Boundedness and compactness of a class of integral operators with power and logarithmic singularity when p≤q$p\leq q$

In this paper, necessary and sufficient conditions for the boundedness and compactness of one class of integral operators with power and logarithmic singularities in weighted Lebesgue spaces are obtained.


Introduction
Let I = (0, ∞) and let v, u be almost everywhere positive and locally integrable functions on the interval I.
Let 1 < p, q < ∞, and p = p p-1 . Let us denote by L p,v ≡ L p (v, I) the set of measurable functions f on I for which Let W be a positive, strictly increasing, and locally absolutely continuous function on the interval I. Let dW (x) dx = w(x) for almost all x ∈ I. Consider the operator where α > 0, β ≥ 0. When β = 0, the operator T α,β has the form u(s)f (s)w(s) ds (W (x) -W (s)) 1-α , (1.2) which is called the fractional integration operator of the function f over the function W for u ≡ 1.
Further, we assume that W is nonnegative on I and lim x→0 + W (x) = 0. The boundedness and compactness of operator (1.2) from L p,w to L q,v is obtained in the paper [5] for α > 1 p , 1 < p ≤ q < ∞, and 0 < q < p < ∞. When α > 1, the results follow from the results in [6]. A criterion for the boundedness and compactness of the dual operator (1.2), when the parameters satisfy the same conditions, was obtained in the paper [7]. The boundedness and compactness of operator (1.2) were obtained in the paper [8] when the upper limit of the integral is a function. When β = 1 and W (x) = x in (1.1), two-sided estimates have been obtained in the paper [9].
The main goal of the paper is to establish the criteria for the boundedness and compactness of operator (1.1) from L p,w to L q,v for the following relations of the space parameters The work is organized as follows. The next section contains the necessary materials to confirm the main results, which are presented in the third and fourth sections. In the third section, we have proved the boundedness of operator (1.1), and the compactness of the operator is proved in the fourth section. The last section contains the corollaries.
Agreements. The uncertainty of the form 0 · ∞ is considered to be zero. We will write A B or B A if there is a number c > 0 and A ≤ cB. The relation A ≈ B means A B and A B. Z is the set of integers, and χ (a,b) is the characteristic function of the interval (a, b) ⊂ I.

Auxiliary statements
Consider the Hardy operator from L p,w to L q,v , where ϕ is a nonnegative measurable function on I.
Theorem 5 of the book [10] implies the following theorem.

Boundedness of the operator T α,β
The main result of this section is the following.
Theorem 3.1 Let 0 < α < 1, 1 α < p ≤ q < ∞, and β ≥ 0. Let the function u be nonincreasing on I. Then the operator T α,β , defined by formula (1.1), is bounded from L p,w to L q,v if and only if Substituting the obtained relations in the expressions of operator (1.1) for f ≥ 0, we obtain The boundedness of the operator T α,β from L p,w to L q,v implies the boundedness of the Hardy operator H α,β from L p,w to L q,v and T α,β H α,β . Then, by Theorem A, the value of A α,β < ∞ and for the norm H α,β of the operator H α,β there is an estimate A α,β H α,β . Then, by virtue of (3.1), Sufficiency. Let A α,β < ∞. Since W is a strictly increasing continuous function such that We have Now we estimate J 1 and J 2 separately.
Hence, based on Theorem A, Now, we estimate J 2 . Using the nonincreasing function u for estimating J 2 and applying Hölder's inequality, we find We replace the variables W (s) = W (x)t in the following expression: In the latter ratio, we used a replacement 1 1-t = e z . Substituting the obtained estimates (3.7) in (3.6), we get Next, we need the following estimation: Substituting the obtained estimate in (3.8) and using Jensen's inequality, by virtue of p ≤ q, we have Substituting the obtained estimates (3.5) and (3.9) in (3.3), we get i.e., the boundedness of the operator T α,β from L p,w to L q,v and the estimate T α,β A α,β holds for the norm T α,β from L p,w to L q,v , which together with (3.2) gives T α,β ≈ A α,β . Theorem 3.1 is proved.

The compactness of the operator T α,β
Assume that

. Let the function u be nonincreasing on the interval I. Then the operator T α,β is compact from L p,w to L q,v if and only if A α,β < ∞ and
Proof of Theorem 4.1 Necessity. Let the operator T α,β be compact from L p,w to L q,v . Then it is bounded from L p,w to L q,v and A α,β < ∞ according to Theorem 3.1. First, let us show the fulfilment of lim z→0 + A α,β (z) = 0. Consider the family of functions {f t } t∈I : Let us note that i.e., f t ∈ L p,w for all t ∈ I. Let us show that f t converges weakly to zero if t → 0 + . For arbitrary Whence it follows that f t weakly converges to zero if t → 0 + . Since the operator T α,β is compact from L p,w to L q,v , then We have Whence and from (4.2) it follows that lim t→0 + A α,β (t) = 0. We now prove that lim t→∞ A α,β (t) = 0. The compactness of the adjoint operator from L q ,v 1-q to L p ,w 1-p follows from the compactness of the operator T α,β from L p,w to L q,v .
Introduce the family of functions {g t } t∈I : It is easy to see that g t ∈ L q ,v 1-q for all t ∈ I. Indeed, Let f ∈ (L q ,v 1-q ) * = L q,v be an arbitrary function. Then Consequently, the family of functions {g t } t∈I ⊂ L q ,v 1-q weakly converges to zero at t → ∞.
Sufficiency. Let A α,β < ∞ and (4.1) be fulfilled. We define P c f = χ (0,c] f , P cd f = χ (c,d] f and Q d f = χ (d,∞) f for 0 < c < d < ∞. Then f = P c f + P cd f + Q d f and, by virtue of P c T α,β P cd ≡ 0, P c T α,β Q d ≡ 0 and P cd T α,β Q d ≡ 0, we obtain T α,β f = P cd T α,β P cd f + P c T α,β P c f + P cd T α,β P c f + Q d T α,β f . (4.4) Let us show that the operator P cd T α,β P cd is compact from L p,w to L q,v . Since P cd T α,β P cd × f (x) = 0 for x ∈ I\(c, d), then it suffices to show that the operator is compact from L p,w (c, d) to L q,v (c, d). This is equivalent to the compactness of the operator Let {x k } k∈Z be a sequence constructed by the function W from Theorem 3.1. Then there exist numbers i and n such that x i ≤ c < x i+1 , x n < d ≤ x n+1 . We will assume that the numbers c, d are chosen so that x i+1 < x n . Proceeding as in Theorem 3.1, we have (4.5) Estimate F 1 and F 2 . Analogously to the estimate of J 1 , Analogously to the estimate of J 2 , Therefore, based on the Kantorovich criterion ( [11], XI, paragraph 3) the operator T is compact from L p (c, d) to L q (c, d), which is equivalent to the compactness of the operator P cd T α,β P cd from L p,w to L q,v . From (4.4) we have T α,β -P cd T α,β P cd ≤ P c T α,β P c + P cd T α,β P c + Q d T α,β . (4.8) Further, we assume that the right-hand side of (4.8) tends to zero as c → 0 and d → ∞. Then the operator T α,β is compact from L p,w to L q,v as the uniform limit of compact operators.