ON THE DEPENDENCE OF ASYMPTOTICS OF S-NUMBERS OF FRACTIONAL INTEGRATION

OPERATORS ON WEIGHT FUNCTIONS

R. Gorenflo^{1)} and S. Samko^{2)}

1) First Mathematical Institute, Free University of Berlin, Arnimallee 2-6, 14195 Berlin, Germany

2) Department of Mathematics and Mechanics, Rostov State University,

Zorge 5, Rostov-na-Donu, 344004, Russia,

currently at Unidade de Ciencias Exactas e Humanas, Universidade de Algarve Campos de Gambelas, Faro 8000, Portugal

Abstract

We show that the singular values σ_{m} of the fractional integration operator of order α > 0 acting from weighted L_{2} spaces of functions defined on the interval (−1, 1) or (0, ∞) may, in both cases, have asymptotics m^{−α} or m^{−α/2}, as m → ∞, depending on the specific choice of weight functions. Some questions of singular value decompositions are also investigated for Riesz potential operators in weighted L_{2} spaces of functions defined on R^{n}.

KEY WORDS: Abel integral operator, Riesz potential operator, singular value decomposition, singular value asymptotics, weighted spaces of squareintegrable functions

MSC (1991): 47B06, 45E10, 44A35

1. Introduction

In the recent paper [7] there was discovered the following effect : singular values of the Riemann-Liouville fractional integration operator I_{0}^{α} : L_{2}(R_{+}^{1}, e^{−x}) ⟶ L_{2}(R_{+}^{1}, x^{α}e^{−x}) have the asymptotics

σ_{m} ∼ m^{−α/2}

as m → ∞ , not σ_{m} ∼ m^{−α} as one could expect from the smoothing properties of the operator I_{0}^{α} .

We show that this effect is not a result of non-compactness of R_{+}^{1} : we may have the same effect on a finite interval, and on the contrary, we may have σ_{m} ∼ m^{−α} on R_{+}^{1}, these effects depending on the special choice of weight functions.

We discuss some arising questions and give some extensions to the multidimensional case, for the Riesz potential operator in weighted L^{2}(R^{n})-spaces. In case n = 1 we explicitly construct singular systems for the potential type operators, both over R^{1} and over a finite interval. We give also some conclusions for the Liouville fractional integration operators over the whole real line as a corollary of our results for the potential type operator.

It should be noted that there exist many investigations on the asymptotics of singular numbers of integral operators. We refer for example to the papers [1] - [3], where some powerful tools were developed for finding the asymptotics of singular numbers of certain pseudo-differential operators of general types. See also [14] where one can find results for the asymptotics in case of fractional integration operators of order α > 1/2 with a certain type of a weight function, where a dependence of the asymptotics on a weight function is given in certain terms. We would like to emphasize that because of the applied aspects of the considered operators we prefer to give the direct construction of a sequence of s-numbers, not only its asymptotics, together with the constructive realization of the ”singular value decomposition”. Besides, even in the case of asymptotics, our weighted considerations are not covered by the known results, up to our knowledge.

2. Preliminaries

1). Fractional integral operators.

We use both the left- and right-sided Riemann-Liouville fractional integrals

I_{a+}^{α}u(x) = \frac{1}{Γ(α)} ∫_{a}^{x}(x − t)^{α−1}u(t)dt , x > a , α > 0 ,  (1)

I_{b−}^{α}u(x) = \frac{1}{Γ(α)} ∫_{x}^{b}(t − x)^{α−1}u(t)dt , x < b , α > 0 ,  (2)

and deal mainly with the cases a = 0 or a = −1, and b = 1.

The corresponding Liouville fractional integrals on the whole real line R^{1} will be denoted as

I_{+}^{α}u(x) = \frac{1}{Γ(α)} ∫_{−∞}^{x}(x − t)^{α−1}u(t)dt , x ∈ R^{1} , α > 0 ,  (3)

I_{−}^{α}u(x) = \frac{1}{Γ(α)} ∫_{x}^{∞}(t − x)^{α−1}u(t)dt , x ∈ R^{1} , α > 0 ,  (4)

The one-dimensional (fractional) Riesz potential has the form

I^{α}u = \frac{1}{2Γ(α) cos \frac{απ}{2}} ∫_{−∞}^{∞} \frac{u(y) dy}{|x − y|^{1−α}} , x ∈ R^{1}, α > 0, α ≠ 1, 3, 5, ... . (5)

We shall also treat its modification for the finite interval

A^{α}u = ∫_{−1}^{1} \frac{u(y) dy}{|x − y|^{1−α}} , |x| < 1.  (6)

The multidimensional version of the operator (5) is

I^{α}u = \frac{1}{γ_{n}(α)} ∫_{R^{n}} \frac{u(y)dy}{|x − y|^{n−α}} , x ∈ R^{n},  (7)

where α > 0, α ≠ n, n + 2, n + 4, ... and

γ_{n}(α) = \frac{2^{α}π^{n/2}Γ(\frac{α}{2})}{Γ(\frac{n−α}{2})}

is the normalizing constant for the Riesz potential, see [19] , section 25.2.

We shall consider the above operators within the frameworks of weighted L^{2}-spaces

L^{2} (Ω; ρ(x)) = {f(x) : ∫_{Ω} |f(x)|^{2}ρ(x)dx < ∞}

over the corresponding set Ω with some special types of weight functions ρ(x).

2). On boundedness of fractional integration operators .

Lemma 1 . Let the real numbers μ, ν and ν_{1} satisfy the conditions μ < 1 and ν_{1} = ν−2α if ν−2α > −1 and ν_{1} > −1 if ν−2α ≤ −1. Then the operator I_{0+}^{α} is bounded from L^{2} ([0, 1]; x^{μ}(1 − x)^{ν}) into L^{2} ([0, 1]; x^{μ−2α}(1 − x)^{ν_{1}}) .

Lemma 2. Let the real numbers μ, μ_{1} and ν satisfy the conditions ν < 1 and μ_{1} = μ − 2α if μ − 2α > −1 and μ_{1} > −1 if μ − 2α ≤ −1. Then the operator I_{1−}^{α} is bounded from L^{2} ([−1, 1]; (1 + x)^{μ}(1 − x)^{ν}) into L^{2} ([−1, 1]; (1 + x)^{μ_{1}}(1 − x)^{ν−2α}) .

Lemma 3. Let the real numbers μ, ν and ν_{1} satisfy the conditions μ < 1 and ν_{1} = ν if μ + ν < 1 and ν_{1} > 1 − μ if μ + ν ≥ 1. Then the operator I_{0+}^{α} is bounded from L^{2} (R_{+}^{1}; x^{μ}(1 + x)^{ν}) into L^{2} (R_{+}^{1}; x^{μ−2α}(1 − x)^{ν_{1}}) .

Lemma 4. The operators (3)-(5) are bounded from L^{2}(R^{1}; (1 + x^{2})^{\frac{μ}{2}} ) into L^{2}(R^{1}; (1 + x^{2})^{\frac{ν}{2}} ) if μ > 2α − 1 and ν < min(μ, 1) − 2α.

Lemmas 1 - 4 are particular cases of more general results on boundedness of operators of fractional integration within the framework of L^{p}-spaces with power-type weights, see [19], Theorems 3.10 and 5.5. We also note that Lemma 2 can be obtained from Lemma 1 by an obvious linear change of variables, while Lemma 3 can be derived from Lemma 2 by means of the connection (23)-(24) given below.

3). On mapping between the spaces on [−1, 1] and R_{+}^{1}.

Obviously, the involutive change of variables

\frac{1 − x}{1 + x} = y , \frac{1 − y}{1 + y} = x  (8)

maps the half-axis R_{+}^{1} onto [−1, 1] and vice versa.

We introduce the operator

M_{α}f(x) = f^{∗}(x) = γ(1 + x)^{−1−α}f (\frac{1 − x}{1 + x}) ,  (9)

associated with the order α of fractional integration. The normalizing factor

γ = 2^{\frac{μ+ν+1}{2}}

is chosen to achieve isometry between weighted spaces considered in Lemma 4 below.

We note that the expression for the operator inverse to M_{α} coincides with the expression for M_{α} up to a constant numerical factor :

M_{α}^{−1}f(x) = 2^{α−μ−ν}M_{α}f(x)  (10)

Lemma 5. Let μ, ν and α be arbitrary real numbers.Then the operator M_{α} maps the space L^{2} ([−1, 1]; (1 + y)^{μ}(1 − y)^{ν}) onto L^{2} (R_{+}^{1}; x^{ν}(1 + x)^{2α−μ−ν}) preserving the scalar product:

∫_{−1}^{1} f_{1}(y)f_{2}(y)(1 + y)^{μ}(1 − y)^{ν}dy = ∫_{−∞}^{∞} f_{1}^{∗}(x)f_{2}^{∗}(x)x^{ν}(1 + x)^{2α−μ−ν}dx ,

Remark 1. In view of (1)-(3) we also have the relation

∫_{−∞}^{∞} f_{1}(x)f_{2}(x)x^{ν}(1+x)^{2α−μ−ν}dx = 2^{2(α−μ−ν)} ∫_{−1}^{1} f_{1}^{∗}(y)f_{2}^{∗}(y)(1+y)^{μ}(1−y)^{ν}dy

4). Singular numbers.

We recall the essentials of the theory of singular numbers of compact linear operators (see details in [4]). Let U and V be infinite-dimensional Hilbert spaces. If A : U → V is an injective compact linear operator, then there exist

(i) an orthonormal basis {u_{j}}_{j∈N_{0}} in U ;

(ii) an orthonormal system {v_{j}}_{j∈N_{0}} in V ;

(iii) a nonincreasing sequence {σ_{j}}_{j∈N_{0}} of positive numbers with limit 0 as j → ∞ , such that

Au_{j} = σ_{j}v_{j}, j ∈ N_{0}.

The numbers σ_{j} are known as singular numbers or s-numbers of the operator A and the system {σ_{j}, u_{j}, v_{j}}_{j∈N_{0}} is called a singular system for the operator A (in view of what is discussed in the paper, it is appropriate to refer to {σ_{j}, u_{j}, v_{j}}_{j∈N_{0}} as a singular system not for the operator A only, but for the operator A and a given pair U and V of the spaces involved.

Remark 2. We note that for any operator A the setting

A : L^{2}(Ω; φ(x)) → L^{2}(Ω; ψ(x))

with a singular system {σ_{m}, u_{m}, v_{m}} is equivalent to the setting

A_{1} = ψ^{\frac{1}{2}} Aφ^{− \frac{1}{2}} : L^{2}(Ω) → L^{2}(Ω)

with the system \{σ_{m}, φ^{\frac{1}{2}} u_{m}, ψ^{\frac{1}{2}} v_{m}\}.

Proposition 1 ([4] , p.27) . For any bounded linear operator B : U → U or B : V → V,

σ_{m}(AB) ≤ ∥B∥_{U} σ_{m}(A),
σ_{m}(BA) ≤ ∥B∥_{V} σ_{m}(A),

respectively.

In the following theorems , obtained in [7], , C_{n}^{λ}(x), P_{n}^{α,β}(x) and L_{n}^{(β)}(x) are the standard notations, see e.g. [8] , for the Gegenbauer, Jacobi and Laguerre polynomials correspondingly and

a_{m} = [\frac{2^{2λ−1}(m + λ)m!}{πΓ(m + 2λ)}]^{1/2} Γ(λ) , b_{m} = [\frac{2^{1−2λ}(m + λ)m!Γ(m + 2λ)}{Γ(m + λ − α + 1/2)Γ(m + λ + α + 1/2)}]^{1/2} ,

A_{m} = (\frac{m!}{Γ(β + m + 1)})^{1/2} , B_{m} = (\frac{Γ(m + β + 1)}{Γ(m + α + β + 1)})^{1/2}

Theorem A [7] . Let λ > α − 1/2, α > 0, λ ≠ 0. Then the operator

I_{−1+}^{α} : L^{2} ([−1, 1]; (\frac{1 − x}{1 + x})^{λ−1/2}) → L^{2} ([−1, 1]; (\frac{1 − x}{1 + x})^{λ−1/2} (1 − x^{2})^{−α})

(11)

has the following singular system:

σ_{m} = [\frac{Γ(m + λ − α + 1/2)}{Γ(m + λ + α + 1/2)}]^{1/2} ∼ m^{−α} ,  (12)

u_{m}(x) = a_{m}(1+x)^{λ−1/2}C_{m}^{λ}(x) , v_{m}(x) = b_{m}(1+x)^{α+λ−1/2}P_{m}^{(λ−α−1/2,λ+α−1/2)}(x) .

(13)

Theorem B [7] . Let β > −1 and α > 0. Then the operator

I_{0+}^{α} : L^{2}(R_{+}^{1}; x^{−β}e^{−x}) → L^{2}(R_{+}^{1}; x^{−α−β}e^{−x})  (14)

has the following singular system

σ_{m} = [\frac{Γ(m + β + 1)}{Γ(m + α + β + 1)}]^{1/2} ∼ m^{−α/2},  (15)

u_{m}(x) = A_{m}x^{β}L_{m}^{(β)}(x) , v_{m}(x) = B_{m}x^{α+β}L_{m}^{(α+β)}(x) .  (16)

As a corollary of Theorem A we also have the following theorem for the adjoint operator

I_{1−}^{α}u(x) = \frac{1}{Γ(α)} ∫_{x}^{1}(t − x)^{α−1}u(t)dt , x < 1 .

Theorem A′ . Let λ > α − 1/2, α > 0, λ ≠ 0. Then the operator

I_{1−}^{α} : L^{2} ([−1, 1]; (\frac{1 − x}{1 + x})^{1/2−λ}) → L^{2} ([−1, 1]; (\frac{1 − x}{1 + x})^{1/2−λ} (1 − x^{2})^{−α})

(17)

has the following singular system:

σ_{m} = [\frac{Γ(m + λ − α + 1/2)}{Γ(m + λ + α + 1/2)}]^{1/2} ∼ m^{−α} ,  (18)

u_{m}(x) = a_{m}(1−x)^{λ−1/2}C_{m}^{λ}(x) , v_{m}(x) = b_{m}(1−x)^{α+λ−1/2}P_{m}^{(λ−α−1/2,λ−α−1/2)}(x) .

(19)

To derive Theorem A′ from Theorem A it is sufficient to remark that

I_{1−}^{α} = QI_{−1+}^{α}Q  (20)

where Qu(x) = u(−x) and note that

C_{m}^{λ}(−x) = (−1)^{m}C_{m}^{λ}(x) , P_{m}^{(α,β)}(−x) = (−1)^{m}P_{m}^{(α,β)}(x)

5). Stereographic projection.

We consider the Euclidean space R^{n} embedded into R^{n+1} , identifying R^{n} with the hyperplane {x ∈ R^{n+1} : x_{n+1} = 0}. The change of variables in R^{n+1} :

ξ = s(x) = {s_{1}(x), ..., s_{n+1}(x)},
s_{k}(x) = \frac{2x_{k}}{1 + |x|^{2}} , k = 1, ..., n, s_{n+1}(x) = \frac{|x|^{2} − 1}{|x|^{2} + 1},  (21)

where x = (x_{1}, ..., x_{n+1}) ∈ R^{n+1}, |x| = (x_{1}^{2} + ... + x_{n+1}^{2})^{1/2}, generates the mapping known as the stereographic projection. It maps the subspace Ṙ^{n} one-to-one onto the unit sphere S^{n} ⊂ R^{n+1}, S^{n} = {ξ ∈ R^{n+1} : ξ_{1}^{2}+...+ξ_{n+1}^{2} = 1}, where by Ṙ^{n} we denote the complementation of the Euclidean space R^{n} by a single infinite point. It is known [11] that the following relations hold

|ξ − σ| = \frac{2|x − y|}{(1 + |x|^{2})^{1/2}(1 + |y|^{2})^{1/2}} ,  (22)

dσ = \frac{2^{n}dy}{(1 + |y|^{2})^{n}} ,  (23)

where ξ = s(x), σ = s(y) , x, y ∈ R^{n+1}.

6). Spherical harmonics.

We refer e.g. to [12],[20] for harmonic analysis on S^{n} and recall only some notations and the Funk-Hekke formula . By Y_{m}(σ), σ ∈ S^{n}, we denote an arbitrary spherical harmonic of order m , that is, the restriction of any homogeneous harmonic polynomial in R^{n+1} onto S^{n}. The space H_{m} of all such harmonics has a dimension

d_{n+1}(m) = \frac{n + 2m − 1}{n + m − 1} ( \begin{matrix}m + n − 1 \\ m \end{matrix} ) .  (24)

Let{Y_{mμ}}_{μ=1,...,d_{n+1}(m)} be an orthonormal basis in H_{m} . It is known ([12],[20]) that the sequence

{Y_{mμ}}_{μ=1,...,d_{n+1}(m);m∈N_{0}}

is a basis in L^{2}(S^{n}).

The formula

∫_{S^{n}} k(ξσ)Y_{m}(σ)dσ = λ_{m}Y_{m}(ξ) , ξ ∈ S^{n},  (25)

known as the Funk-Hekke formula [9], states that an arbitrary spherical harmonic Y_{m}(ξ) is an eigen-function for any operator K , defined as

Ku = ∫_{S^{n}} k(ξσ)u(σ)dσ , ξ ∈ S^{n},  (26)

with the kernel depending only on the scalar product ξσ. In (25)

λ_{m} = \frac{|S^{n−1}|}{( \begin{matrix}m + n − 2 \\ m \end{matrix} )} ∫_{−1}^{1}(1 − t^{2})^{\frac{n−1}{2}} C_{m}^{λ}(t)k(t)dt  (27)

where λ = \frac{n−1}{2}, C_{m}^{λ}(t) is the Gegenbauer polynomial and |S^{n−1}| is the surface area of the unit sphere in R^{n} , |S^{n−1}| = \frac{2π^{n/2}}{Γ( \frac{n}{2} )} .

3. Statements and comments

Theorems 1 and 2 below show that the effect of σ_{m} ∼ m^{−α/2} is not a result of the consideration of the non-compact set R_{+}^{1} instead of a compact interval. We show that the same operator I_{0+}^{α} has different asymptotics of s-numbers in two different settings on a finite interval:

I_{0}^{α} : L^{2}([0, 1]; ρ_{1}(x)) → L^{2}([0, 1]; r_{1}(x)) , σ_{m} ∼ m^{−α} ,  (28)
I_{0}^{α} : L^{2}([0, 1]; ρ_{2}(x)) → L^{2}([0, 1]; r_{2}(x)) , σ_{m} ∼ m^{−α/2} .  (29)

The case (28) is contained in Theorem A , while the case (29) is covered by Theorem 2 below.

Similarly, we may have both situations on the half-axis R_{+}^{1} :

I_{0}^{α} : L^{2}(R_{+}^{1}; φ_{1}(x)) → L^{2}(R_{+}^{1}; ψ_{1}(x)) , σ_{m} ∼ m^{−α} ,  (30)
I_{0}^{α} : L^{2}(R_{+}^{1}; φ_{2}(x)) → L^{2}(R_{+}^{1}; ψ_{2}(x)) , σ_{m} ∼ m^{−α/2},  (31)

the case (31) being contained in Theorem B , while the case (30) is covered by Theorem 1 below.

Theorem 1 . The asymptotics σ_{m} ∼ m^{−α} takes place for the operator I_{0+}^{α}, α > 0, on the half-axis R_{+}^{1} in the setting (30) under the following choice of the weight-functions:

φ_{1}(x) = x^{\frac{1}{2} −λ}(1 + x)^{2α} , ψ_{1}(x) = x^{\frac{1}{2} −λ−α},  (32)

where λ > α − 1/2, λ ≠ 0. The singular system is given by the same σ_{m} as in (12) and

u_{m}(x) = 2^{λ}a_{m}x^{λ−1/2}(1 + x)^{−λ−α−1/2}C_{m}^{λ} (\frac{1 − x}{1 + x}) ,  (33)

v_{m}(x) = 2^{λ}b_{m}x^{λ+α−1/2}(1 + x)^{−λ−α−3/2}P_{m}^{(λ−α−1/2,λ+α−1/2)} (\frac{1 − x}{1 + x}) (34)

Theorem 2 . The asymptotics σ_{m} ∼ m^{−α/2} takes place for the operator I_{−1}^{α}, α > 0, on the interval [−1, 1] in the setting (29) under the following choice of the weight-functions:

ρ_{2}(x) = (1 + x)^{−β}(1 − x)^{2α+β}e^{− \frac{1+x}{1−x}} , r_{2}(x) = (1 + x)^{−β−α}(1 − x)^{β−α}e^{− \frac{1+x}{1−x}} ,

where β > −1. The singular system is given by the same σ_{m} as in (15) and

u_{m}(x) = 2^{1/2}A_{m}(1 + x)^{β}(1 − x)^{−α−β−1}L_{m}^{(β)} (\frac{1 + x}{1 − x}) ,

v_{m}(x) = 2^{1/2}B_{m}(1 + x)^{α+β}(1 − x)^{−β−1}L_{m}^{(α+β)} (\frac{1 + x}{1 − x})

Theorems 1 and 2 lead to the following Question:

Question. Given s ∈ (0, α], does there exist a pair of weight functions ρ(x) and r(x) , such that s-numbers of the operator

I_{0}^{α} : L^{2}(Ω; ρ(x)) → L^{2}(Ω; r(x)) ,

with Ω = [0, 1] or Ω = R_{+}^{1} have the asymptotic

σ_{m} ∼ m^{−s} ?

Because of Theorems 1 and 2, Question seems to be especially natural for values s ∈ [α/2, α].

In Theorem 3 below we give some version of Theorem 1 for the multidimensional Riesz potential operator (7) using the notation x̃ = (x, x_{n+1}) ∈ R^{n+1}, where x ∈ R^{n}, and s(x̃) for the stereographic projection in the Euclidean space R^{n+1} , which maps the hyperplane R^{n} = {x̃ ∈ R^{n+1} : x_{n+1} = 0}, onto the unit sphere S^{n} ⊂ R^{n+1} , see (21). By Y_{mμ}(ξ), ξ ∈ S^{n}, we denote basic spherical harmonics (see Section 1, no 6).).

Theorem 3. The operator

I^{α} : L^{2} (R^{n}; (1 + |x|^{2})^{α}) → L^{2} (R^{n}; (1 + |x|^{2})^{−α}) , α > 0, α ≠ n, n+2, n+4, ...,

(35)

has singular values

σ_{m} = \frac{1}{2^{α}} \frac{Γ(m + \frac{n−α}{2})}{Γ(m + \frac{n+α}{2})} ∼ m^{−α}  (36)

and a singular system is {σ_{m}, u_{mμ}(x), v_{mμ}(x)} with

u_{mμ}(x) = \frac{2^{\frac{n}{2}} Ỹ_{mμ}(x)}{(1 + |x|^{2})^{\frac{n+α}{2}}} , v_{mμ}(x) = \frac{2^{\frac{n}{2}} Ỹ_{mμ}(x)}{(1 + |x|^{2})^{\frac{n−α}{2}}} ,  (37)
μ = 1, 2, ..., d_{n+1}(m); m = 0, 1, 2, ...

where Ỹ_{mμ}(x) = Y_{mμ}[s(x̃)] |_{x_{n+1}=0} .

Corollary 1 (of Theorem 3) . The one-dimensional Riesz potential (5) in the setting

I^{α} : L^{2}(R^{1}; (1 + x^{2})^{α}) → L^{2}(R^{1}; (1 + x^{2})^{−α}, 0 < α < 1,  (38))

has a singular system {σ_{m}, u_{mμ}, v_{mμ}}_{μ=1,2;m∈N_{0}} where σ_{m} is given by (36) with n = 1 and

u_{m1}(x) = \frac{T_{m} (\frac{2x}{1+x^{2}} )}{(1 + x^{2})^{\frac{1+α}{2}}} , v_{m1}(x) = \frac{T_{m} (\frac{2x}{1+x^{2}} )}{(1 + x^{2})^{\frac{1−α}{2}}} ,

where m = 0, 1, 2, ... and

u_{m2}(x) = \frac{(1 − x^{2})U_{m−1} (\frac{2x}{1+x^{2}} )}{(1 + x^{2})^{\frac{3+α}{2}}} , v_{m2}(x) = \frac{(1 − x^{2})U_{m−1} (\frac{2x}{1+x^{2}} )}{(1 + x^{2})^{\frac{3−α}{2}}}

where m = 1, 2, ... , T_{m}(x) and U^{m}(x) being the Chebyshev polynomials of the 1st and 2nd kind, respectively.

Remark 3. According to the above corollary the singular system is found explicitly in the one -dimensional case in terms of Chebyshev polynomials. If n > 1, the functions Y_{mμ}(x), x ∈ S^{n}, we use in (37) can be only described as restrictions of basic harmonic homogeneous polynomials Y_{mμ}(x̃), x̃ ∈ R^{n+1} onto S^{n}. In this connection we refer to the paper [17] where there is given a criterion, in terms of coefficients, for an arbitrary homogeneous polynomial ∑_{|j|=m} a_{j}x^{j} to be harmonic.

Corollary 2 (of Theorem 3) . Let 0 < α < 1/2. The singular values of the Liouville fractional integration operators (3)-(4) considered in the setting (38) , have the asymptotics σ_{m} ∼ m^{−α}.

Theorem 4. The potential-type operator (6) in the setting

A^{α} : L^{2} ([−1, 1]; (1 − x^{2})^{α}) → L^{2} ([−1, 1]; (1 − x^{2})^{−α}) ,  (39)

0 < α < 1, has the singular system {σ_{m}, u_{m}, v_{m}}_{m∈N_{0}} with

σ_{m} = \frac{πΓ(m + 1 − α)}{m! sin \frac{απ}{2}Γ(1 − α)} ∼ \frac{1}{m^{α}}

and

u_{m}(x) = c_{m}(1 − x^{2})^{−α}Cn^{\frac{1−α}{2}} (x) , v_{m}(x) = c_{m}Cn^{\frac{1−α}{2}} (x)  (40)

with c_{m} = Γ (\frac{1−α}{2}) 2^{− \frac{α}{2}} (\frac{m!(n+ \frac{1−α}{2} )}{πΓ(n+1−α)})^{\frac{1}{2}} , C_{m}^{λ}(x), λ = \frac{1−α}{2}, being the Gegenbauer polynomials.

4. Proofs of Theorems.

P r o o f o f T h e o r e m 1 .

Naturally, we base our proof on reduction to Theorem A′ . Therefore, we use the change of variables (8) and the mapping operator (9) to reduce the case of the half-axis R_{+}^{1} to the case of the interval [-1,1] and then apply Theorem A′ . By (1) we have

∫_{0}^{x} \frac{u(t)dt}{(x − t)^{1−α}} = 2^{α}(1 + y)^{1−α} ∫_{y}^{1} \frac{u (\frac{1−τ}{1+τ} )}{(1 + τ)^{1+α}} \frac{dτ}{(τ − y)^{1−α}}  (41)

with y = \frac{1−x}{1+x}. This relation is crucial for our purposes (compare with the relation (5.33) in [19]).

Because of Theorem A′ we choose the space

L^{2} ([−1, 1]; (\frac{1 − x}{1 + x})^{1/2−λ})

for functions on [-1,1] . So, we base our consideration on Lemma 5 with μ = −ν = λ − 1/2. For the mapping (9), by Lemma 5 we see that

u^{∗}(τ) = 2^{1/2}(1 + τ)^{−1−α}u (\frac{1 − τ}{1 + τ} ) ∈ L^{2} ([−1, 1]; (\frac{1 − τ}{1 + τ} )^{1/2−λ}) (42)

if u(x) ∈ L^{2} (R_{+}^{1}; x^{−λ+1/2}(1 + x)^{2α}) . In terms of the mapping (9) the relation (41) can be represented as

I_{0+}^{α}u(x) = 2^{α−1/2}(1 + y)^{1−α}I_{1−}^{α}u^{∗}(y),  (43)

which can be also given symmetrically as

I_{1−}^{α}v(y) = 2^{α−1/2}(1 + x)^{1−α}I_{0+}^{α}v^{∗}(x).  (44)

The latter can be verified directly in view of (10).

To apply Theorem A′ , we must be sure that our consideration of the operator I_{0+}^{α} as

I_{0+}^{α} : L^{2} (R_{+}^{1}; x^{−λ+1/2}(1 + x)^{2α}) → L^{2} (R_{+}^{1}; x^{−λ−α+1/2})  (45)

exactly corresponds to the setting (17) of the operator I_{1−} in Theorem A′. (We note that the consideration (17) under the condition λ > α − 1/2 is in accordance with Lemma 2 . (Lemma 2 gives even a stronger assertion).

Checking this correspondence , in view of (43) we have:

∥I_{0+}^{α}u(x)∥_{L_{2}(R_{+}^{1};x^{−λ−α+1/2)}} = 2^{α−1/2} \{∫_{0}^{∞} |(1 + y)^{1−α}I_{1−}^{α}u^{∗}(y)|_{y= \frac{1−x}{1+x}}^{2} x^{−λ−α+1/2}dx\}^{1/2}

= 2^{α} \{∫_{−1}^{1} |(1 + y)^{−α}I_{1−}^{α}u^{∗}(y)|^{2} (\frac{1 − y}{1 + y} )^{−λ−α+1/2} dy\}^{1/2}

= 2^{α}∥I_{1−}^{α}u^{∗}(y)∥_{L^{2}([−1,1];(\frac{1−y}{1+y})^{−λ+1/2}(1−y^{2})^{−α})}.

Then, by Theorem A′ we have

I_{1−}u_{m}(y) = σ_{m}v_{m}(y)

where σ_{m} is given by (18) and u_{m}(y), v_{m}(y) are defined in (19). Applying the connection (43), we obtain

2^{α−1/2}(1 + x)^{1−α}I_{0+}^{α}u_{m}(x) = σ_{m}v_{m} (\frac{1 − x}{1 + x})

or

I_{0+}^{α}u_{m}^{∗}(x) = σ_{m}v_{m}^{#}(x)  (46)

where

v_{m}^{#}(x) = \frac{(1 + x)^{2α}}{2^{α}} v_{m}^{∗}(x).

Here the sequence {u_{m}(x)} is an orthonormal basis in the space L^{2} (R_{+}^{1}; x^{−λ+1/2}(1 + x)^{2α}) by Lemma 5. The orthogonality also holds for

v_{m}^{#}(x) in L^{2} (R_{+}^{1}; x^{−λ−α+1/2}) . Really,

∫_{0}^{∞} v_{k}^{#}(x)v_{m}^{#}(x)x^{−λ−α+1/2}dx =

∫_{−1}^{1} v_{k}(y)v_{m}(y) (\frac{1 − y}{1 + y} )^{−λ+1/2} (1 − y^{2})^{−α}dy = δ_{mk}.

Then (46) means that

{σ_{m}, u_{m}^{∗}(x), v_{m}^{#}(x)}

is a singular system for the operator (45). The direct calculation shows that u_{m}^{∗}(x) and v_{m}^{#}(x) coincide with functions given in (33)-(34). □

P r o o f o f T h e o r e m 2 .

The proof of Theorem 2 is completely analogous to that of Theorem 1 : we must

1) use the relation (41) = (43) to pass from the interval to the half-axis;

2) to apply then Theorem A for the half-axis and return back to the interval by means of (8) , recalculating u_{m}(x) and v_{m}(x) ;

3) to verify that the new constructed sequences u_{m}(x) and v_{m}(x) are really orthonormal in the corresponding spaces.

P r o o f o f T h e o r e m 3 .

We base the proof on the following two facts:

a) there exists an explicit relation between the Riesz potential (7) over R^{n} and the similar spherical potential over S^{n} ⊂ R^{n+1};

b) the L^{2}(S^{n})-basis of spherical harmonics is the set of eigen-functions of the spherical potential operator.

The relation mentioned in a) is the following

∫_{S^{n}} \frac{f(σ)dσ}{|ξ − σ|^{n−α}} = 2^{α}(|x|^{2} + 1)^{\frac{n−α}{2}} ∫_{R^{n}} \frac{f[s(y)]dy}{|x − y|^{n−α}(1 + |y|^{2})^{\frac{n+α}{2}}} ,  (47)

where σ = s(y) is the stereographic projection (21) of R^{n} onto the unit sphere S^{n}. The equality (47) immediately follows from (22)-(23) (see [15] and references there).

As regards the fact b), it is given precisely by the relation

\frac{1}{γ_{n}(α)} ∫_{S^{n}} \frac{Y_{m}(σ)dσ}{|ξ − σ|^{n−α}} = λ_{m} Y_{m}(ξ) , ξ ∈ S^{n},  (48)

where Y_{m}(ξ) is an arbitrary spherical harmonic of order m and

λ_{m} = \frac{Γ (m + \frac{n−α}{2})}{Γ (m + \frac{n+α}{2})}.  (49)

The formula (48)-(49) is known: it can be found e.g. in [18] in a non-direct form. We mention that this formula is a consequence of the Funk-Hekke formula (25). Really, since

|ξ − σ|^{2} = 2(1 − ξσ)

for ξ, σ ∈ S^{n}, the spherical potential operator, that is the left-hand side in (47) , has the form (26) with

k(t) = \frac{(1 − t)^{\frac{α−n}{2}}}{2^{\frac{n−α}{2}} γ_{n}(α)}.

Then the Funk-Hekke formula (25) is applicable, which gives (48) and, to obtain (49), it remains to calculate the integral (27) for the above kernel k(t) :

λ_{m} = \frac{2^{1+ \frac{α−n}{2}} π^{\frac{n}{2}}}{Γ (\frac{n}{2}) γ_{n}(α) ( \begin{matrix}m + n − 2 \\ m \end{matrix} )} ∫_{−1}^{1}(1−t)^{\frac{α−2}{2}} (1+t)^{λ− \frac{1}{2}} C_{m}^{λ}(t)dt , λ = \frac{n − 1}{2},

which is known:

∫_{−1}^{1}(1 + t)^{β}(1 − t)^{λ− \frac{1}{2}} C_{m}^{λ}(t)dt =

\frac{2^{β+λ+1/2}Γ)β + 1)γ(λ + 1/2)Γ(2λ + m)Γ(β − λ + 3/2)}{m!Γ(2λ)Γ(β − λ − m + 3/2)Γ(β + λ + m + 3/2)},

see e.g. [8], 7.311.3. Hence, after easy calculations we arrive at (49). Now, we use the relation (47) which allows to transform (48) to

\frac{2^{α}}{γ_{n}(α)} ∫_{R^{n}} \frac{Y_{m}[s(y)]}{(1 + |y|^{2})^{\frac{n+α}{2}}} \frac{dy}{|x − y|^{n−α}} = λ_{m} \frac{Y_{m}[s(x)]}{(1 + |x|^{2})^{\frac{n−α}{2}}} .

This is valid for an arbitrary spherical harmonic Y_{m}. We take in particular basic harmonics and then this may be rewritten as

I^{α}u_{mμ}(x) = \frac{λ_{m}}{2^{α}} v_{mμ}(x)

with the functions from (37). These functions form basises in the spaces

L^{2}(R^{n}; (1 + |x|^{2})^{α}) , L^{2}(R^{n}; (1 + |x|^{2})^{−α}) ,  (50)

respectively. Checking that, for example, for the former of these spaces, by (23) we have

∫_{R^{n}} u_{mμ}(x)u_{kν}(x)(1 + |x|^{2})^{α}dx =

2^{n} ∫_{R^{n}} \frac{Y_{mμ}[s(x)]Y_{kν}[s(x)]}{(1 + |x|^{2})^{n}}dx = ∫_{S^{n}} Y_{mμ}(ξ)Y_{kν}(ξ)dξ = δ_{mμ,kν}

where δ_{mμ,kν} = 1 if both m = k and μ = ν and δ_{mμ,kν} = 0 otherwise . This gives the orthogonality condition. It remains to refer to the fact that {Y_{mμ}(σ)}_{μ=1,...,d_{n+1}(m);m=0,1,2,...} is a basis in L^{2}(S^{n}) and so the functions (37) form basis in the corresponding weighted spaces (50) as well.

Finally, it remains to observe that the operator I^{α} is bounded in the setting (35) which follows from the relation (47) and boundedness of the operator in the left hand side of (47) in the space L^{2}(S^{n}). □

P r o o f o f C o r o l l a r y 1

The operator I^{α} is bounded in the setting (38) by Lemma 4. So, we should only calculate the functions Y_{mμ}(x) in (37) , taking into account that μ = 1, 2 in case n = 1. Evidently,

Y_{m1}(x̃) |_{S^{1}}= \frac{cos mθ}{\sqrt{2}} , Y_{m2}(x̃) |_{S^{1}}= \frac{sin mθ}{\sqrt{2}} , 0 < θ < 2π

for n + 1 = 2(x̃ = (x, x_{2}) ∈ S^{1} = {x̃ ∈ R^{2} : x^{2} + x_{2}^{2} = 1}; θ = arccos x). Since

s(x̃) = (\frac{2x}{1 + x^{2} + x_{2}^{2}} , \frac{x^{2} + x_{2}^{2} − 1}{x^{2} + x_{2}^{2} + 1}) ,

we obtain

Y_{m1}[s(x̃)] |_{x_{2}=0}= \frac{1}{\sqrt{2}}T_{m} (\frac{2x}{1 + x^{2}} ) ,

Y_{m2}[s(x̃)] |_{x_{2}=0}= \frac{1}{\sqrt{2}} \frac{1 − x^{2}}{1 + x^{2}} U_{m−1} (\frac{2x}{1 + x^{2}} ) ,

which proves the corollary.

P r o o f o f C o r o l l a r y 2 .

It is known ([19], Section 12 ) that the Riesz potential operator (5) and the Liouville fractional integration operators (3)-(4) are connected with each other by means of the relation

I_{±}^{α} = I^{α}B_{±} = B_{±}I^{α}  (51)

where

B_{±} = cos \frac{απ}{2} E ∓ sin \frac{απ}{2} S ,

E being the identity operator and

Su = \frac{1}{π} ∫_{−∞}^{∞} \frac{u(t)dt}{t − x}.

It is well known that S^{2} = −E , see [13]. Then, evidently, the operators B_{±} are invertible and (B_{±})^{−1} = B_{∓}, so that

I^{α} = B_{m}I_{±}^{α}  (52)

We apply Proposition 1 (Section 1, no 4) to (51) and (52) and obtain

σ_{m}(I_{±}^{α}) ≤ c(α)σ_{m}(I^{α}) ,
σ_{m}(I^{α}) ≤ c(α)σ_{m}(I_{±}^{α}) ,

where c(α) = | cos \frac{απ}{2}| + | sin \frac{απ}{2}|∥S∥_{L^{2}(R^{1};(1+x^{2})^{α})}.

It remains to refer to the boundedness of the operator S in the weighted spaces L^{2}(R^{1}; (1 + x^{2})^{±α}) in case 0 < α < 1/2.

P r o o f o f T h e o r e m 4 .

The following relation holds

∫_{−1}^{1} |x − y|^{α−1}(1 − y^{2})^{− \frac{α}{2}} Cm^{\frac{1−α}{2}} (y)dy = σ_{m}Cm^{\frac{1−α}{2}} (x), |x| ≤ 1,

where 0 < α < 2, σ_{m} = \frac{2Γ(α)Γ(m+1−α)}{m! cos \frac{απ}{2}}, see [16 ] (where it is given in a nondirect form). This may be rewritten as

A^{α}u_{m} = σ_{m} v_{m},

where u_{m} and v_{m} are the functions given in (40). Both {u_{m}} and {v_{m}} form a basis in the corresponding space. This is checked directly due to the well known orthogonality relations for the Gegenbauer polynomials. It remains to refer to the boundedness of the operator A^{α} = Γ(α)(I_{−1+}^{α} + I_{1−}^{α}) in the setting (39) which follows from Lemmas 1-2 in the case 0 < α < 1.

5. Returning to the Question.

Let us reformulate the Question put in Section 3 in the following form. Given s ∈ (0, α] , does there exist a pair of Hilbert spaces U and V of

functions defined on [0, 1] or R_{+}^{1} such that singular values of the operator I_{0+}^{α} : U → V have the asymptotics

σ_{m} ∼ m^{−s} ?

Naturally, in such a general setting this question is already very easy, definitely having a positive answer. Really, admitting spaces U and V of types completely independent of each other, we can always take the space V with better smoothness properties, which will immediately diminish order of decreasing of singular numbers. This is quite understandable in general, but to be precise, we give below the exact statement (Theorem 5).

Let H^{s} = H^{s}(R^{1}) = H^{s,2}(R^{1}) be the well known Sobolev space of fractional smoothness, known also as the space of Bessel potentials. It is defined as the completion of C_{0}^{∞} with respect to the norm

∥u∥_{H^{s}} = ∥u∥_{2} + ∥F^{−1}(1 + |ξ|^{2})^{1/2}F u∥_{2}  (53)

where F stands for the Fourier transform operator.

By H_{0}^{s}([0, 1]) we denote the space of restrictions of functions u(x) ∈ H^{s}(R^{1}) onto [0,1] satisfying the conditions

u^{(k)}(0) = 0, k = 0, 1, ..., [s − 1/2].

(H_{0}^{0} = L^{2}).

Theorem 5. Given any s ∈ [0, α], s ≠ α − 1/2, α − 3/2, ..., the singular numbers of the operator

I_{0+}^{α} : L^{2}([0, 1]) → H_{0}^{α−s}([0, 1])  (54)

have the asymptotics σ_{m} ∼ m^{−s}.

Proof. The key moment in the proof is in the following fact: if s ≠ 1/2, 3/2, ..., [α − 1/2], the space H_{0}^{s}([0, 1]) coincides with the range I_{0+}^{s}(L^{2}) of the fractional integration operator I_{0+}^{s} , considered in L^{2}([0, 1]), up to equivalence of the norm (53) to the natural norm in the range defined by

∥u∥_{I_{0+}^{s}(L^{2})} = ∥D_{0+}^{s}u∥_{L^{2}},

where D_{0+}^{α} is the fractional differentiation operator, left inverse to I_{0+}^{α} (see [19], Theorem 18.3 and Remark 18.1).

Therefore, the range H_{0}^{α−s}([0, 1]) used in (54) can be treated as a Hilbert space with the scalar product

(u, v)_{V} = (D_{0+}^{α−s}u, D_{0+}^{α−s}v)_{L^{2}}.  (55)

The operator I_{0+}^{s} : L^{2}([0, 1]) → L^{2}([0, 1]) is known to have the singular system {σ_{m}, u_{m}, v_{m}} with

σ_{m} ∼ m^{−s}

and some {u_{m}(x)} and {v_{m}(x)} :

I_{0+}^{s}u_{m}(x) = σ_{m}v_{m}(x).

Then we take U_{m}(x) = u_{m}(x) , V_{m}(x) = I_{0+}^{α−s}v_{m}(x). Obviously,

I_{0+}^{α}U_{m}(x) = σ_{m}V_{m}(x)

and it remains to note that the sequence V_{m}(x) is orthonormal in H_{0}^{α−s} with respect to the scalar product (55). □

6. Further comments.

In view of Theorems 1 and 2 it is interesting to find whether there exist weight function ρ_{λ}(x) and r_{λ}(x) depending, say, on a parameter λ such that the operator

I_{0+}^{α} : L^{2} (Ω; ρ_{λ}(x)) → L^{2} (Ω; r_{λ}(x))  (56)

has singular values with behaviour at infinity explicitly depending on the parameter λ.

This remains open for the fractional integration operator . Lemma 6 below shows that such an effect happens for an integral operator of another kind, with essentially better behaviour of the kernel. Anyhow, we find it important to show that such things really may happen in principle.

Let us consider the example

Ku(x) = ∫_{−∞}^{∞} e^{−(x−y)^{2}}u(y)dy, x ∈ R^{1}.  (57)

Since the kernel e^{−x^{2}} is infinitely differentiable, its s-numbers vanish at infinity more rapidly than any power. It is natural to expect that

σ_{m} ∼ q^{m}  (58)

for some q ∈ (0, 1). The following lemma shows that the number q determining the asymptotics (58), explicitly depends on the choice of weight functions.

We treat the operator (58) as

K : L^{2}(R^{1}; e^{−β^{2}x^{2}}) → L^{2}(R^{1}; e^{−γ^{2}x^{2}})  (59)

where β is an arbitrary number in (0,1) and γ = \frac{β}{(1−β^{2})^{1/2}} .

Lemma 6. The singular values of the operator (59) are equal to

σ_{n} = π^{1/2}q^{n+1/2}  (60)

where q = (1 − β^{2})^{1/2} and

u_{m}(x) = (\frac{β}{2^{m}m!π^{1/2}} )^{1/2} H_{m}(βx),  (61)

v_{m}(x) = (\frac{γ}{2^{m}m!π^{1/2}} )^{1/2} H_{m}(γx),  (62)

the H_{n}(x) being the Hermite polynomials.

Proof. The Hermite polynomials form an orthogonal basis in the space L^{2}(R^{1}; e^{−x^{2}}) :

∫_{−∞}^{∞} e^{−x^{2}}H_{k}(x)H_{m}(x)dx = c_{m}δ_{mk} , c_{m} = 2^{m}m!π^{1/2}.

Hence it is easily derived that the sequences (61) - (62) are orthonormal basises in the spaces L^{2}(R^{1}; e^{−β^{2}x^{2}}) and L^{2}(R^{1}; e^{−γ^{2}x^{2}}), respectively. So, it remains to check the relation Ku_{m} = σ_{m}v_{m} that is

∫_{−∞}^{∞} e^{−(x−y)^{2}}H_{m}(βy)dy = π^{1/2}(1 − β^{2})^{m/2}H_{m}(γx)  (63)

which is known, see [8], 7.374.8, but may be also checked directly. Really, we may make use of the generating function for the Hermite polynomials:

e^{−t^{2}+2ty} = ∑_{k=0}^{∞} \frac{t^{k}}{k!}H_{k}(y)  (64)

(see [8], 8.957.1). Replacing here y by αy , multiplying by e^{−(x−y)^{2}} and integrating and then using the generating series (64) again, we arrive at (63). □

Remark 4. It is well known that convolution operators with a summable kernel definitely are not compact in L^{p}(R^{1}), 1 ≤ p ≤ ∞, see e.g. [9]. Since lim_{n→∞}σ_{n} = 0 if and only if an operator is compact ([4], p.62, Corollary 7.1), this means that the operator K in the setting (59) should be certainly compact. The latter may be checked directly. Really, by Remark 2, the setting (59) is equivalent to K_{1} : L^{2}(R^{1}) → L^{2}(R^{1}) for

K_{1}u = e^{− \frac{γ^{2}x^{2}}{2}} ∫_{−∞}^{∞} e^{\frac{β^{2}t^{2}}{2}} e^{−(x−t)^{2}}u(t)dt .

We transform K_{1} to

K_{1}u = e^{−c^{2}x^{2}} ∫_{−∞}^{∞} e^{−(bx−ay)^{2}}u(t)dt

with a = (1 − \frac{β^{2}}{2} )^{1/2}, b = 1/a, c = β^{2} [2(1 − β^{2})(2 − β^{2})]^{−1/2} . Then the operator K_{1} is compact because of the factor e^{−c^{2}x^{2}} (see e.g. [9]).

Acknowledgements The second named author was supported by a DAAD grant during his visit at Freie Universität Berlin in November and December 1995. From the side of the first named author this work is also embedded in the project ”Fractional Order Systems” , supported by NATO Collaborative Research Grant 940 843 (R.Gorenflo and R.Rutman) and in the research project ”Convolutions”, supported by the Research Commission of Free University of Berlin.

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Addresses of authors:

1) Prof. Dr. Rudolf Gorenflo First Mathematical Institute Free University of Berlin Arnimallee 2-6 D 14195 Berlin , Germany e-mail: gorenflo@math.fu-berlin.de

2) Prof. Dr. Stefan Samko Department of Mathematics and Mechanics Rostov State University Zorge 5 , Rostov-na-Donu, 344004, Russia Currently at Unidade de Ciencias Exactas e Humanas Universidade de Algarve Campos de Gambelas Faro 8000 , Portugal