On Boundedness and Compactness OF A CERTAIN CLASS KERNEL OPERATORS

New conditions for Lp[0,∞)-Lq[0,∞) boundedness and compactness (1<p, q<∞) of the map f→w(x)∫a(x)b(x)k(x,y)f(y)v(y)dy with locally integrable weight functions v,w and a positive continuous kernel k(x,y) from the Oinarov’s class are obtained.

where positive constants α 1 , α 2 depend only on the parameters p, q and Similar to (1.5) and (1.8) two-sided estimates for the case 1 < q < p < ∞ were obtained in [9] in a discrete form. This fact together with double supremum in the definitions of A a and A b may be rather inconvenient for further development.
In this work we obtain new necessary and new sufficient conditions for L p − L q boundedness and compactness of K (see Sections 3 and 4) which become a criterion either under some additional requirements on weight functions or when a kernel satisfies (1.3) and (1.4) simultaneously (see Section 5). Unlike to the above results the new criteria have one supremum if 1 < p ≤ q < ∞ and a continuous form for 1 < q < p < ∞. We start the paper with Preliminaries (Section 2) and conclude by Examples (Section 6).
Throughout the paper products of the form 0 ·∞ are taken to be equal to 0. Relations A B mean A ≤ cB with some constants c depending only on parameters of summations and, possibly, on the constants of equivalence in the inequalities of the type (1.3). We write A ≈ B instead of A B A or A = cB . Z denotes the set of all integers and χ E stands for a characteristic function (indicator) of a subset E ⊂ R + . Also we make use of marks : = and = : for introducing new quantities and suppose p := p/(p − 1) for 1 < p < ∞ and r := pq/(p − q) for 1 < q < p < ∞.

Preliminaries
Here we collect some statements, which are necessary to prove our results. We start with Lemma 2.1 on a block-diagonal operator. Lemma 2.1 ([10, Lemma 1]). Let 0 < q < p < ∞, let U = k U k and V = k V k be unions of non-overlapping measurable sets and let The next Theorem 2.1 is a known result for the Hardy type operator with the kernel k(x, y) ≥ 0 satisfying the Oinarov's condition of the form: there exists a constant D ≥ 1 independent on x, y, z such that The result of Theorem 2.1 can be extended to a more general then (2.1) with a differentiable and strictly increasing on [c, d) function b(x) ≥ 0 and a non-negative kernel k(x, y) satisfying the condition: there exists a constant D ≥ 1 such that (2.4) Corollary 2.1. Let 1 < q < p < ∞, r = p q/(p − q) and the operator K b be given by (2.3) with a differentiable and strictly increasing on [c, d) function b(x) ≥ 0 and a non-negative kernel k(x, y) ∈ (2.4). Then By duality and Theorem 2.1 we can obtain norm estimates for the operator with a differentiable and strictly increasing on [c, d) function a(x) ≥ 0 and a non-negative kernel k(x, y) satisfying the condition: there exists a constant D ≥ 1 such that (2.9) Corollary 2.2. Let 1 < q < p < ∞, r = p q/(p − q) and the operator K a be defined by (2.8) with a differentiable and strictly increasing on Now we can state the next auxiliary result of the paper.
where B b,0 and B b,1 are defined by (2.6) and (2.7) respectively.
For the proof of Lemma 2.2, we refer first to the paper [5].   Proof. [Proof of Lemma 2.2.] By the substitution τ = b −1 (t) we reduce B b,0 , B b,1 to the constants B 0 and B 1 respectively. Analogously we arrive to B 0 := B b(x)=x,0 and B 1 := B b(x)=x,1 . Now it is sufficient to prove (2.10), (2.11) for the case b(x) = x only. Note that under this condition the kernel k(x, t) = k(x, b(τ )) =: k b (x, τ ) becomes satisfying (2.2). Moreover, without loss of generality we can assume that k b (x, τ ) is non-decreasing with respect to the variable x and non-increasing with respect to y. Otherwise we can consider the kernelk b (x, τ ) = sup τ ≤s≤x k b (s, τ ), where k b (s, τ ) = sup τ ≤t≤s k b (s, t), which satisfies the pointed monotonicity properties and For the proof of (2.10) we put and denote Z 1 := {k ∈ Z : t k < t k+1 } . Observe that the function W q (t) := d t k q (x, t)w q (x)dx is continuous, non-increasing and such that Put V (t) := t c v p (y)dy. We have Applying Proposition 2.1(b) we find that Analogously, to prove (2.11) we put dy is continuous, non-decreasing and such that Applying Proposition 2.1(a) we obtain Analogous result is true for the constants B a,0 and B a,1 .
Suppose that a function a(x) ≥ 0 is differentiable and strictly increasing on [c, d), and a continuous function k(x, y) ≥ 0 on R a := {(x, y): x > 0, a(x) < y < a(d)} is from the class (2.9). Then where B a,0 , B a,1 are defined by (2.10) and (2.10) respectively.
We conclude this section by the following two statements. (2.17)

The main result
Let functions σ(x) and ρ(y) on on R + such that for any fixed x > 0 the function k p (x, y)v p (y) is locally integrable on R + with respect to the variable y as well as for any y > 0 the function k q (x, y)w q (x) is locally integrable on R + with respect to x, we define two fairways -the functions By assumptions of the definition the fairways σ(x) and ρ(y) are continuous functions. Put and denote The main result of the paper is proved in Section 4 and reads Analogously we obtain a similar result for K with k(x, y) satisfying the condition (1.4):

Proof of Theorem 3.1
(a) The lower estimate. Let 1 < p ≤ q < ∞. It follows from (1.5) of Theorem 1.1 that On the strength of (4 For the opposite estimate we put τ 0 := ρ(a(t)) and write .
(b) Now we consider the case 1 < q < p < ∞. We prove first the upper estimate in (3.4). To this end we take a point sequence Breaking the semiaxis (0, ∞) by points {ξ k } k∈Z we decompose the operator K into the sum of block-diagonal operators T and S such that where Kernels k(x, y) of the operators T k and S k satisfy the condition (1. respectively. Since k∈Z [ξ k , ξ k+1 ) = (0, ∞) it follows from (4.4) -(4.5) that To estimate a norm of the operator S k we take into account two key points )) and consider three only possible variants: In the case (i) we have Applying Corollary 2.1 with K b = S k,1 , c = ξ k , d = s ρ and Lemma 2.2 we obtain To estimate B b,1 note that in view of (1.3) we have k(x, y) Next, we decompose the operator H k,1 into a sum by using (1. By Hölder's inequality and (1.3) Denote H k,2 := H k,2 Lp(b(ξ k ),b(sρ))→Lq(sσ ,ξ k+1 ) . By Hölder's inequality and (1.3)  1 (σ(t)). To estimate S k,3 we use again Corollary 2.1 with c = s σ , d = ξ k+1 and Lemma 2.2: (4.20) The estimate 1 (σ(t)). To estimate the last operator norm H k,3 := H k,3 Lp(b(sρ),b(sσ ))→Lq(sσ,ξ k+1 ) we make a decomposition Thus, by (4.8) -(4.23) it holds for the case (i) that In the case (iii) we have The estimate can be obtained analogously to the case (i). The next operator H k,1 should be decomposed by (1. Putting H k,1 := H k,1 Lp(b(ξ k ),b(sσ ))→Lq(sσ,sρ) , by Hölder's inequality and (1.3) we obtain on the strength of a(t) < a(ξ k+1 ) = b(ξ k ) and a(s ρ ) < b(ξ k ) = σ(s σ ) ≤ σ(t) =⇒ s ρ < a −1 (σ(t)). For S k,2 we use Corollary 2.1 with c = s σ and d = s ρ : Analogously to the case (i) it holds that (4.33) Now from (4.25) -(4.33) we have the estimate (4.24) for the case (iii) too. The case (ii) is clear from either (i) or (iii). Now we obtain from (4.5) by Lemma 2.1 To estimate the norm of the operator T k we decompose it by (1.3), (4.6) into the sum Thus and from (4.34) the upper estimate in (3.4) follows in view of (4.4).

The lower estimate. Suppose that the inequality (4.43)
Kf q ≤ K f p holds. To prove ∞) and (4.43) holds we have Using the explicit form of the operator K we find that Note that the kernel k(x, y) is satisfying a condition following from (1.3) : Break the semiaxis (0, ∞) by the point sequence (4.45) and put and (4.48), we have It follows from (4.49) that k(x, z) Thus and from (4.50) (4.51) Combining (4.44) and (4.51) we obtain the lower estimate in (3.4). The assertion about compactness for q < p is a direct corollary of the obtained boundedness criterion and Ando's theorem.

Criterion cases
Here we consider two cases when the results of Theorems 3.1 and 3.2 give criteria.

and K is compact if and only if
and K is compact if and only if B ρ , B σ < ∞.
Proofs of Theorems 5.1 and 5.2 easy follow from Theorems 3.1 and 3.2.

Examples
We conclude the paper by illustrating some of our results by examples. The first of them is about a criterion case from Theorem 5.1.
According to (3.1) Integrating by parts we find x.