c Pleiades Publishing, Ltd., 2016. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2016, Vol. 293, pp. 255–271.  c R. Oinarov, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 293, pp. 263–279. Original Russian Text 

Boundedness and Compactness of a Class of Convolution Integral Operators of Fractional Integration Type R. Oinarov a Received October 6, 2015

Abstract—For a class of convolution integral operators whose kernels may have integrable singularities, boundedness and compactness criteria in weighted Lebesgue spaces are obtained. DOI: 10.1134/S0081543816040180

1. INTRODUCTION Let 1 < p < ∞, 0 < q < ∞, 1/p + 1/p = 1, and Lp := Lp (I) be the space of all measurable functions f on I := (0, ∞) with finite norm ⎛∞ ⎞1/p  f p = ⎝ |f (x)|p dx⎠ . 0

Let u(·) and v(·) be nonnegative weight functions that are locally integrable on I. In this paper, we consider the Lp → Lq boundedness and compactness of convolution integral operators of the form x K f (x) = v(x) +

K− f (x) = v(x)

K(x − s)u(s)f (s) ds,

x > 0,

(1.1)

K(s − x)u(s)f (s) ds,

x > 0.

(1.2)

0 ∞

x

Here and in what follows, K(·) is a nonnegative nonincreasing function on I. In the case of K(x) = xα−1 , 0 < α < 1, the operators (1.1) and (1.2) turn into the Riemann– Liouville and Weyl operators, respectively, with weight functions u and v: x Iα f (x) = v(x) 0 ∞

Wα f (x) = v(x) x

u(s)f (s) ds , (x − s)1−α

x > 0,

(1.3)

u(s)f (s) ds , (s − x)1−α

x > 0.

(1.4)

The problem of boundedness and compactness of operators (1.3) and (1.4) from Lp to Lq was studied by many authors. A brief survey of these studies is presented in [1]. a L.N. Gumilyov Eurasian National University, Satpayev Str. 2, Astana, 010008 Kazakhstan.

E-mail address: [email protected]

255

256

R. OINAROV

In 1997, under the assumptions that limx→∞ K(x) = 0 and K(x) ≤ CK(2x) for all x ∈ I with some constant C > 0, Lorente [2] established a boundedness criterion for the operator (1.1) from Lp to Lq for 1 < p ≤ q < ∞, which implies an Lp → Lq boundedness criterion for the Riemann– Liouville operator (1.3) in the same range of parameters p and q. However, due to the implicit character of the conditions of Lp → Lq boundedness of the operator (1.1) in [2], these conditions are difficult to verify. Therefore, the goal of subsequent studies was to obtain explicit criteria for the Lp → Lq boundedness of the operator (1.3). In the case of 0 < q < ∞, 1 < p < ∞, α > 1/p, and u(·) ≡ 1, explicit Lp → Lq boundedness criteria for the operators (1.3) and (1.4) were obtained independently by Meskhi [3] and Prokhorov [4]. A generalization of these results to the case when u(·) is a nonincreasing function was given by Farsani [5]. In [1], Prokhorov and Stepanov established criteria for the Lp → Lq boundedness and compactness of the operator (1.3) for 1 < p ≤ q < ∞ in the following cases: (a) 1 − q  /p < α ≤ 1, and u is a nondecreasing function; (b) 1 − p/q < α ≤ 1, and u is a nonincreasing function. Rautian [6] extended these Lp → Lq boundedness criteria for the operator (1.3) to the case of the convolution (1.1). The goal of the present study is to generalize the results of [3–5], which correspond to the case of 1/p < α < 1, to the operators (1.1) and (1.2). The results obtained are new for the operators (1.3) and (1.4) as well. The paper is organized as follows. In Section 2, we gather auxiliary statements that are needed to prove the main results; in Sections 3 and 4, we obtain boundedness criteria for the operators (1.1) and (1.2), respectively; in Section 5, we give necessary and sufficient conditions for the compactness of the operators (1.1) and (1.2). In the concluding Section 6, we apply the results to the problem of boundedness and compactness of the operators (1.3) and (1.4). Everywhere below we assume that indeterminacy of the form 0 · ∞ equals zero. The relation A  B means that A ≤ cB, where the constant c > 0 may only depend on inessential parameters. We write A ≈ B if A  B  A. By χ(c,d) (·) we denote the characteristic function of an interval (c, d) ⊂ I. In addition, without loss of generality, we assume that the functions f are nonnegative. 2. AUXILIARY STATEMENTS Consider the weighted Hardy inequalities ⎞q ⎞1/q ⎛∞ ⎞1/p ⎛ ∞⎛  x  ⎝ ⎝v(x)K(x) u(s)f (s) ds⎠ dx⎠ ≤ C ⎝ f p (t) dt⎠ , 0

0

(2.1)

0

⎞q ⎞1/q ⎛∞ ⎞1/p ⎛ ∞⎛  ∞  ⎝ ⎝v(x) K(s)u(s)f (s) ds⎠ dx⎠ ≤ C ⎝ f p (t) dt⎠ . 0

x

(2.2)

0

The following results on Hardy inequalities are well known (see [7] and remarks in [8]). Theorem 2.1. For 1 < p ≤ q < ∞, inequality (2.1) is valid if and only if ⎛ A+ = sup ⎝ z>0

∞ z

⎞1/q ⎛ z ⎞1/p   v q (x)K q (x) dx⎠ ⎝ up (s) ds⎠ < ∞; 0

moreover, C ≈ A+ , where C is the best constant in (2.1). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Theorem 2.2. For 0 < q < p < ∞ and p > 1, inequality (2.1) is valid if and only if ⎛ ∞⎛ ∞ ⎞q/(p−q) ⎛ z ⎞q(p−1)/(p−q) ⎞(p−q)/(pq)    ⎝ up (s) ds⎠ v q (z)K q (z) dz ⎠ < ∞; B + = ⎝ ⎝ v q (x)K q (x) dx⎠ z

0

0

B+,

where C is the best constant in (2.1). moreover, C ≈ Theorem 2.3. For 1 < p ≤ q < ∞, inequality (2.2) is valid if and only if ⎞1/q ⎛ ∞ ⎞1/p ⎛ z     A− = sup ⎝ v q (x) dx⎠ ⎝ up (s)K p (s) ds⎠ < ∞; z>0

z

0

A− ,

where C is the best constant in (2.2). moreover, C ≈ Theorem 2.4. For 0 < q < p < ∞ and p > 1, inequality (2.2) is valid if and only if ⎛ ∞⎛ z ⎞q/(p−q) ⎛ ∞ ⎞q(p−1)/(p−q) ⎞(p−q)/(pq)    ⎝ up (s)K p (s) ds⎠ v q (z) dz ⎠ < ∞; B − = ⎝ ⎝ v q (x) dx⎠ 0

z

0

B−,

where C is the best constant in (2.2). moreover, C ≈ Definition 2.1. We say that the pair of functions (u, K) satisfies the (γ + , δ+ ) condition if there exist numbers γ + > 1 and δ+ > 0 such that x



+  x/γ  x u(s)K(x − s) ds ≤ δ+ K x − + u(s) ds γ

Let

0

(2.3)

0

x/γ +

x

for a.e. x ∈ I.

u(s)K(x − s) ds < ∞ for a.e. x > 0. Let a function σ(x) be such that x

σ(x) 

u(s)K(x − s) ds =

u(s)K(x − s) ds

for a.e. x ∈ I.

(2.4)

0

σ(x)

The function σ(·) is called a fairway function [9]. Lemma 2.1. If γ + = ess inf x>0 (x/σ(x)) > 1, then the pair of functions (u, K) satisfies the (γ + , δ+ ) condition with δ+ = 1. Indeed, it follows from the definition of the number γ + that σ(x) ≤ x/γ + for a.e. x > 0. Then, since K is a nonincreasing function, from (2.4) we have x

x u(s)K(x − s) ds ≤

x/γ +

σ(x) 

u(s)K(x − s) ds =

u(s)K(x − s) ds 0

σ(x) + x/γ 



x u(s)K(x − s) ds ≤ K x − + γ

≤ 0

 x/γ 

+

u(s) ds 0

for a.e. x > 0; i.e., (2.3) is satisfied. Denote by 1 the function identically equal to one. Lemma 2.2. If the pair of functions (1, K) satisfies the (γ + , δ+ ) condition, u1 (·) is a nonincreasing function, and u(s) = u1 (s)sβ , β ≥ 0, then the pair of functions (u, K) satisfies the (γ + , δ1+ ) condition with δ1+ = δ+ (β + 1)(γ + )β . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Using the fact the u1 does not increase, we have       x x x x x x β β + β u1 (s)s K(x − s) ds ≤ u1 K(x − s) ds ≤ δ u1 x x K x− + + + γ γ γ γ+ x/γ +

x/γ +

 = δ+ (β + 1)(γ + )β u1

   x/γ  x x tβ dt K x− + γ+ γ +

0

 x ≤ δ+ (β + 1)(γ + )β K x − + γ

  x/γ  x + β u1 (t)t dt = δ1 K x − + u1 (t)tβ dt. γ

+  x/γ 

+

0

0

(γ + , δ+ )

condition, then, for any Remark 2.1. If the pair of functions (u, K) satisfies the d > 0, the pair of functions (χ(0,d) u, K) also satisfies the (γ + , δ+ ) condition. Definition 2.2. We say that the pair of functions (u, K) satisfies the (γ − , δ− ) condition if there exist numbers γ − > 1 and δ− > 0 such that xγ 

−

u(s)K(s − x) ds ≤ δ

xγ −

x

∞

Let

x

∞

−



s u(s)K s − − γ

 ds

for a.e. x > 0.

(2.5)

u(s)K(s − x) ds < ∞ for a.e. x > 0 and − (x) σ

∞

u(s)K(s − x) ds =

u(s)K(s − x) ds

for a.e. x > 0.

(2.6)

σ− (x)

x

Lemma 2.3. If γ − = ess inf x>0 (σ − (x)/x) > 1, then the pair (u, K) satisfies the (γ − , δ− ) condition with δ− = 1. Indeed, the definition of γ − implies that xγ − ≤ σ − (x) for a.e. x > 0. Then, since K does not increase, from (2.6) we have − xγ 

∞

u(s)K(s − x) ds ≤

∞ u(s)K(s − x) ds ≤

xγ −

x

xγ −



s u(s)K s − − γ

 ds;

i.e., (2.5) holds with δ− = 1. Lemma 2.4. Suppose that u0 (s) = s−β , β > 0, and u1 (·) is a nondecreasing function. If the pair of functions (u0 , K) satisfies the (γ − , δ− ) condition, then the pair of functions (u, K), where u(s) = u1 (s)s−β , also satisfies the (γ − , δ− ) condition. Indeed, − xγ 

u1 (s)s

−β

−

− xγ 

K(s − x) ds ≤ u1 (xγ )

x

s

−β

≤δ

−

u1 (s)s xγ −

−

∞

K(s − x) ds ≤ δ u1 (xγ )

s

−β

xγ −

x

∞

−

−β



s K s− − γ

 ds

  s K s − − ds. γ

Remark 2.2. If the pair of functions (u, K) satisfies the (γ − , δ− ) condition, then, for any d > 0, the pair of functions (χ(d,∞) u, K) also satisfies the (γ − , δ− ) condition. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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3. BOUNDEDNESS OF THE OPERATOR K+ Suppose there exist numbers 0 < α < 1 and C0 > 0 such that sup K(xλ)λ1−α ≤ C0 K(x)

for a.e. x > 0.

(3.1)

0<λ<1

Condition (3.1) determines the behavior of the function K near zero, and the operator K+ may have an integrable singularity in the neighborhood of zero. Here and in what follows, we assume that the function K satisfies condition (3.1).   Theorem 3.1. Suppose that 1 < p ≤ q < ∞ and the pair of functions (up , K p ) satisfies the (γ + , δ+ ) condition. Then the operator K+ is bounded from Lp to Lq if and only if A+ < ∞; moreover, K+ p→q ≈ A+ , where K+ p→q is the norm of the operator K+ from Lp to Lq . Proof. Necessity. Suppose the operator K+ is bounded from Lp to Lq , i.e., the inequality ⎛ x ⎞q ⎞1/q ⎞1/p ⎛∞ ⎛∞    ⎝ v q (x)⎝ K(x − s)u(s)f (s) ds⎠ dx⎠ ≤ K+ p→q ⎝ f p (t) dt⎠ (3.2) 0

0

0

is valid for all 0 ≤ f ∈ Lp . Then, since K is a nonincreasing function, inequality (2.1) holds with the estimate C ≤ K+ p→q for the best constant C in (2.1). Hence, by Theorem 2.1 we have K+ p→q A+ .

(3.3)

Sufficiency. Suppose the hypotheses of Theorem 3.1 are satisfied and A+ < ∞. Set γ + = μ2 . Since μ > 1, it follows that

(μk , μk+1 ]. (3.4) I= k∈Z

In view of (3.4), we have ⎛ x ⎞q ⎛ x ⎞q μk+1 ∞     v q (x)⎝ K(x − s)u(s)f (s) ds⎠ dx = v q (x)⎝ K(x − s)u(s)f (s) ds⎠ dx 0

k

0

0

μk

⎛ μk−1 ⎞q  x v q (x)⎝ K(x − s)u(s)f (s) ds + K(x − s)u(s)f (s) ds⎠ dx

μ  

k+1

=

k

0

μk

μk−1

⎛ μk−1 ⎞q ⎛ x ⎞q μk+1    v q (x)⎝ K(x − s)u(s)f (s) ds⎠ dx + v q (x)⎝ K(x − s)u(s)f (s) ds⎠ dx

μ 

k+1



k

k

0

μk

μk

μk−1

(3.5)

= J1 + J2 .

Let us estimate J1 and J2 separately. Using the fact that K is a nonincreasing function and condition (3.1), we obtain ⎛ μk−1 ⎞q μk+1    v q (x)⎝ K(μk − μk−1 )u(s)f (s) ds⎠ dx J1 ≤ k

0

μk

μ  

k+1

=

k

μk

v q (x)K q



⎞q ⎛  μk−1 μ − 1 ⎝ u(s)f (s) ds⎠ dx μk+1 2 μ 0

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 ≤ C0q

⎛ μk−1 ⎞q q(α−1)  μk+1  μ−1 v q (x)K q (x)⎝ u(s)f (s) ds⎠ dx μ2 k

 

⎛

μk+1



k

v q (x)K q (x)⎝

μk

x

0

⎞q

⎞q ⎛ ∞ x u(s)f (s) ds⎠ dx ≤ ⎝v(x)K(x) u(s)f (s) ds⎠ dx.

0

μk

0

Hence, by Theorem 2.1 we have

⎛ J1  (A+ )q ⎝

∞

(3.6)

0

⎞q/p f p (t) dt⎠

(3.7)

.

0

To estimate J2 , we apply H¨older’s inequality and μk−1 ≥ x/μ2 = x/γ + for μk ≤ x ≤ μk+1 : ⎛ x ⎞q/p ⎛ x ⎞q/p μk+1       v q (x)⎝ K p (x − s)up (s) ds⎠ ⎝ f p (t) dt⎠ dx J2 ≤ k

μk

 

⎛

μk+1

≤

k

μk−1

x

v q (x)⎝

μk−1

⎞q/p 



K p (x − s)up (s) ds⎠

⎛

dx⎝

x/γ +

μk

⎞q/p

μk+1

f p (t) dt⎠

μk−1

(apply the (γ + , δ+ ) condition) μ   



k+1

≤ (δ+ )q/p

k

v q (x)K q x −

x γ+



⎞q/p ⎛ μk+1 ⎞q/p ⎛ x/γ +    ⎝ up (s) ds⎠ ⎝ f p (t) dt⎠ 0

μk

μk−1

(using condition (3.1), apply K(x(γ + − 1)/γ + ) ≤ C0 ((γ + − 1)/γ + )α−1 K(x)) ⎞q/p μk+1 ⎛ x/γ + ⎞q/p ⎛ μk+1      ⎝ f p (t) dt⎠ v q (x)K q (x)⎝ up (s) ds⎠ dx  k

μk−1

0

μk

(employ the fact that μk+1 /γ + = μk+1 /μ2 = μk−1 ≥ x/γ + for μk ≤ x ≤ μk+1 ) ⎞q/p μk+1 ⎛ μk−1 ⎞q/p ⎛ μk+1      ⎝ f p (t) dt⎠ v q (x)K q (x) dx⎝ up (s) ds⎠ ≤ k

μk−1

(3.8)

0

μk

⎞q/p ∞ ⎛ μk ⎞q/p ⎛ μk+1      ⎝ f p (t) dt⎠ v q (x)K q (x) dx⎝ up (s) ds⎠ ≤ k

μk−1

0

μk

⎞q/p ⎞q/p ⎛ μk+1 ⎛ μk+1     ⎝ ≤ (A+ )q f p (t) dt⎠ ≤ (A+ )q ⎝ f p (t) dt⎠ k

⎛ + q⎝

μk−1

∞

 (A )

k

⎞q/p

f p (t) dt⎠

μk−1

(3.9)

.

0

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From (3.5), (3.7), and (3.9), we find that inequality (3.2) holds, i.e., we obtain the boundedness of the operator K+ from Lp to Lq and the estimate K+ p→q  A+ , which, combined with (3.3), yields K+ p→q ≈ A+ . Theorem 3.1 is proved.    Theorem 3.2. Suppose that 0 < q < p < ∞, p > 1, and the pair of functions (up , K p ) satisfies the (γ + , δ+ ) condition. Then the operator K+ is bounded from Lp to Lq if and only if B + < ∞; moreover, K+ p→q ≈ B + . Proof. Necessity. Proceeding as in Theorem 3.1, on the basis of Theorem 2.2 we obtain K+ p→q B + .

(3.10)

Sufficiency. Let B + < ∞. From the proof of Theorem 3.1, we have relations (3.5), (3.6), and (3.8). By Theorem 2.2, it follows from (3.6) that ⎞q/p ⎛∞  (3.11) J1  (B + )q ⎝ f p (t) dt⎠ . 0

In (3.8), we apply H¨older’s inequality with exponents p/q and p/(p − q). Then ⎛ ⎞q/p ⎛ ⎛ μk+1 ⎞p/(p−q) ⎛ μk−1 ⎞q(p−1)/(p−q) ⎞(p−q)/p μk+1       ⎝ ⎠ f p (t) dt⎠ ⎝ ⎝ v q (x)K q (x) dx⎠ up (s) ds⎠ J2 ≤ ⎝ k

k

μk−1

0

μk

⎞q/p ⎛∞  ≤ J21 ⎝ f p (t) dt⎠ ,

(3.12)

0

where

⎛

J21

⎞p/(p−q) ⎛ μk−1 ⎞q(p−1)/(p−q) ⎞(p−q)/p ⎛ μk+1     ⎝ ⎠ =⎝ ⎝ v q (x)K q (x) dx⎠ up (s) ds⎠ . k

0

μk

Using the relation ⎞p/(p−q) ⎛ μk+1 ⎞q/(p−q) ⎛ μk+1 μk+1   p−q ⎝ v q (x)K q (x) dx⎠ = v q (x)K q (x)⎝ v q (t)K q (t) dt⎠ dx, p μk p/(p−q)

we estimate J21

μ  

:

k+1

p/(p−q)

J21



x

μk

k

⎛

v q (x)K q (x)⎝

μk

∞

⎞q/(p−q) ⎛ v q (t)K q (t) dt⎠

x

⎝

x

⎞q(p−1)/(p−q) p

u (s) ds⎠

dx

0

⎞q/(p−q) ⎛ x ⎞q(p−1)/(p−q) ⎛ ∞ ∞  ⎝ up (s) ds⎠ v q (x)K q (x) dx = (B + )pq/(p−q) . ≤ ⎝ v q (t)K q (t) dt⎠ 0

x

0

Hence, from (3.12) we have ⎞q/p ⎛∞  J2  (B + )q ⎝ f p (t) dt⎠ . 0

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This and estimates (3.5) and (3.11) imply that the operator K+ is bounded from Lp to Lq and its norm satisfies the estimate K+ p→q  B + , which, combined with (3.10), yields K+ p→q ≈ B + . Theorem 3.2 is proved.  

Suppose that the function K satisfies condition (3.1) with α > 1/p. Then the pair (1, K p ) satisfies the (γ + , δ+ ) condition for any γ + > 1 with δ+ = C0 (γ + − 1)/(1 + p (α − 1)). Indeed, for any γ + > 1 we have x

γ+ − 1 K (x − s) ds = x γ+ p

1 K

p



 x x − + t dt γ

0

x/γ +

≤ C0 (γ − 1)K +

p



x x− + γ



x γ+

1 t

p (α−1)

0

  γ+ − 1 x x p K x− + dt = C0 . 1 + p (α − 1) γ γ+

Therefore, Lemma 2.2 and Theorems 3.1 and 3.2 imply the following statement: Corollary 3.1. Suppose that α > 1/p, u(s) = u1 (s)sβ , β ≥ 0, and u1 is a nonincreasing function. Then the operator K+ is bounded from Lp to Lq if and only if (a) A+ < ∞ for 1 < p ≤ q < ∞; (b) B + < ∞ for 0 < q < p < ∞ and p > 1. 4. BOUNDEDNESS OF THE OPERATOR K− 



Theorem 4.1. Suppose that 1 < p ≤ q < ∞ and the pair of functions (up , K p ) satisfies the (γ − , δ− ) condition. Then the operator K− is bounded from Lp to Lq if and only if A− < ∞; moreover, K− p→q ≈ A− . Proof. Necessity. Suppose the operator K− is bounded from Lp to Lq , i.e., the inequality ⎛ ⎝

∞

⎛ v q (x)⎝

∞

⎞q

⎞1/q

⎞1/p ⎛∞  ≤ K− p→q ⎝ f p (t) dt⎠

K(s − x)u(s)f (s) ds⎠ dx⎠

x

0

(4.1)

0

is valid. Since K(s − x) ≥ K(s) for s ≥ x, inequality (4.1) implies inequality (2.2) and the estimate C ≤ K− p→q for the best constant C in (2.2). Therefore, by Theorem 2.3, we have K− p→q A− .

(4.2)

Sufficiency. Suppose A− < ∞ and the hypotheses of Theorem 4.1 are satisfied. Set γ − = μ2 . Then μ > 1 and (3.4) holds. Therefore, ∞

⎛ v q (x)⎝

∞

K(s − x)u(s)f (s) ds⎠ dx =

μ  

k

k

μ  

⎛

k

k

x

0



⎞q

v q (x) ⎝

∞

⎞q K(s − x)u(s)f (s) ds⎠ dx

x

μk−1

⎛ μk+1 ⎞q ⎛ ∞ ⎞q μk     v q (x)⎝ K(s − x)u(s)f (s) ds⎠ dx + v p (x)⎝ K(s − x)u(s)f (s) ds⎠ dx x

μk−1

k

μk−1

μk+1

= J1− + J2− . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Applying H¨older’s inequality, the (γ − , δ− ) condition, and condition (3.1), we estimate J1− : ⎛ μk+1 ⎞q/p ⎛ μk+1 ⎞q/p μk       v q (x)⎝ K p (s − x)up (s) ds⎠ dx⎝ f p (t) dt⎠ J1− ≤ k

x

μk−1

x

(μk+1 ≤ xμ2 = xγ − for μk−1 ≤ x ≤ μk ) ⎛ xγ − ⎞q/p ⎛ μk+1 ⎞q/p μk       v q (x)⎝ K p (s − x)up (s) ds⎠ dx⎝ f p (t) dt⎠ ≤ k

x

μk−1 μ   

k

≤ (δ− )q/p

k

μk−1

⎛

∞

v q (x)⎝

xγ −

μk−1

⎞q/p ⎛ μk+1 ⎞q/p    s   K p s − − up (s) ds⎠ dx⎝ f p (t) dt⎠ γ μk−1









(in view of (3.1) we have K p (s − s/γ − ) = K p (s(γ − − 1)/γ − ) ≤ C0 ((γ − − 1)/γ − )p (α−1) K p (s)) ⎛ ∞ ⎞q/p ⎛ μk+1 ⎞q/p μk       v q (x)⎝ K p (s)up (s) ds⎠ dx⎝ f p (t) dt⎠  k

xγ −

μk−1

μk−1

(xγ − = xμ2 > μk for μk−1 ≤ x ≤ μk ) ⎞ μk ⎛ ∞ ⎞q/p ⎛ μk+1       ⎝ f p (t) dt⎠ v q (x) dx⎝ K p (s)up (s) ds⎠ ≤ k

μk−1

μk−1

(4.4)

μk

⎞q/p ⎞q/p ⎛ ⎛∞ μk+1    f p (t) dt⎠  (A− )q ⎝ f p (t) dt⎠ . ≤ (A− )q ⎝ k

(4.5)

0

μk−1

Now, let us estimate J2− . Using the fact that K is a nonincreasing function and condition (3.1), we have     μ − 1 α−1 μ−1 ≤ C0 K(s) K(s − x) ≤ K s μ μ for s ≥ μk+1 and x ≤ μk . Therefore, ⎛ ∞ ⎞q ⎛ ∞ ⎞q μk μk       v q (x)⎝ K(s − x)u(s)f (s) ds⎠ dx  v q (x)⎝ K(s)u(s)f (s) ds⎠ dx J2− = k

μk−1

μ  

k

≤

k

μk−1

⎛ v q (x)⎝

μk+1

∞

k

⎞q

K(s)u(s)f (s) ds⎠ dx ≤

∞

x

0

μk−1

⎛ v q (x)⎝

∞

μk+1

⎞q

K(s)u(s)f (s) ds⎠ dx.

(4.6)

x

From A− < ∞, applying Theorem 2.3, we find that inequality (2.2) holds with the estimate C ≤ A− for the best constant C in (2.2). Then, from (4.6) we have ⎞q/p ⎛∞  (4.7) J2−  (A− )q ⎝ f p (t) dt⎠ . 0

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Relations (4.3), (4.5), and (4.7) imply inequality (4.1), i.e., the boundedness of the operator K− from Lp to Lq with the estimate K− p→q  A− , which, together with (5.2), yields K− p→q ≈ A− . Theorem 4.1 is proved.  



Theorem 4.2. Suppose that 0 < q < p < ∞, p > 1, and the pair of functions (up , K p ) satisfies the (γ − , δ− ) condition. Then the operator K− is bounded from Lp to Lq if and only if B − < ∞; moreover, K− p→q ≈ B − . Proof. Necessity. Inequality (4.1) implies inequality (2.2); therefore, by Theorem 2.4, K− p→q ≥ B − .

(4.8)

Sufficiency. Let us apply relations (4.3), (4.4), and (4.6). By Theorem 2.4, relation (4.6) immediately implies ⎞q/p ⎛∞  (4.9) J2−  (B − )q ⎝ f p (t) dt⎠ . 0

From (4.4), proceeding in the same way as in (3.8) when proving Theorem 3.2, we obtain ⎛ J1−  (B − )q ⎝

∞

⎞q/p f p (t) dt⎠

(4.10)

.

0

Relations (4.3), (4.9), and (4.10) imply inequality (4.1) with the estimate K− p→q  B − , which, combined with (4.8), yields the relation K− p→q ≈ B − . Theorem 4.2 is proved.  Passing to the adjoint operator, from Theorems 3.1, 3.2, 4.1, and 4.2 we obtain the following Lp → Lq boundedness criteria for the operators K− and K+ for 1 < p, q < ∞. Theorem 4.3. Suppose that the pair of functions (v q , K q ) satisfies the (γ + , δ+ ) condition. Then the operator K− is bounded from Lp to Lq if and only if (i) for 1 < p ≤ q < ∞ ⎛ E + ≡ A− = sup⎝ z>0

∞

⎞1/p ⎛ z ⎞1/q  u (x)K (x) dx⎠ ⎝ v q (s) ds⎠ < ∞; p

p

z

0

(ii) for 1 < q < p < ∞ ⎛ ∞⎛ ∞ ⎞p(q−1)/(p−q) ⎛ z ⎞p/(p−q) ⎞(p−q)/(pq)        ⎝ v q (s) ds⎠ up (z)K p (z) dz⎠ < ∞; E − = ⎝ ⎝ up (x)K p (x) dx⎠ 0

z

0

moreover, K− p→q ≈ E ± , where K− p→q is the norm of the operator K− from Lp to Lq . Theorem 4.4. Suppose that the pair of functions (v q , K q ) satisfies the (γ − , δ− ) condition. Then the operator K+ is bounded from Lp to Lq if and only if (i) for 1 < p ≤ q < ∞ ⎛ F + ≡ A+ = sup⎝

z

z>0

0

⎞1/p ⎛ 

up (x) dx⎠

⎝

∞

⎞1/q v q (s)K q (s) ds⎠

< ∞;

z

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(ii) for 1 < q < p < ∞ ⎛ ∞⎛ z ⎞p(q−1)/(p−q) ⎛ ∞ ⎞p/(p−q) ⎞(p−q)/(pq)      ⎝ v q (s)K q (s) ds⎠ up (z) dz ⎠ < ∞; F − = ⎝ ⎝ up (x) dx⎠ 0

z

0

moreover, K+ p→q ≈ F ± , where K+ p→q is the norm of the operator K+ from Lp to Lq . 5. COMPACTNESS OF THE OPERATORS K+ AND K− Theorem 5.1. Suppose that 1 < p ≤ q < ∞ and the hypotheses of Theorem 3.1 are satisfied. Then the operator K+ is compact from Lp to Lq if and only if A+ < ∞ and lim A+ (z) = lim A+ (z) = 0,

where

⎛ A+ (z) = ⎝

∞

(5.1)

z→∞

z→0

⎞1/q ⎛ v q (x)K q (x) dx⎠

⎝

z

z

⎞1/p 

up (s) ds⎠

.

0

Proof. Necessity. Suppose that the operator K+ is compact from Lp to Lq . Then it is bounded; hence, A+ < ∞ by Theorem 3.1. First, let us show that limz→0 A+ (z) = 0. For z > 0, consider the family of functions ⎛ z ⎞−1/p    . fz (t) = χ(0,z) (t)up −1 (t)⎝ up (s) ds⎠ 0

The family of functions fz , z > 0, is jointly bounded in Lp because ⎛ z ⎞−1 ∞ z    fzp (t) dt = up (t) dt⎝ up (s) ds⎠ ≡ 1. 0

0

0

Let us show that fz tends weakly to zero in Lp as z → 0. For any g ∈ Lp , we have ∞ ⎛ z ⎞1/p ⎛ z ⎞1/p ⎛ z ⎞1/p z     g(t)fz (t) dt ≤ |g(t)|fz (t) dt ≤ ⎝ |g(t)|p dt⎠ ⎝ fzp (t) dt⎠ = ⎝ |g(t)|p dt⎠ . 0

0

0

0

0

Hence the family of functions fz tends weakly to zero in Lp as z → 0. Then it follows from the compactness of the operator K+ from Lp to Lq that lim K+ fz q = 0.

(5.2)

z→0

We have

∞ K+ fz qq =

⎛ v q (x)⎝

0

∞ ≥

⎛ v q (x)⎝

z

∞ ≥

0

z 0

⎞q K(x − s)u(s)fz (s) ds⎠ dx ⎞q

⎛ z ⎞−q/p    K(x − s)up (s) ds⎠ dx⎝ up (s) ds⎠ ⎛

v (x)K (x) dx⎝ q

z

x

z

q

0

⎞q/p 

up (s) ds⎠

= (A+ (z))q .

0

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From this and (5.2) we obtain limz→0 A+ (z) = 0. The Lp → Lq compactness of the operator K+ for 1 < p ≤ q < ∞ implies the Lq → Lp compactness of the adjoint operator ∞

+ ∗

K(x − s)v(x)g(x) dx.

(K ) g(s) = u(s) s

For any z > 0, we set ⎛ gz (x) = χ(z,∞) (x)v q−1 (x)K q−1 (x)⎝

∞

⎞−1/q v q (x)K q (x) dx⎠

.

z

Then

∞

 gzq (x) dx

∞ z

0

and for any φ ∈ Lq we have



gzq (x) dx = 1,

=

⎛ ∞ ∞ ⎞1/q   φ(x)gz (x) dx ≤ ⎝ |φ(x)|q dx⎠ . z

0

This implies that the family of functions gz , z > 0, converges weakly to zero in Lq as z → ∞. Therefore, due to the compactness of the operator (K+ )∗ , lim (K+ )∗ gz  = 0.

(5.3)

z→∞

We have p

(K+ )∗ gz p ≥

z

⎛ 

up (s)⎝

≥

⎞p

⎛

p

u (s) ds⎝

0

⎛

K(x − s)K q−1 (x)v q (x) dx⎠ ds⎝

z

0

z

∞

∞

⎞−p /q v q (x)K q (x) dx⎠

z

⎞p /q K q (x)v q (x) dx⎠

∞



= (A+ (z))p .

z

Hence, from (5.3) we obtain limz→∞ A+ (z) = 0. Sufficiency. Suppose that A+ < ∞ and (5.1) holds. For 0 < c < d < ∞, we set Pc f = χ(0,c] f,

Pc,d f = χ(c,d] f,

and

Qd f = χ(d,∞) f.

Then f = Pc f + Pc,d f + Qd f and K+ f = Pc,d K+ Pc,d f + Pc K+ Pc f + Pc,d K+ Pc f + Qd K+ f,

(5.4)

since Pc K+ Pc,d ≡ 0, Pc K+ Qd ≡ 0, and Pc,d K+ Qd ≡ 0. Let us show that the operator Pc,d K+ Pc,d is compact from Lp to Lq . Since Pc,d K+ Pc,d f (x) = 0 for x ∈ I \ (c, d), it suffices to show that the operator Pc,d K+ Pc,d is compact from Lp (c, d) to Lq (c, d), which is equivalent to the compactness of the operator d Kf (x) =

K(x, s)f (s) ds c

with the kernel K(x, s) = v(x)K(x − s)u(s)χ(c,x) (s) from Lp (c, d) to Lq (c, d). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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There exist integers n and i, n > i, such that μi ≤ c < μi+1 and μn < d ≤ μn+1 . Then ⎞q/p ⎛ x ⎞q/p ⎛ d d d  ⎝ |K(x, s)|p ds⎠ dx = v q (x)⎝ K p (x − s)up (s) ds⎠ dx c

≤

c n  k=i

μk

c





v q (x)⎝

x





K p (x − s)up (s) ds⎠

dx

0

x/γ +

⎛

⎞q/p

+ x/γ 

K p (x − s)up (s) ds +

μk

n μ  k=i

x

v q (x)⎝

k+1



c

⎛

μk+1

⎞q/p   K p (x − s)up (s) ds⎠

n μ 

⎛ x/γ + ⎞q/p    v p (x)⎝ K p (x − s)up (s) ds⎠ dx

k+1

dx +

k=i

x/γ +

0

μk

⎛ μk−1 ⎞q/p k+1  n μ   v q (x)K q (x) dx⎝ up (s) ds⎠ ≤ (n − i + 1)(A+ )q < ∞.  k=i

0

μk

Thus, the operator K satisfies the Kantorovich condition [10, Ch. 11]; hence, the operator K is compact from Lp (c, d) to Lq (c, d). From (5.4) we have

+

K − Pc,d K+ Pc,d ≤ Pc K+ Pc p→q + Pc,d K+ Pc p→q + Qd K+ p→q . (5.5) p→q Let us show that the right-hand side of (5.5) tends to zero as c → 0 and d → ∞; then, as a uniform limit of compact operators, the operator K+ is compact from Lp to Lq . By Remark 2.1 and Theorem 3.1, we have ⎛∞ ⎛ x ⎞q ⎞1/q   Pc K+ Pc f  = ⎝ (Pc v)q (x)⎝ (Pc u)(s)K(x − s)f (s) ds⎠ dx⎠ 0

0

⎞1/q ⎛ z ⎞1/p ⎛ ∞ ⎞1/p ∞     sup⎝ (Pc v)q (x)K q (x) dx⎠ ⎝ (Pc u)p (s) ds⎠ ⎝ f p (t) dt⎠ ⎛

z>0

z

⎛

≤ sup A (z)⎝

∞

+

0

⎞1/p f p (t) dt⎠

0

.

0
0

Hence, lim Pc K+ Pc p→q  lim sup A+ (z) = lim A+ (c) = 0.

c→0

c→0 0
(5.6)

c→0

Proceeding in exactly the same way, we obtain lim Pc,d K+ Pc p→q  lim A+ (c) = 0,

c→0

(5.7)

c→0

lim Qd K+ p→q  lim sup A+ (z) = lim A+ (d) = 0.

d→∞

d→∞ d
(5.8)

d→∞

It follows from (5.6)–(5.8) that the right-hand side of (5.5) tends to zero as c → 0 and d → ∞. Theorem 5.1 is proved.  PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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The following theorem is proved by an analogous method. Theorem 5.2. Suppose that 1 < p ≤ q < ∞ and the hypotheses of Theorem 4.1 are satisfied. Then the operator K− is compact from Lp to Lq if and only if A− < ∞ and lim A− (z) = lim A− (z) = 0, z→∞

z→0

where

⎛ A− (z) = ⎝

z

⎞1/q ⎛ v q (x) dx⎠ ⎝

∞

⎞1/p 



up (s)K p (s) ds⎠

.

z

0

Theorem 5.3. Let 1 < q < p < ∞. Then (i) under the hypotheses of Theorem 3.2, the operator K+ is compact from Lp to Lq if and only if B + < ∞; (ii) under the hypotheses of Theorem 4.2, the operator K− is compact from Lp to Lq if and only if B − < ∞. Proof. Necessity. Suppose that the operator K+ (K− ) is compact from Lp to Lq . Then it is bounded; hence, by Theorem 3.2 (Theorem 4.2), B + < ∞ (B − < ∞). Sufficiency in assertion (i). Let B + < ∞ and 0 < d < ∞. Then K+ f = Pd K+ Pd f + Qd K+ f because Pd K+ Qd ≡ 0. Hence, K+ − Pd K+ Pd p→q ≤ Qd K+ p→q .

(5.9)

The operator Pd K+ Pd can be assumed to act from Lp (0, d) to Lq (0, d). Therefore, in view of [11, Theorem 5.5], it is compact. If we show that limd→∞ Qd K+ p→q = 0, then (5.9) implies the compactness of the operator K+ as a uniform limit of compact operators. By Theorem 3.2, we have ⎛

Qd K+ p→q

⎞q/(p−q) ⎛ z ⎞q(p−1)/(p−q) ⎞(p−q)/(pq) ⎛ ∞ ∞  ⎝ up (s) ds⎠  ⎝ ⎝ v q (x)K q (x) dx⎠ v q (z)K q (z) dz⎠ . d

z

0

Therefore, since B + < ∞, we have limd→∞ Qd K+ p→q = 0. Sufficiency in assertion (ii). Here we have K− f = Pd K− Pd f + Pd K− Qd f + Qd K− f. Hence, K− − Pd K− Pd p→q ≤ Pd K− Qd p→q + Qd K− p→q .

(5.10)

Since Pd K− Pd f ≤ K− f for f ≥ 0, the operator Pd K+ Pd is bounded from Lp (0, d) to Lq (0, d). Then, by [11, Theorem 5.5], it is compact. Let us show that limd→∞ Pd K− Qd p→q = 0; the equality limd→∞ Qd K− p→q = 0 is proved in the same way as the equality limd→∞ Qd K+ p→q = 0. In view of the remark from [8], ⎛

⎞p/(p−q) ⎛ ∞ ⎞p(q−1)/(p−q) ⎞(p−q)/(pq) ⎛ ∞ z    ⎝ up (s)K p (s) ds⎠ up (z)K p (z) dz ⎠ . (5.11) B − ≈ ⎝ ⎝ v q (x) dx⎠ 0

0

z

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We have ⎛ ⎝

⎞1/q ⎛

d 0

v q (x) dx⎠ ⎝

∞

⎞1/p 



up (s)K p (s) ds⎠

d

⎛

⎞p/(p−q) ⎛ ∞ ⎞p(q−1)/(p−q) ⎞(p−q)/(pq) ⎛ ∞ z      ⎝ up (s)K p (s) ds⎠  ⎝ ⎝ v q (x) dx⎠ up (z)K p (z) dz ⎠ . (5.12) d

z

0

On the basis of Remark 2.2, Theorem 4.2, and (5.11), we obtain

Pd K− Qd p→q

⎛ d ⎞1/q ⎛ ∞ ⎞1/p      ⎝ v q (x) dx⎠ ⎝ up (s)K p (s) ds⎠ . 0

d

Therefore, from (5.12) we have limd→∞ Pd K− Qd p→q = 0. Hence, the right-hand side of (5.10) tends to zero as d → ∞. Then the operator K− is compact from Lp (0, d) to Lq (0, d) as a uniform limit of compact operators.  6. BOUNDEDNESS AND COMPACTNESS OF THE RIEMANN–LIOUVILLE AND WEYL OPERATORS Set ⎛ ⎝ A+ α (z) =

∞

⎞1/q ⎛ v q (x)xq(α−1) dx⎠ ⎝

z

⎞1/p 

up (s) ds⎠

,

+ A+ α = sup Aα (z), z>0

z

0

⎛ z ⎞1/q ⎛ ∞ ⎞1/p     ⎝ v q (x) dx⎠ ⎝ up (s)sp (α−1) ds⎠ , A− α (z) =

− A− α = sup Aα (z),

z

0

z>0

⎛

⎞q/(p−q) ⎛ z ⎞q(p−1)/(p−q) ⎞(p−q)/(pq) ⎛ ∞ ∞   ⎝ up (s) ds⎠ v q (z)z q(α−1) dz ⎠ , Bα+ = ⎝ ⎝ v q (x)xq(α−1) dx⎠ 0

z

0

⎛

⎞q/(p−q) ⎛ ∞ ⎞q(p−1)/(p−q) ⎞(p−q)/(pq) ⎛ ∞ z    ⎝ up (s)sp (α−1) ds⎠ v q (z) dz ⎠ . Bα− = ⎝ ⎝ v q (x) dx⎠ 0

0

z

Below we assume that K(x) ≡ xα−1 , 0 < α < 1. Since it is a nonincreasing function satisfying condition (3.1), we obtain the following theorems for the weighted Riemann–Liouville operator Iα and weighted Weyl operator Wα from the results of Sections 3–5. 



Theorem 6.1. Suppose that 0 < α < 1, 1 < p ≤ q < ∞, and the pair of functions (up , K p ) satisfies the (γ + , δ+ ) condition. Then + (i) the operator Iα is bounded from Lp to Lq if and only if A+ α < ∞; moreover, Iα p→q ≈ Aα ; + (ii) the operator Iα is compact from Lp to Lq if and only if A+ α < ∞ and limz→0 Aα (z) = + limz→∞ Aα (z) = 0.

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Theorem 6.2. Let 0 < α < 1, 0 < q < p < ∞, and p > 1. Suppose that the pair of   functions (up , K p ) satisfies the (γ + , δ+ ) condition. Then the operator Iα is bounded (compact for 1 < q < p < ∞) from Lp to Lq if and only if B + < ∞. 



Theorem 6.3. Suppose that 0 < α < 1, 1 < p ≤ q < ∞, and the pair of functions (up , K p ) satisfies the (γ − , δ− ) condition. Then (i) the operator Wα is bounded from Lp to Lq if and only if A− α < ∞; − (ii) the operator Wα is compact from Lp to Lq if and only if A− α < ∞ and limz→0 Aα (z) = − limz→∞ Aα (z) = 0.

Theorem 6.4. Let 0 < α < 1, 0 < q < p < ∞, and p > 1. Suppose that the pair of   functions (up , K p ) satisfies the (γ − , δ− ) condition. Then the operator Wα is bounded (compact for 1 < q < p < ∞) from Lp to Lq if and only if Bα− < ∞. Corollary 3.1 and Theorems 6.1 and 6.2 imply Corollary 6.1. Let 1/p < α ≤ 1 and u(s) = u1 (s)sβ , where β ≥ 0 and u1 is a nonincreasing function. Then all the assertions of Theorems 6.1 and 6.2 are valid. 



Let β > α − 1/p and up0 (x) = x−p β . Then, as above, one can easily establish that the pair   (up0 , K p ) satisfies the (γ − , δ− ) condition for any γ − > 1. Therefore, by Lemma 2.4, Theorems 6.3 and 6.4 imply Corollary 6.2. Let 1/p < α ≤ 1, β > α − 1/p, and u(s) = u1 (s)s−β , where u1 is a nondecreasing function. Then all the assertions of Theorems 6.3 and 6.4 are valid. Remark 6.1. The assertion of Corollary 6.1 extends the results of [5, Theorems 1–4], where the case of β = 0 was considered, and even in this case Corollary 6.1 improves these theorems because the criteria in [5] employ more cumbersome expressions. Remark 6.2. In [5], Farsani also considered the Weyl operator Wα in the case when u is a nondecreasing function; 5 and 6 in [5] are invalid, because the criteria given there ∞however, Theorems  contain the integral t/2 (u(t)tα−1 )p dt, which diverges in the case 1/p < α even for the constant function u ≡ 1. ACKNOWLEDGMENTS I am grateful to the referee for valuable remarks, which helped to improve the content of the paper. This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan, project no. 5499/GF4 in the priority field “Intellectual Potential of the Country.” REFERENCES 1. D. V. Prokhorov and V. D. Stepanov, “Weighted estimates for the Riemann–Liouville operators and applications,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 243, 289–312 (2003) [Proc. Steklov Inst. Math. 243, 278–301 (2003)]. 2. M. Lorente, “A characterization of two weight norm inequalities for one-sided operators of fractional type,” Can. J. Math. 49 (5), 1010–1033 (1997). 3. A. Meskhi, “Solution of some weight problems for the Riemann–Liouville and Weyl operators,” Georgian Math. J. 5 (6), 565–574 (1998). 4. D. V. Prokhorov, “On the boundedness and compactness of a class of integral operators,” J. London Math. Soc., Ser. 2, 61 (2), 617–628 (2000). 5. S. M. Farsani, “On boundedness and compactness of Riemann–Liouville fractional operators,” Sib. Mat. Zh. 54 (2), 468–479 (2013) [Sib. Math. J. 54, 368–378 (2013)]. 6. N. A. Rautian, “On the boundedness of a class of fractional type integral operators,” Mat. Sb. 200 (12), 81–106 (2009) [Sb. Math. 200, 1807–1832 (2009)]. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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7. A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy Inequality: About Its History and Some Related Results (Vydavatelsk´ y Servis, Plzeˇ n, 2007). 8. G. Sinnamon and V. D. Stepanov, “The weighted Hardy inequality: New proofs and the case p = 1,” J. London Math. Soc., Ser. 2, 54 (1), 89–101 (1996). 9. V. D. Stepanov and E. P. Ushakova, “On integral operators with variable limits of integration,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 232, 298–317 (2001) [Proc. Steklov Inst. Math. 232, 290–309 (2001)]. 10. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon, Oxford, 1982). 11. M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Nauka, Moscow, 1966; Noordhoff, Leyden, 1976).

Translated by I. Nikitin

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