Georgian Mathematical Journal Volume 8 (2001), Number 2, 323–332

ASYMPTOTIC BEHAVIOR OF SINGULAR AND ENTROPY

NUMBERS FOR SOME RIEMANN–LIOUVILLE TYPE OPERATORS

A. MESKHI

Abstract. The asymptotic behavior of the singular and entropy numbers is established for the Erdelyi–Köber and Hadamard integral operators (see, e.g., [15]) acting in weighted L^{2} spaces. In some cases singular value decompositions are obtained as well for these integral transforms.

2000 Mathematics Subject Classification: 47B06, 47B34, 47H50. Key words and phrases: Singular numbers, entropy numbers, Riemann– Liouville operators.

In this paper, we investigate the asymptotic behavior of singular and entropy numbers for the following integral operators:

I_{α,σ}f(x) = \frac{1}{Γ(α)} ∫_{0}^{x}(x^{σ} − y^{σ})^{α−1}f(y) dy, x > 0, α > 0, σ > 0,

(Erdelyi–Köber operator) and

H_{α}f(x) = \frac{1}{Γ(α)} ∫_{1}^{x} ( ln \frac{x}{y})^{α−1}f(y) dy, x > 1, α > 0,

(Hadamard operator) in some weighted L^{2} spaces. We get singular value decompositions for these integral transforms.

Analogous problems for the Riemann–Liouville operator

R_{α}f(x) = \frac{1}{Γ(α)} ∫_{0}^{x}(x − y)^{α−1}f(y) dy, α > 0,

were studied in [1]–[6]. We refer also to [7]–[8], where some powerful tools were developed for establishing the asymptotics of singular numbers of certain pseudo-differential operators (see also [9] for some properties of singular numbers for the weighted Riemann–Liouville operator R_{α,v}f(x) ≡ v(x)R_{α}f(x), where α > 1/2).

Two-sided estimates of singular (approximation) numbers for the weighted

Hardy operator H_{v,w}f(x) = v(x) ∫_{0}^{x} f(y)w(y) dy were given in [10]–[12] (for some

related topics concerning the weighted Volterra integral operators see [13], [14]).

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Note that some mapping properties of the operators I_{α,σ} and H_{α} were established in [15].

Let A and B be infinite-dimensional Hilbert spaces. It is known that if K : A → B is an injective compact linear operator, then there exist:

(a) an orthonormal basis {u_{j}}_{Z_{+}} in A;

(b) an orthonormal basis {v_{j}}_{Z_{+}} in B;

(c) a nonincreasing sequence {s_{j}(K)}_{Z_{+}} of positive numbers with limit 0 as j → +∞ such that

Ku_{j} = s_{j}(K)v_{j}, j ∈ Z_{+}.

The numbers s_{j}(K) are known as singular numbers or s-numbers of the operator K, the system {s_{j}(K), u_{j}, v_{j}}_{j∈Z_{+}} is called a singular system of K. For the operator K the singular value decomposition

Kf = ∑_{j=0}^{∞} s_{j}(K)(f, u_{j})_{A}v_{j}, f ∈ A,

is valid.

Let w be a measurable a.e. positive function on Ω ⊂ R_{+}. We denote by L_{w}^{2}(Ω) the class of all measurable functions f : Ω → R_{+} for which

∥f∥_{L_{w}^{2}(Ω)} = ( ∫_{Ω} |f(x)|^{2}w(x) dx)^{1/2} < ∞.

In the sequel by writing a_{n} ≈ b_{n} for sequences of positive numbers a_{n} and b_{n} we mean that there exist positive constants c_{1} and c_{2} such that c_{1} ≤ a_{n}/b_{n} ≤ c_{2} for all n ∈ ℕ.

The following result is well-known (see [5]):

Theorem A. Let α > 0, β > −1, φ(t) = t^{−β}e^{−t}, ψ(t) = t^{−(α+β)}e^{−t}. Then the singular system {s_{j}(R_{α}), u_{j}, v_{j}}_{j∈Z_{+}} of the operator R_{α} : L_{φ}^{2}(R_{+}) → L_{ψ}^{2}(R_{+}) is given by

s_{n}(R_{α}) = (\frac{Γ(n + β + 1)}{Γ(n + α + β + 1)})^{1/2},  (1)
u_{n}(t) = (\frac{n!}{Γ(n + β + 1)})^{1/2}t^{β}L_{n}^{(β)}(t),
v_{n}(t) = (\frac{n!}{Γ(n + α + β + 1)})^{1/2}t^{α+β}L_{n}^{(α+β)}(t),

and s_{n}(R_{α})/n^{−α/2} → 1 as n → ∞, where L_{n}^{(γ)} is the Laguerre polynomial:

L_{n}^{(γ)}(x) = ∑_{k=0}^{n}(−1)^{k}\binom{n + γ}{n − k}\frac{x^{k}}{k!} , γ > −1, n ∈ Z_{+}.

ASYMPTOTIC BEHAVIOR OF SINGULAR AND ENTROPY NUMBERS 325

Theorem B ([4]). Let α > 0, λ > α − 1/2, λ ≠ 0. Then the operator R_{α} : L_{φ}^{2}(R_{+}) → L_{ψ}^{2}(R_{+}), where φ(x) = x^{1/2−λ}(1 + x)^{2α}, ψ(x) = x^{1/2−λ−α}, has the following singular system:

s_{n}(R_{α}) = (\frac{Γ(n + λ − α + 1/2)}{Γ(n + λ + α + 1/2)})^{1/2},  (2)
u_{n}(t) = 2^{λ}a_{n}t^{λ−1/2}(1 + t)^{−λ−α−1/2}C_{n}^{λ}(\frac{1 − t}{1 + t}),
v_{n}(t) = 2^{λ}b_{n}t^{λ+α−1/2}(1 + t)^{−λ−α−3/2}P_{n}^{(λ−α−1/2,λ+α−1/2)}(\frac{1 − t}{1 + t}),

where

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

C_{n}^{λ}(t) is the Gegenbauer polynomial

C_{n}^{λ}(t) = \frac{1}{Γ(α)} ∑_{j=0}^{[n/2]}(−1)^{j} \frac{Γ(α + n − j)}{j!(n − 2j)!}(2t)^{n−2j},

and P_{m}^{(α,β)} is the Jacobi polynomial

P_{n}^{(α,β)}(t) = 2^{−n} ∑_{m=0}^{n} \binom{n + α}{m}\binom{n + β}{n − m}(t − 1)^{n−m}(t + 1)^{m}, n ∈ Z_{+}.

Moreover, lim_{n→∞} s_{n}(R_{α})/n^{−α} = 1.

Theorem C ([6]). The singular values of the operator R_{α} : L^{2}(0, 1) → L_{x^{−γ}}^{2}(0, 1) have the following asymptotics:

s_{n}(R_{α}) ≈ n^{−α}, 0 ≤ γ < α.

When γ = 0, the upper estimate in the previous statement was derived in [1], [2], while the lower estimate was given in [2].

The following lemma follows immediately:

Lemma 1. Let φ, ψ, v and w be measurable a.e. positive functions on Ω ⊆ R_{+}. Then the operator A is compact from L_{φ}^{2}(Ω) to L_{ψ}^{2}(Ω) if and only if the operator A_{1}f(x) = v^{1/2}(x)A(fw^{−1/2})(x) is compact from L_{φw^{−1}}^{2}(Ω) to L_{ψv^{−1}}^{2}(Ω).

Taking into account the definition of the singular system of the operator, we easily derive the next statement.

Lemma 2. Let v and w be a.e. positive measurable functions on Ω ⊆ R_{+}. A system {s_{j}(A), u_{j}, v_{j}}_{j∈Z_{+}} is a singular system for the operator A : L_{φ}^{2}(Ω) → L_{ψ}^{2}(Ω) if and only if the operator A_{1} : L_{φw^{−1}}^{2}(Ω) → L_{ψv^{−1}}^{2}(Ω) has the singular system {s_{j}(A_{1}), w^{1/2}u_{j}, v^{1/2}v_{j}}_{j∈Z_{+}}, where A_{1}f(x) = v^{1/2}(x)A(fw^{−1/2})(x) and s_{j}(A_{1}) = s_{j}(A).

Let I_{α,σ}f(x) = I_{α,σ}(fρ)(x), where ρ(y) = y^{σ−1}, α > 0, σ > 0 and x > 0. From the definition of compactness we easily deduce

Lemma 3. Let α > 0, σ > 0 and let Ω = (0, 1) or Ω = (0, ∞). Assume that v and w are measurable a.e. positive functions on Ω. Then the operator I_{α,σ} is compact from L_{w}^{2}(Ω) to L_{v}^{2}(Ω) if and only if R_{α} is compact from L_{W}^{2} (Ω) to L_{V}^{2} (Ω), where W (x) = w(x^{1/σ})x^{1/σ−1}, V (x) = v(x^{1/σ})x^{1/σ−1}.

Now we prove the following statement:

Lemma 4. Let α > 0, σ > 0 and let v and w be measurable a.e. positive functions on Ω, where Ω = (0, ∞) or Ω = (0, 1). Then for the singular system \{s_{j}(I_{α,σ}\}, \overline{u}_{j}, \overline{v}_{j}\}_{j∈Z_{+}} of the operator I_{α,σ} : L_{w}^{2}(Ω) → L_{v}^{2}(Ω) we have s_{j}(I_{α,σ}) = σ^{−1}s_{j}(R_{α}), \overline{u}_{j}(x) = σ^{1/2}u_{j}(x^{σ}), \overline{v}_{j}(x) = σ^{1/2}v_{j}(x^{σ}), where \{s_{j}(R_{α}), u_{j}, v_{j}\}_{j∈Z_{+}} is a singular system for the operator R_{α} : L_{W}^{2} (0, ∞) → L_{V}^{2} (0, ∞), with W (x) = w(x^{1/σ})x^{1/σ−1} and V (x) = v(x^{1/σ})x^{1/σ−1}.

Proof. Let Ω = (0, ∞). Using the change of variable y = t^{1/σ}, we have

(I_{α,σ}\overline{u}_{j})(x) = \frac{1}{Γ(α)} ∫_{0}^{x}(x^{σ} − y^{σ})^{α−1}y^{σ−1}\overline{u}_{j}(y) dy

= \frac{σ^{1/2}}{Γ(α)} ∫_{0}^{x}(x^{σ} − y^{σ})^{α−1}u_{j}(y^{σ})y^{σ−1} dy = \frac{σ^{−1/2}}{Γ(α)} ∫_{0}^{x^{σ}}(x^{σ} − t)^{α−1}u_{j}(t) dt

= σ^{−1/2}(R_{α}u_{j})(x^{σ}) = s_{j}(R_{α})σ^{−1/2}v_{j}(x^{σ}) = σ^{−1}s_{j}(R_{α})\overline{v}_{j}(x).

Further, the change of variable yields

∫_{0}^{∞}\overline{v}_{j}(x)\overline{v}_{i}(x)v(x) dx = σ ∫_{0}^{∞}v_{j}(x^{σ})v_{i}(x^{σ})V (x^{σ})x^{σ−1}dx

= ∫_{0}^{∞}v_{j}(x)v_{i}(x)V (x)dx = δ_{ij},

where δ_{ij} denotes Kronecker’s symbol.

Analogously, we have

∫_{0}^{∞}\overline{u}_{j}(x)\overline{u}_{i}(x)w(x) dx = ∫_{0}^{∞}u_{j}(x)u_{i}(x)W (x) dx = δ_{ij},

Hence \{\overline{v}_{j}\} and \{\overline{u}_{j}\} are orthonormal systems in L_{v}^{2}(R_{+}) and L_{w}^{2}(R_{+}), respectively.

The case Ω = (0, 1) follows in a similar way.

Theorem 1. Let α > 0, σ > 0 and 0 ≤ γ < α. Then there exist positive constants c_{1} and c_{2} depending on α, σ and γ such that for the singular numbers of the operator I_{α,σ} : L_{x^{1−σ}}^{2}(0, 1) → L_{x^{σ−1−γσ}}^{2}(0, 1) we have s_{n}(I_{α,σ}) ≈ n^{−α}.

Proof. By Lemma 2 we have that s_{j}(I_{α,σ}) = s_{j}(I_{α,σ}), where I_{α,σ} acts from L_{x^{σ−1}}^{2}(0, 1) to L_{x^{σ−1−γσ}}^{2}(0, 1), while Lemma 4 yields s_{j}(I_{α,σ}) = 1/σs_{j}(R_{α}), where R_{α} is the Riemann–Liouville operator acting from L^{2}(0, 1) to L_{x^{−γ}}^{2}(0, 1). Theorem C completes the proof.

Theorem 2. Let α > 0, σ > 0, λ > α − 1/2 and λ ≠ 0. Assume that w(x) = x^{1−σ/2−σλ}(1 + x^{σ})^{2α}, v(x) = x^{3σ/2−σλ−σα−1}. Then the operator I_{α,σ} : L_{w}^{2}(0, ∞) → L_{v}^{2}(0, ∞) has a singular system \{s_{n}(I_{α,σ}), \overline{u}_{n}, \overline{v}_{n}\}_{n∈Z_{+}}, where

s_{n}(I_{α,σ}) = 1/σ(\frac{Γ(n + λ − α + 1/2)}{Γ(n + λ + α + 1/2)})^{1/2},
\overline{u}_{n}(x) = σ^{1/2}2^{λ}a_{n}x^{σ(λ+1/2)−1}(1 + x^{σ})^{−λ−α−1/2}C_{n}^{λ}(\frac{1 − x^{σ}}{1 + x^{σ}} ),
\overline{v}_{n}(x) = σ^{1/2}2^{λ}b_{n}x^{σ(λ+α−1/2)}(1 + x^{σ})^{−λ−α−3/2}P_{n}^{(λ−α−1/2,λ+α−1/2)}(\frac{1 − x^{σ}}{1 + x^{σ}} ),

C_{n}^{λ}(x) and P_{n}^{(α,β)} are Gegenbauer and Jacobi polynomials, respevetively (see Theorem B), and a_{n}, b_{n} are the constants defined in Theorem B. Moreover,

lim_{n→∞} s_{n}(I_{α,σ})/n^{−α} = 1/σ.

Proof. Lemma 2 implies that the singular system \{s_{m}(I_{α,σ}), \overline{u}_{m}, \overline{v}_{m}\}_{m∈Z_{+}} of the map I_{α,σ}:L_{w}^{2}(0, ∞)→L_{v}^{2}(0, ∞) coincides with the singular system {s_{m}(I_{α,σ}), ũ_{m}, ṽ_{m}}_{m∈Z_{+}} of the map I_{α,σ} : L_{W}^{2} (0, ∞) → L_{V}^{2} (0, ∞), where W (x) = w(x)x^{2(σ−1)}, V (x) = v(x), ũ_{m}(x) = x^{1−σ}u_{m}(x), ṽ_{m}(x) = \overline{v}_{m}(x). Further, by Lemma 4 we have that the operator R_{α} : L_{φ}^{2}(0, ∞) → L_{ψ}^{2}(0, ∞) (φ(x) = x^{1/2−λ}(1 + x)^{2α}, ψ(x) = x^{1/2−λ−α}) has a singular system {s_{m}(R_{α}), u_{m}, v_{m}}_{m∈Z_{+}}, where

s_{m}(R_{α})=σs_{m}(I_{ασ}) ≈ m^{−α}, \overline{u}_{m}(x)=σ^{1/2}x^{σ−1}u_{m}(x^{σ}), \overline{v}_{m}(x)=σ^{1/2}v_{m}(x^{σ}).

Analogously, we have

Theorem 3. Let α > 0, σ > 0, β > −1, w(y) = y^{−σβ−σ+1}e^{−y^{σ}} and v(y) = y^{−σ(α+β)+σ−1}e^{−y^{σ}}. Then the operator I_{α,σ} : L_{w}^{2}(0, ∞) → L_{v}^{2}(0, ∞) has a singular system \{s_{m}(I_{α,σ}), \overline{u}_{m}, \overline{v}_{m}\}_{m∈Z_{+}} defined by

s_{n}(I_{α,σ}) = 1/σ(\frac{Γ(n + β + 1)}{Γ(n + α + β + 1)})^{1/2},
\overline{u}_{n}(x) = σ^{1/2}x^{σ−1+σβ}(\frac{n!}{Γ(n + β + 1)})^{1/2}L_{n}^{(β)}(x^{σ}),
\overline{v}_{n}(x) = σ^{1/2}(\frac{n!}{Γ(n + α + β + 1)})^{1/2}x^{σ(α+β)}L_{n}^{(α+β)}(x^{σ}),

where L_{n}^{(γ)}(x) is a Laguerre polynomial (see Theorem A). Moreover,

lim_{n→∞} s_{n}(I_{α,σ})/n^{−α/2} = 1/σ.

Now we consider the operator of Hadamard’s type H_{α}.

The following lemma holds:

Lemma 5. Let α > 0 and (v, w) be a pair of weights defined on (1, ∞). Then \{s_{m}(L_{α}), \overline{u}_{m}, \overline{v}_{m}\}_{m∈Z_{+}} is a singular system for the operator L_{α} : L_{w}^{2}(1, ∞) → L_{v}^{2}(1, ∞), where

L_{α}f(x) = \frac{1}{Γ(α)} ∫_{1}^{x} ( ln \frac{x}{y})^{α−1}f(y) \frac{dy}{y} ,

if and only if the Riemann–Liouville operator R_{α} : L_{W}^{2} (0, ∞) → L_{V}^{2} (0, ∞) has a singular system {s_{m}(R_{α}), ũ_{m}, ṽ_{m}}_{m∈Z_{+}}, where W (x) = w(e^{x})e^{x}, V (x) = v(e^{x})e^{x}, s_{m}(R_{α}) = s_{m}(L_{α}), ũ_{m}(x) = \overline{u}_{m}(e^{x}), ṽ_{m}(x) = \overline{v}_{m}(e^{x}).

Proof. Using the change of variable y = e^{z} we have

(L_{α}\overline{u}_{m})(x) = \frac{1}{Γ(α)} ∫_{1}^{x} ( ln \frac{x}{y})^{α−1}\overline{u}_{m}(y) \frac{dy}{y}

= \frac{1}{Γ(α)} ∫_{0}^{ln x}(ln x − z)^{α−1}ũ_{m}(z) dz = (R_{α}ũ_{m})(ln x) = ṽ(ln x)s_{j}(R_{α}).

On the other hand,

∫_{0}^{∞}ũ_{i}(x)ũ_{j}(x)W (x) dx = ∫_{0}^{∞}\overline{u}_{i}(e^{x})\overline{u}_{j}(e^{x})w(e^{x})e^{x} dx = δ_{ij},

∫_{0}^{∞}ṽ_{i}(x)ṽ_{j}(x)V (x) dx = ∫_{1}^{∞}\overline{v}_{i}(y)\overline{v}_{j}(y)v(y) dy = δ_{ij},

where δ_{ij} is Kronecker’s symbol.

Lemmas 2 and 5 yield the following statements:

Theorem 4. Let α>0, β >−1, w(x)=ln^{−β} x, v(x)=x^{−2} ln^{−(α+β)} x. Then the operator H_{α}:L_{w}^{2}(1, ∞)→L_{v}^{2}(1, ∞) has a singular system {s_{n}(H_{α}), ũ_{n}, ṽ_{n}}_{n∈Z_{+}}, where s_{n}(H_{α}) = s_{n}(R_{α}) (s_{m}(R_{α}) is defined by (1)),

ũ_{n}(x) = x^{−1}(\frac{n!}{Γ(n + β + 1)})^{1/2}L_{n}^{(β)}(ln x) ln^{β} x,
ṽ_{n}(x) = (\frac{n!}{Γ(n + α + β + 1)})^{1/2}L_{n}^{(α+β)}(ln x) ln^{α+β} x,

and L_{n}^{(γ)} is the Laguerre polynomial. Moreover,

lim_{n→∞} s_{n}(H_{α})/n^{−α/2} = 1.

Theorem 5. Let λ>α−\frac{1}{2}, λ≠0. Then the operator H_{α}:L_{w}^{2}(1, ∞)→L_{v}^{2}(1, ∞) has a singular system {s_{n}(H_{α}), ũ_{m}, ṽ_{n}}_{m∈Z_{+}}, where v(x) = x^{−1} ln^{1/2−λ−α} x, w(x) = (1 + ln x)^{2α}x ln^{1/2−λ} x, s_{n}(H_{α}) = s_{n}(R_{α}) (s_{n}(R_{α}) is defined by (2)),

ũ_{n}(x) = 2^{λ}a_{n}(1 + ln x)^{−λ−α−1/2}C_{n}^{λ}(\frac{1 − ln x}{1 + ln x})x^{−1} ln^{λ−1/2} x,
v_{n}(x) = 2^{λ}b_{n}(1 + ln x)^{−λ−α−3/2}P_{n}^{(λ−α−1/2,λ+α−1/2)}(\frac{1 − ln x}{1 + ln x}) ln^{λ+α−1/2} x.

Moreover,

lim_{n→∞} s_{n}(H_{α})/n^{−α} = 1.

Definition 1. Let X and Y be Banach spaces and let T be a bounded linear map from X to Y. Then for all k ∈ N, the k^{th} entropy number e_{k}(T) of T is defined by

e_{k}(T) = inf {ε > 0 : T (U_{X}) ⊂ ⋃_{j=1}^{2^{k−1}}(b_{i} + εU_{Y}) for some b_{1}, …, b_{2^{k−1}} ∈ Y },

where U_{X} and U_{Y} are the closed unit balls in X and Y, respectively.

It is easy to verify that ∥T ∥ = e_{1}(T) ≥ e_{2}(T) ≥ ⋯ ≥ 0.

For other properties of the entropy numbers see, e.g., [16].

It is known (see, e.g., [15]), that if T is a compact linear map of a Hilbert space X into a Hilbert space Y, then s_{n}(T) ≈ n^{−λ} if and only if e_{n}(T) ≈ n^{−λ}. Hence we can get asymptotics of the entropy numbers for the operators I_{α,σ} and H_{α}. In particular, Theorems 1, 2 and 3 yield

Proposition 1. Let α > 0 and σ > 0. Then the following statements are valid:

(a) If 0 ≤ γ < α, then the asymptotic formula

e_{n}(I_{α,σ}) ≈ n^{−α}  (3)

holds for the operator I_{α,σ} : L_{x^{1−σ}}^{2}(0, 1) → L_{x^{σ−1−γσ}}^{2}(0, 1).

(b) Assume that λ > α − 1/2 and λ ≠ 0. Then the asymptotic formula (3) is valid for the map I_{α,σ} : L_{w}^{2}(0, ∞) → L_{v}^{2}(0, ∞), where w(x) = x^{−σ/2−σλ+1}(1 + x^{σ})^{2α} and v(x) = x^{3σ/2−σλ−σα−1}.

(c) For the entropy numbers e_{n}(I_{α,σ}) of the operator I_{α,σ} : L_{w}^{2}(0, ∞) → L_{v}^{2}(0, ∞) (w(y) = y^{−σβ−σ+1}e^{−y^{σ}}, v(y) = y^{−σ(α+β)+σ−1}e^{−y^{σ}}, β > −1) we have

e_{n}(I_{α,σ}) ≈ n^{−α/2}.

Let T : L_{w}^{2} → L_{v}^{2} be a compact linear operator. We shall denote by n(t, T) the distribution function of singular values for the operator T, i.e.,

n(t, T) ≡ ♯{k : s_{k}(T) > t}.

Theorem 6. Let α > 1/2 and σ > 0. Assume that v is a measurable a.e. positive function of (0, ∞) satisfying the condition

∑_{k∈ℤ}( ∫_{2^{k/σ}}^{2^{(k+1)/σ}}v(y)y^{(2α−1)σ} dy)^{1/(2α)} < ∞.  (4)

Then for the operator I_{α,σ} : L_{w}^{2}(R_{+}) → L_{v}^{2}(R_{+}), where w(x) = x^{1−σ}, the asymptotic formula

lim_{t→0} t^{1/α}n(t, I_{α,σ}) = \frac{σ^{−1/α+1}}{π} ∫_{0}^{∞}v^{1/(2α)}(y)y^{(1−σ)(1/(2α)−1)} dy

holds.

Proof. Condition (4) implies that

∑_{k∈ℤ}( ∫_{2^{k}}^{2^{k+1}}\overline{v}^{2}(y)y^{2α−1} dy)^{1/(2α)} < ∞,  (5)

where \overline{v}(x) ≡ [v(x^{1/σ})x^{1/σ−1}]^{1/2}. By virtue of Theorem 1 from [9] we have that for the operator R_{α,\overline{v}} : L^{2}(R_{+}) → L^{2}(R_{+}), where R_{α,\overline{v}}f(x) ≡ \overline{v}(x)R_{α}f(x), the asymptotic formula

lim_{t→0} t^{1/α}n(t, R_{α,\overline{v}}) = π^{−1} ∫_{R_{+}} \overline{v}^{1/α}(x) dx

holds. Further, using Lemmas 1, 2 and 3 we obtain that s_{k}(R_{α,\overline{v}}) = σ · s_{k}(I_{α,σ}). Consequently,

lim_{t→0} t^{1/α}n(t, I_{α,σ}) = σ^{−1/α} lim_{t→0} t^{1/α}n(t, R_{α,\overline{v}})

= σ^{−1/α} \frac{1}{π} ∫_{0}^{∞}(\overline{v}(x))^{1/α} dx = \frac{σ^{−1/α+1}}{π} ∫_{0}^{∞}(v(y))^{1/(2α)}y^{(1−σ)(1/(2α)−1)} dy.

Theorem 7. Let α > 1/2 and σ > 0. Suppose that v is a measurable a.e. positive function on (0, 1) satisfying the condition

∑_{k∈ℤ}( ∫_{a_{k}}^{a_{k+1}}v(x)x^{−σ+2ασ}(1 − x^{σ})^{−1} dx)^{1/(2α)} < ∞, a_{k} = (2^{k}/(2^{k} + 1))^{1/σ}. (6)

Then for the operator I_{α,σ} acting from L_{w}^{2}(0, 1) into L_{v}^{2}(0, 1), where w(x) = (1 − x^{σ})^{2α}x^{1−σ}, we have

lim_{t→0} t^{1/α}n(t, I_{α,σ}) = \frac{σ^{−1/α+1}}{π} ∫_{0}^{1} v^{1/(2α)}(x)x^{(1−σ)(1/(2α)−1)}(1 − x^{σ})^{−1} dx.

Proof. Using Lemmas 1–4 we have that s_{n}(I_{α,σ}) = 1/σs_{n}(R_{α}), where R_{α} is the Riemann–Liouville operator acting from L_{w_{1}}^{2}(0, 1) into L_{v_{1}}^{2}(0, 1), with

w_{1}(x) = w(x^{1/σ})x^{1−1/σ}, v_{1}(x) = v(x^{1/σ})x^{1/σ−1}.

Further, by the change of variable x = y/(1 − y) we obtain that the operator \overline{R}_{α} : L_{w_{2}}^{2}(R_{+}) → L_{v_{2}}^{2}(R_{+}) has singular numbers s_{n}(\overline{R}_{α}) = σs_{n}(I_{α,σ}), where w_{2}(x) = w_{1}(x/(x + 1))(x + 1)^{−2}, v_{2}(x) = v_{1}(x/(x + 1))(x + 1)^{−2} and \overline{R}_{α}f(x) = ψ(x)R_{α}(fφ)(x) with ψ(x) = (x + 1)^{−α+1}, φ(x) = (x + 1)^{−1−α}. Hence for the singular numbers of the Riemann–Liouville operator R_{α} : L_{w_{3}}^{2}(R_{+}) → L_{v_{3}}^{2}(R_{+}) we derive s_{n}(R_{α}) = σs_{n}(I_{α,σ}), where w_{3}(x) = w_{2}(x)(x + 1)^{2α+2} = 1 and v_{3}(x) = v_{2}(x)(x + 1)^{2−2α}. Further, condition (6) implies (5) with v_{3} instead of v. Thus, taking into account Theorem 1 from [9], we arrive at

lim_{t→0} t^{1/α}n(t, I_{α,σ}) = σ^{−1/α} lim_{t→0} t^{1/α}n(t, R_{α})

= σ^{−1/α} \frac{1}{π} ∫_{0}^{∞}v_{4}^{1/α}(x) dx = \frac{σ^{−1/α+1}}{π} ∫_{0}^{1}(v(y))^{1/(2α)}y^{(1−σ)(1/(2α)−1)}(1 − y^{σ})^{−1} dy.

In the last equality we used the change of variable twice.

Finally, we have

Theorem 8. Let α > 1/2 and let v be a measurable a.e. positive function on (1, ∞) satisfying the condition

∑_{k∈ℤ}( ∫_{a_{k}}^{a_{k+1}}v(x) ln^{2α−1} x dx)^{1/(2α)} < ∞, a_{k} = e^{2^{k}}.  (7)

Then for the operator H_{α} : L_{w}^{2}(1, ∞) → L_{v}^{2}(1, ∞), where w(x) = e^{x}, the asymptotic formula

lim_{t→0} t^{1/α}n(t, H_{α,σ}) = \frac{1}{π} ∫_{1}^{∞}v^{1/(2α)}(x)x^{1/(2α)−1} dy  (8)

holds.

Proof. Taking into account Lemmas 2 and 5 we obtain that s_{n}(R_{α}) = s_{n}(H_{α}), where R_{α} is the Riemann–Liouville operator acting from L^{2}(R_{+}) into L_{v_{1}}^{2}(R_{+}), v_{1}(x) = v(e^{x})e^{x}. By condition (7), Theorem 1 from [9] and the change of variable x = e^{y} we conclude that (8) holds.

Acknowledgement

The work was partially supported by Grant No. 1.7 of the Georgian Academy of Sciences.

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(Received 20.03.2001)

Author’s address: A. Razmadze Mathematical Institute, Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093, Georgia E-mail: meskhi@rmi.acnet.ge