicative Sonine pair: additive convolution is converted into multiplication in the operational field (Luchko, 2021).

The paper provides several explicit Sonine pairs. The classical one is

$(0,b)$10

A Bessel-type pair due to Sonine is

$(0,b)$11

A tempered example is

$(0,b)$12

with associate

$(0,b)$13

Another nonclassical pair is

$(0,b)$14

These examples show that multiplicative reciprocity in transform space is compatible with oscillatory, tempered, and Mittag-Leffler kernels, not only with power laws (Luchko, 2021).

The generalized Sonine condition of Zheng makes this calculus more flexible. If

$(0,b)$15

then $(0,b)$16 is already a genuine Sonine kernel. The variable-exponent Abel kernel

$(0,b)$17

admits the associated kernel

$(0,b)$18

under the generalized condition, and Theorem 3.5 then implies the original Sonine condition. This shows that exact constant convolution can often be recovered from a more tractable normalized relation (Zheng, 2024).

6. Stochastic inverse problems, transmutation, and broader significance

Sonine pairs also govern inverse representations for Gaussian Volterra processes. Consider

$(0,b)$19

If $(0,b)$20 forms a Sonine pair with $(0,b)$21, then the forward and backward fractional-type operators

$(0,b)$22

satisfy

$(0,b)$23

This yields explicit inverses for the derivative-level operators associated with $(0,b)$24 (Mishura et al., 2020).

Under the paper’s assumptions, the Wiener process itself can be recovered from $(0,b)$25: $(0,b)$26 where

$(0,b)$27

This is presented as a natural extension of the filtration-equivalence mechanism known for fractional Brownian motion with $(0,b)$28, where the relevant Sonine pair is given by normalized power kernels (Mishura et al., 2020).

Beyond convolution and stochastic calculus, Sonine terminology also appears in transmutation theory. The paper on the Bessel-Struve operator defines the Bessel-Struve Sonine transform $(0,b)$29 on $(0,b)$30, proves that it is a transmutation operator from $(0,b)$31 into $(0,b)$32, establishes a corresponding result for its dual on $(0,b)$33, and derives a relation between the Bessel-Struve transforms $(0,b)$34 and $(0,b)$35 by invoking the Weyl integral transform and the dual Sonine transform $(0,b)$36 on $(0,b)$37 (Kamoun et al., 2010).

Taken together, these works show that multiplicative Sonine pairs are best understood as a unifying theme across several research areas. In one-variable Volterra theory they are inverse kernels whose transforms satisfy reciprocal product laws. In weighted Weyl-Sonine theory they become Mellin-type ratio kernels after logarithmic rectification. On symmetric cones they are beta kernels acting through the Jordan-multiplicative map $(0,b)$38. In Dunkl theory they are positive or distributional index-shift measures between kernels of different multiplicities. The main substantive lesson is that multiplicativity is real but context-dependent: it may mean reciprocal spectral symbols, Mellin convolution, Jordan multiplication, or positive integral representation. Equally important, the higher-rank theories show that such pairs are not ubiquitous; positivity and measure-valued realization are often restricted by critical thresholds, discrete exceptional sets, or Wallach-type admissibility conditions (Rösler et al., 2018, Rösler et al., 2019).