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

Abstract

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.

1 Introduction

Let $I=(0, \infty )$ and let v, u be almost everywhere positive and locally integrable functions on the interval I.

Let $1< p,q<\infty$, and $p'=\frac{p}{p-1}$. Let us denote by $L_{p,v}\equiv L_{p }(v,I)$ the set of measurable functions f on I for which

$$ \Vert f \Vert _{p,v}= \biggl( \int _{0}^{\infty } \bigl\vert f(x) \bigr\vert ^{p}v(x)\,dx \biggr)^{ \frac{1}{p}}< \infty .$$

Let W be a positive, strictly increasing, and locally absolutely continuous function on the interval I. Let $\frac{dW(x)}{dx}=w(x)$ for almost all $x\in I$.

Consider the operator

$$ T_{\alpha , \beta }f(x)= \int _{0}^{x} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }u(s)f(s)w(s)\,ds}{(W(x)-W(s))^{1-\alpha }}, \quad x\in I, $$

(1.1)

where $\alpha >0$, $\beta \geq 0$.

When $\beta =0$, the operator $T_{\alpha , \beta }$ has the form

$$ T_{\alpha }f(x)= \int _{0}^{x} \frac{u(s)f(s)w(s)\,ds}{(W(x)-W(s))^{1-\alpha }}, $$

(1.2)

which is called the fractional integration operator of the function f over the function W for $u\equiv 1$.

Operator (1.2) becomes the Riemann–Liouville fractional integration operator for $u\equiv 1$, $W(x)=x$, which were investigated in papers [1–4]. We obtain the Hadamard fractional integration operator from (1.2) for $u\equiv 1$, $W(x)=\ln x$.

Further, we assume that W is nonnegative on I and $\lim_{x\rightarrow 0^{+}} W(x)=0$.

The boundedness and compactness of operator (1.2) from $L_{p,w}$ to $L_{q,v}$ is obtained in the paper [5] for $\alpha >\frac{1}{p}$, $1< p\leq q<\infty$, and $0< q< p<\infty$. When $\alpha >1$, 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 $\beta =1$ 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 $1< p\leq q<\infty$.

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 $0\cdot \infty$ is considered to be zero. We will write $A\ll B$ or $B\ll A$ if there is a number $c>0$ and $A\leq cB$. The relation $A\approx B$ means $A\ll B$ and $A\gg B$. Z is the set of integers, and $\chi _{(a,b)}$ is the characteristic function of the interval $(a,b)\subset I$.

2 Auxiliary statements

Consider the Hardy operator

$$ Hf(x)=\varphi (x) \int _{0}^{x}u(s)f(s)w(s)\,ds$$

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.

Theorem A

Let $1< p\leq q<\infty$. Then the Hardy operator H is bounded from $L_{p,w}$ to $L_{q,v}$ if and only if

$$ A=\sup_{z>0} \biggl( \int _{z}^{\infty }\varphi ^{q}(x)v(x)\,dx \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{z}u^{p'}(s)w(s)\,ds \biggr)^{\frac{1}{p'}}< \infty ;$$

moreover $\|H\|\approx A$, where $\|H\|$ is the norm of the operator H from $L_{p,w}$ to $L_{q,v}$.

Now let us consider the properties of the function $\ln \frac{W(x)}{W(x)-W(s)}$:

$$ \frac{W(s)}{W(x)-W(s)}>\ln \frac{W(x)}{W(x)-W(s)}= \int _{0}^{x} \frac{w(s)\,ds}{W(x)-W(s)}> \frac{W(s)}{W(x)}, \quad x>s>0.$$

The function $\frac{1}{W(s)}\cdot \ln \frac{W(x)}{W(x)-W(s)}$ increases with respect to $s\in (0,x)$. Indeed

$$ \frac{\partial }{\partial s} \biggl(\frac{1}{W(s)}\cdot \ln \frac{W(x)}{W(x)-W(s)} \biggr)= \frac{w(s)}{W^{2}(s)} \biggl( \frac{W(s)}{W(x)-W(s)}- \ln \frac{W(x)}{W(x)-W(s)} \biggr)>0$$

for $s\in (0,x)$.

3 Boundedness of the operator $T_{\alpha , \beta }$

The main result of this section is the following.

Theorem 3.1

Let $0<\alpha <1$, $\frac{1}{\alpha }< p\leq q<\infty$, and $\beta \geq 0$. Let the function u be nonincreasing on I. Then the operator $T_{\alpha , \beta }$, defined by formula (1.1), is bounded from $L_{p,w}$ to $L_{q,v}$ if and only if

$$ A_{\alpha ,\beta }=\sup_{z>0} \biggl( \int _{z}^{\infty }v(x)W^{q( \alpha -\beta -1)}(x)\,dx \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{z}W^{ \beta p'}(s)u^{p'}(s)w(s) \,ds \biggr)^{\frac{1}{p'}}< \infty ;$$

moreover $\|T_{\alpha , \beta }\|\approx A_{\alpha ,\beta }$, where $\|T_{\alpha , \beta }\|$ is the norm of operator (1.1) from $L_{p,w}$ to $L_{q,v}$.

Proof of Theorem 3.1

Necessity. Let operator (1.1) be bounded from $L_{p,w}$ to $L_{q,v}$. Using the properties of the function $\ln \frac{W(x)}{W(x)-W(s)}$ for $x>s>0$, we have

$$ \frac{1}{ (W(x)-W(s) )^{1-\alpha }}\geq \frac{1}{ (W(x) )^{1-\alpha }} \quad \text{for almost all } x\in I. $$

Substituting the obtained relations in the expressions of operator (1.1) for $f\geq 0$, we obtain

$$ T_{\alpha ,\beta }f(x)\geq W^{\alpha -\beta -1}(x) \int _{0}^{x} W^{\beta }(s)u(s)f(s)w(s) \,ds\equiv H_{\alpha ,\beta }f(x). $$

(3.1)

The boundedness of the operator $T_{\alpha ,\beta }$ from $L_{p,w}$ to $L_{q,v}$ implies the boundedness of the Hardy operator $H_{\alpha ,\beta }$ from $L_{p,w}$ to $L_{q,v}$ and $\|T_{\alpha ,\beta }\|\gg \|H_{\alpha ,\beta }\|$. Then, by Theorem A, the value of $A_{\alpha ,\beta }<\infty$ and for the norm $\|H_{\alpha ,\beta }\|$ of the operator $H_{\alpha ,\beta }$ there is an estimate $A_{\alpha ,\beta }\ll \|H_{\alpha ,\beta }\|$. Then, by virtue of (3.1),

$$ A_{\alpha ,\beta }\ll \Vert T_{\alpha ,\beta } \Vert . $$

(3.2)

Sufficiency. Let $A_{\alpha ,\beta }<\infty$. Since W is a strictly increasing continuous function such that $\lim_{x\rightarrow 0^{+}} W(x)=0$, then for $k\in Z$ define $x_{k}=\sup \{x: W(x)\leq 2^{k}, x\in I\}$.

Let $k_{\infty }=\inf \{k\in Z: \sup_{x>0}W(x)\leq 2^{k}\}$. Then $0< x_{k}< x_{k+1}$ for $k+1\leq k_{\infty }$. Then, without limiting generality, we put $k_{\infty }=\infty$. Then $I=\bigcup_{k \in Z}[x_{k}, x_{k+1})$. Let $f\geq 0$.

We have

$$\begin{aligned} \Vert T_{\alpha ,\beta }f \Vert ^{q}_{q,v}&= \sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{0}^{x} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}u(s)f(s)w(s) \,ds \biggr)^{q}\,dx \\ &=\sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \biggl[ \biggl( \int _{0}^{x_{k-1}}+ \int _{x_{k-1}}^{x} \biggr) \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}u(s)f(s)w(s) \,ds \biggr]^{q}\,dx \\ &\ll J_{1}+J_{2}. \end{aligned}$$

(3.3)

Now we estimate $J_{1}$ and $J_{2}$ separately.

$$ J_{1}=\sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{0}^{x_{k-1}} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}u(s)f(s)w(s) \,ds \biggr)^{q}\,dx $$

(using the monotonicity of the function $\frac{1}{W(s)}\ln \frac{W(x)}{W(x)-W(s)}$)

$$\begin{aligned}& =\sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{0}^{x_{k-1}} \frac{ (\frac{1}{W(s)}\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }} W^{\beta }(s)u(s)f(s)w(s)\,ds \biggr)^{q}\,dx \\& \leq \sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \frac{ (\frac{1}{W(x_{k-1})}\ln \frac{W(x)}{W(x)-W(x_{k-1})} )^{q\beta }}{(W(x)-W(x_{k-1}))^{q(1-\alpha )}} \biggl( \int _{0}^{x_{k-1}}W^{\beta }(s)u(s)f(s)\,ds \biggr)^{q}\,dx. \end{aligned}$$

(3.4)

Since $W(x_{k-1})=2^{k-1}=\frac{1}{2}W(x_{k})\leq \frac{1}{2}W(x)$ and $W(x_{k-1})=\frac{1}{4}W(x_{k+1})\geq \frac{1}{4}W(x)$ for $x_{k}\leq x\leq x_{k+1}$, then from (3.4) it follows

$$ J_{1}\leq 2^{-(2q\beta +q(1-\alpha ))}\ln ^{q\beta }2 \biggl( \int _{0}^{\infty }v(x)W^{q(\alpha -\beta -1)}(x) \biggl( \int _{0}^{x}W^{\beta }(s)u(s)f(s)w(s) \,ds \biggr)^{q}\,dx \biggr). $$

Hence, based on Theorem A,

$$ J_{1}\ll A_{\alpha ,\beta }^{q} \Vert f \Vert ^{q}_{p,w}. $$

(3.5)

Now, we estimate $J_{2}$. Using the nonincreasing function u for estimating $J_{2}$ and applying Hölder’s inequality, we find

$$\begin{aligned} J_{2}={}&\sum_{k} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{x_{k-1}}^{x} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}u(s)f(s)w(s) \,ds \biggr)^{q}\,dx \\ ={}&\sum_{k} \int _{x_{k}}^{x_{k+1}}v(x)W^{-q\beta }(x) \biggl( \int _{x_{k-1}}^{x} \frac{ (\frac{W(x)}{W(s)}\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }} W^{\beta }(s)u(s)f(s)w(s)\,ds \biggr)^{q}\,dx \\ \leq{}& \sum_{k}u^{q}(x_{k-1})W^{q\beta }(x_{k+1}) \\ &{}\times \int _{x_{k}}^{x_{k+1}}v(x)W^{-q \beta }(x) \biggl( \int _{x_{k-1}}^{x} \frac{ (\frac{W(x)}{W(s)}\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }} f(s)w(s)\,ds \biggr)^{q}\,dx \\ \leq {}&\sum_{k}u^{q}(x_{k-1})W^{q\beta }(x_{k+1}) \int _{x_{k}}^{x_{k+1}}v(x)W^{-q \beta }(x) \biggl( \int _{x_{k-1}}^{x} \frac{ (\frac{W(x)}{W(s)}\ln \frac{W(x)}{W(x)-W(s)} )^{p'\beta }}{(W(x)-W(s))^{p'(1-\alpha )}}w(s)\,ds \biggr)^{\frac{q}{p'}} \\ &{}\times \biggl( \int _{x_{k-1}}^{x_{k+1}}f^{p}(s)w(s)\,ds \biggr)^{ \frac{q}{p}}\,dx. \end{aligned}$$

(3.6)

We replace the variables $W(s)=W(x)t$ in the following expression:

$$\begin{aligned}& \int _{x_{k-1}}^{x} \frac{ (\frac{W(x)}{W(s)}\ln \frac{W(x)}{W(x)-W(s)} )^{p'\beta }}{(W(x)-W(s))^{p'(1-\alpha )}}w(s)\,ds \\& \quad = W^{p'(\alpha -1)+1}(x) \int _{2^{k-1}W^{-1}(x)}^{1} \frac{ (\frac{1}{t}\ln \frac{1}{1-t} )^{p'\beta }}{(1-t)^{p'(1-\alpha )}}\,dt \\& \quad \leq W(x_{k+1}) W^{p'(\alpha -1)}(x) \int _{2^{k-1}W^{-1}(x_{k+1})}^{1} \frac{ (\frac{1}{t}\ln \frac{1}{1-t} )^{p'\beta }}{(1-t)^{p'(1-\alpha )}}\,dt \\& \quad \leq 2^{2p'\beta }W(x_{k+1}) W^{p'(\alpha -1)}(x) \int _{0}^{1} \frac{ (\ln \frac{1}{1-t} )^{p'\beta }}{(1-t)^{p'(1-\alpha )}}\,dt \\& \quad = \gamma 2^{2p'\beta }W(x_{k+1}) W^{p'(\alpha -1)}(x) \end{aligned}$$

(3.7)

for $x_{k}\leq x\leq x_{k+1}$, where $\gamma =\int _{0}^{1} \frac{ (\ln \frac{1}{1-t} )^{p'\beta }}{(1-t)^{p'(1-\alpha )}}\,dt= \int _{1}^{\infty }z^{p'\beta }e^{-z p'(\alpha -\frac{1}{p})}\,dz< \infty$.

In the latter ratio, we used a replacement $\frac{1}{1-t}=e^{z}$. Substituting the obtained estimates (3.7) in (3.6), we get

$$\begin{aligned} J_{2} \ll& \sum_{k} u^{q}(x_{k-1})W^{q\beta +\frac{q}{p'}}(x_{k+1}) \\ &{}\times \int _{x_{k}}^{x_{k+1}}v(x)W^{q(\alpha -\beta -1)}(x) \biggl( \int _{x_{k-1}}^{x_{k+1}} \bigl\vert f(s) \bigr\vert ^{p}w(s)\,ds \biggr)^{ \frac{q}{p}}\,dx. \end{aligned}$$

(3.8)

Next, we need the following estimation:

$$\begin{aligned} u^{q}(x_{k-1})W^{q\beta +\frac{q}{p'}}(x_{k+1})&=2^{2(q\beta + \frac{q}{p'})} \bigl( u^{p'}(x_{k-1})W^{p'\beta +1}(x_{k-1}) \bigr)^{ \frac{q}{p'}} \\ &=2^{2q(\beta +\frac{1}{p'})}\bigl(p'\beta +1\bigr)^{\frac{q}{p'}} \biggl( u^{p'}(x_{k-1}) \int _{0}^{x_{k-1}}W^{p'\beta }(s)w(s)\,ds \biggr)^{\frac{q}{p'}} \\ &\ll \biggl( \int _{0}^{x_{k-1}}u^{p'}(s)W^{p'\beta }(s)w(s) \,ds \biggr)^{\frac{q}{p'}}. \end{aligned}$$

Substituting the obtained estimate in (3.8) and using Jensen’s inequality, by virtue of $p \le q$, we have

$$\begin{aligned} J_{2}\ll {}&\sum_{k} \biggl( \int _{0}^{x_{k-1}}u^{p'}(s)W^{p' \beta }(s)w(s) \,ds \biggr)^{\frac{q}{p'}} \biggl( \int _{x_{k-1}}^{ \infty }v(x)W^{q(\alpha -\beta -1)}(x)\,dx \biggr) \\ &{}\times\biggl( \int _{x_{k-1}}^{x_{k+1}} \bigl\vert f(s) \bigr\vert ^{p}w(s)\,ds \biggr)^{\frac{q}{p}} \\ \leq{}& A_{\alpha ,\beta }^{q} \sum_{k} \biggl( \int _{x_{k-1}}^{x_{k+1}} \bigl\vert f(s) \bigr\vert ^{p}w(s)\,ds \biggr)^{\frac{q}{p}}\,dx \ll A_{\alpha ,\beta }^{q} \Vert f \Vert ^{q}_{p,w}. \end{aligned}$$

(3.9)

Substituting the obtained estimates (3.5) and (3.9) in (3.3), we get

$$ \Vert T_{\alpha ,\beta }f \Vert _{q,v}\ll A_{\alpha ,\beta } \Vert f \Vert _{p,w}, $$

i.e., the boundedness of the operator $T_{\alpha ,\beta }$ from $L_{p,w}$ to $L_{q,v}$ and the estimate $\|T_{\alpha ,\beta }\|\ll A_{\alpha ,\beta }$ holds for the norm $\|T_{\alpha ,\beta }\|$ from $L_{p,w}$ to $L_{q,v}$, which together with (3.2) gives $\|T_{\alpha ,\beta }\|\approx A_{\alpha ,\beta }$. Theorem 3.1 is proved. □

4 The compactness of the operator $T_{\alpha , \beta }$

Assume that

$$ A_{\alpha ,\beta }(z)= \biggl( \int _{z}^{\infty }v(x)W^{q( \alpha -\beta -1)}(x)\,dx \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{z}W^{ \beta p'}(s)u^{p'}(s)w(s) \,ds \biggr)^{\frac{1}{p'}}.$$

Theorem 4.1

Let $0<\alpha <1$, $\frac{1}{\alpha }< p\leq q<\infty$, and $\beta \ge 0$. Let the function u be nonincreasing on the interval I. Then the operator $T_{\alpha , \beta }$ is compact from $L_{p,w}$ to $L_{q,v}$ if and only if $A_{\alpha ,\beta }<\infty$ and

$$ \lim_{z\rightarrow 0^{+}}A_{\alpha ,\beta }(z)=\lim_{z \rightarrow \infty }A_{\alpha ,\beta }(z)=0. $$

(4.1)

Proof of Theorem 4.1

Necessity. Let the operator $T_{\alpha , \beta }$ be compact from $L_{p,w}$ to $L_{q,v}$. Then it is bounded from $L_{p,w}$ to $L_{q,v}$ and $A_{\alpha ,\beta }<\infty$ according to Theorem 3.1. First, let us show the fulfilment of $\lim_{z\rightarrow 0^{+}}A_{\alpha ,\beta }(z)=0$. Consider the family of functions $\{f_{t}\}_{t\in I}$:

$$f_{t}(x)=\chi _{(0,t)}(x)u^{p'-1}(x)W^{(p'-1)\beta }(x) \biggl(\int _{0}^{t}u^{p'}(s)W^{\beta p'}(s)w(s)\,ds \biggr)^{-\frac{1}{p}}, \quad x,t\in I. $$

Let us note that

$$\begin{aligned} \Vert f_{t} \Vert _{p,w}&= \biggl( \int _{0}^{t} \bigl\vert f_{t}(x) \bigr\vert ^{p}w(x)\,dx \biggr)^{\frac{1}{p}} \\ &= \biggl( \int _{0}^{t}u^{p'}(x)W^{\beta p'}(x)w(x) \,dx \biggr)^{ \frac{1}{p}} \biggl( \int _{0}^{t}u^{p'}(s)W^{\beta p'}(s)w(s) \,ds \biggr)^{-\frac{1}{p}}=1, \end{aligned}$$

i.e., $f_{t}\in L_{p,w}$ for all $t\in I$. Let us show that $f_{t}$ converges weakly to zero if $t\rightarrow 0^{+}$. For arbitrary $g\in (L_{p,w})^{\ast }=L_{p',w^{1-p'}}$, we have

$$\begin{aligned} \int _{0}^{\infty }f_{t}(x)g(x)\,dx&\leq \biggl( \int _{0}^{t} \bigl\vert f_{t}(x) \bigr\vert ^{p}w(x)\,dx \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{t} \bigl\vert g(s) \bigr\vert ^{p'}w^{1-p'}(s)\,ds \biggr)^{\frac{1}{p'}}\\ &= \biggl( \int _{0}^{t} \bigl\vert g(s) \bigr\vert ^{p'}w^{1-p'}(s)\,ds \biggr)^{\frac{1}{p}}. \end{aligned}$$

Whence it follows that $f_{t}$ weakly converges to zero if $t\rightarrow 0^{+}$. Since the operator $T_{\alpha , \beta }$ is compact from $L_{p,w}$ to $L_{q,v}$, then

$$ \lim_{t\rightarrow 0^{+}} \Vert T_{\alpha ,\beta }f_{t} \Vert _{q,v}=0. $$

(4.2)

We have

$$\begin{aligned} \Vert T_{\alpha ,\beta }f_{t} \Vert _{q,v}^{q} &\geq \int _{t}^{\infty }v(x) \biggl( \int _{0}^{t} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}u(s)f_{t}(s)w(s) \,ds \biggr)^{q}\,dx \\ &\geq \int _{t}^{\infty }v(x)W^{q(\alpha -\beta -1)}(x) \biggl( \int _{0}^{t}W^{\beta }(s)u(s)f_{t}(s)w(s) \,ds \biggr)^{q}\,dx \\ &= \int _{t}^{\infty }v(x)W^{q(\alpha -\beta -1)}(x)\,dx \biggl( \int _{0}^{t}W^{p'\beta }(s)u^{p'}(s)w(s) \,ds \biggr)^{ \frac{q}{p'}}= \bigl( A_{\alpha ,\beta }(t) \bigr)^{\frac{1}{q}}. \end{aligned}$$

Whence and from (4.2) it follows that $\lim_{t\rightarrow 0^{+}}A_{\alpha ,\beta }(t)=0$. We now prove that $\lim_{t\rightarrow \infty }A_{\alpha ,\beta }(t)=0$. The compactness of the adjoint operator

$$ T_{\alpha ,\beta }^{\ast }g(s)= u(s)w(s) \int _{s}^{\infty } \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}g(x)\,dx $$

from $L_{q',v^{1-q'}}$ to $L_{p',w^{1-p'}}$ follows from the compactness of the operator $T_{\alpha , \beta }$ from $L_{p,w}$ to $L_{q,v}$.

Introduce the family of functions $\{g_{t}\}_{t\in I}$:

$$ g_{t}(x)=\chi _{(t,\infty )}(x)W^{(q-1)(\alpha -\beta -1)}(x)v(x) \biggl( \int _{t}^{\infty }W^{q(\alpha -\beta -1)}(s)v(s)\,ds \biggr)^{-\frac{1}{q'}}. $$

It is easy to see that $g_{t}\in L_{q',v^{1-q'}}$ for all $t\in I$. Indeed,

$$\begin{aligned} \Vert g_{t} \Vert _{q',v^{1-q'}} =& \biggl( \int _{t}^{\infty } \bigl\vert W^{(q-1)( \alpha -\beta -1)}(x)v(x) \bigr\vert ^{q'}v^{1-q'}(x)\,dx \biggr)^{\frac{1}{q'}} \biggl( \int _{t}^{\infty }W^{q(\alpha -\beta -1)}(s)v(s)\,ds \biggr)^{-\frac{1}{q'}}\\ =&1. \end{aligned}$$

Let $f\in (L_{q',v^{1-q'}})^{\ast }=L_{q,v}$ be an arbitrary function. Then

$$\begin{aligned} \int _{0}^{\infty }g_{t}(x)f(x)\,dx \leq& \biggl( \int _{t}^{ \infty } \bigl\vert g_{t}(x) \bigr\vert ^{q'}v^{1-q'}(x)\,dx \biggr)^{\frac{1}{q'}} \biggl( \int _{t}^{\infty } \bigl\vert f(s) \bigr\vert ^{q}v(s)\,ds \biggr)^{\frac{1}{q}}\\ =& \biggl( \int _{t}^{\infty } \bigl\vert f(x) \bigr\vert ^{q}v(x)\,dx \biggr)^{ \frac{1}{q}}. \end{aligned}$$

This implies that $\lim_{t\rightarrow \infty }\int _{0}^{\infty }g_{t}(x)f(x)\,dx=0$ for all $f\in L_{q,v}$. Consequently, the family of functions $\{g_{t}\}_{t\in I}\subset L_{q',v^{1-q'}}$ weakly converges to zero at $t\rightarrow \infty$.

Then, from the compactness $T_{\alpha ,\beta }^{\ast }: L_{q',v^{1-q'}}\rightarrow L_{p',w^{1-p'}}$, we have

$$ \lim_{t\rightarrow \infty } \bigl\Vert T_{\alpha ,\beta }^{\ast }g_{t} \bigr\Vert _{p',w^{1-p'}}=0. $$

(4.3)

Since

$$\begin{aligned} \bigl\Vert T_{\alpha ,\beta }^{\ast }g_{t} \bigr\Vert _{p',w^{1-p'}} \geq& \biggl( \int _{0}^{t}u^{p'}(s) w(s) \biggl( \int _{t}^{\infty } \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}g_{t}(x) \,dx \biggr)^{p'}\,ds \biggr)^{\frac{1}{p'}} \\ \geq& \biggl( \int _{0}^{t}u^{p'}(s) W^{p'\beta }(s)w(s) \biggl( \int _{t}^{\infty }W^{(\alpha -\beta -1)}(x)g_{t}(x) \,dx \biggr)^{p'}\,ds \biggr)^{\frac{1}{p'}} \\ = &\biggl( \int _{0}^{t}u^{p'}(s) W^{p'\beta }(s)w(s) \,ds \biggr)^{ \frac{1}{p'}} \biggl( \int _{t}^{\infty }W^{q(\alpha -\beta -1)}(x)v(x)\,dx \biggr) \\ &{}\times\biggl( \int _{t}^{\infty }W^{q(\alpha -\beta -1)}(x)v(x)\,dx \biggr)^{-\frac{1}{q'}}= A_{\alpha ,\beta }(t), \end{aligned}$$

then (4.3) implies that $\lim_{t\rightarrow \infty }A_{\alpha ,\beta }(t)=0$. The necessity has been proven.

Sufficiency. Let $A_{\alpha ,\beta }<\infty$ and (4.1) be fulfilled. We define $P_{c}f=\chi _{(0,c]}f$, $P_{cd}f=\chi _{(c,d]}f$ and $Q_{d}f=\chi _{(d,\infty )}f$ for $0< c< d<\infty$. Then $f=P_{c}f+P_{cd}f+Q_{d}f$ and, by virtue of $P_{c}T_{\alpha ,\beta }P_{cd}\equiv 0$, $P_{c}T_{\alpha ,\beta }Q_{d}\equiv 0$ and $P_{cd}T_{\alpha ,\beta }Q_{d}\equiv 0$, we obtain

$$ T_{\alpha ,\beta }f= P_{cd}T_{\alpha ,\beta }P_{cd}f+P_{c}T_{\alpha , \beta }P_{c}f+P_{cd}T_{\alpha ,\beta }P_{c}f+Q_{d}T_{\alpha ,\beta }f. $$

(4.4)

Let us show that the operator $P_{cd}T_{\alpha ,\beta }P_{cd}$ is compact from $L_{p,w}$ to $L_{q,v}$. Since $P_{cd}T_{\alpha ,\beta }P_{cd}\times f(x)=0$ for $x\in I\backslash (c,d)$, 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

$$ Tf(x)= \int _{c}^{d}K(x,s)f(s)\,ds $$

from $L_{p}(c,d)$ to $L_{q}(c,d)$ with the kernel

$$ K(x,s)=u(s)v^{\frac{1}{q}}(x)\chi _{(c,d)}(x-s)w^{\frac{1}{p'}}(s) \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{\beta }}{(W(x)-W(s))^{1-\alpha }}. $$

Let $\{x_{k}\}_{k\in Z}$ be a sequence constructed by the function W from Theorem 3.1. Then there exist numbers i and n such that $x_{i}\leq c< x_{i+1}$, $x_{n}< d\leq x_{n+1}$. 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

$$\begin{aligned}& \int _{c}^{d} \biggl( \int _{c}^{d} \bigl\vert K(x,s) \bigr\vert ^{p'}\,ds \biggr)^{\frac{q}{p'}}\,dx \\& \quad \leq \sum_{k=i} ^{n} \int _{x_{k}}^{x_{k+1}}v(x) \biggl[ \biggl( \int _{0}^{x_{k-1}}+ \int _{x_{k-1}}^{x} \biggr) \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{p'\beta }}{(W(x)-W(s))^{p'(1-\alpha )}}u^{p'}(s)w(s) \,ds \biggr]^{\frac{q}{p'}}\,dx \\& \quad \ll F_{1}+F_{2}. \end{aligned}$$

(4.5)

Estimate $F_{1}$ and $F_{2}$. Analogously to the estimate of $J_{1}$,

$$\begin{aligned} F_{1}&=\sum_{k=i} ^{n} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{0}^{x_{k-1}} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{p'\beta }}{(W(x)-W(s))^{p'(1-\alpha )}}u^{p'}(s)w(s) \,ds \biggr)^{\frac{q}{p'}}\,dx \\ &\leq \sum_{k=i} ^{n} \int _{x_{k}}^{x_{k+1}}v(x) \frac{ (\frac{1}{W(x_{k-1})}\ln \frac{W(x)}{W(x)-W(x_{k-1})} )^{q\beta }}{(W(x)-W(x_{k-1}))^{q(1-\alpha )}}\,dx \biggl( \int _{0}^{x_{k-1}}W^{p'\beta }(s)u^{p'}(s)w(s) \,ds \biggr)^{\frac{q}{p'}} \\ &\ll \sum_{k=i} ^{n} A_{\alpha ,\beta }^{q}(x_{k-1}) \leq (n-i+1)A_{\alpha ,\beta }^{q}. \end{aligned}$$

(4.6)

Analogously to the estimate of $J_{2}$,

$$\begin{aligned} F_{2}&=\sum_{k=i} ^{n} \int _{x_{k}}^{x_{k+1}}v(x) \biggl( \int _{x_{k-1}}^{x} \frac{ (\ln \frac{W(x)}{W(x)-W(s)} )^{p'\beta }}{(W(x)-W(s))^{p'(1-\alpha )}}u^{p'}(s)w(s) \,ds \biggr)^{\frac{q}{p'}}\,dx \\ &\ll \sum_{k=i} ^{n}u^{q}(x_{k-1})W^{q(\beta + \frac{1}{p'})}(x_{k-1}) \int _{x_{k}}^{x_{k+1}}v(x) W^{q( \alpha -\beta -1)}(x)\,dx \biggl( \int _{0}^{1} \frac{ (\ln \frac{1}{1-t} )^{p'\beta }}{(1-t)^{p'(1-\alpha )}}\,dt \biggr)^{\frac{q}{p'}} \\ &\ll (n-i+1)A_{\alpha ,\beta }. \end{aligned}$$

(4.7)

Substituting (4.6) and (4.7) into (4.5), we obtain

$$ \int _{c}^{d} \biggl( \int _{c}^{d} \bigl\vert K(x,s) \bigr\vert ^{p'}\,ds \biggr)^{\frac{q}{p'}}\,dx< \infty . $$

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 $P_{cd}T_{\alpha ,\beta }P_{cd}$ from $L_{p,w}$ to $L_{q,v}$. From (4.4) we have

$$ \Vert T_{\alpha ,\beta }-P_{cd}T_{\alpha ,\beta }P_{cd} \Vert \leq \Vert P_{c}T_{ \alpha ,\beta }P_{c} \Vert + \Vert P_{cd}T_{\alpha ,\beta }P_{c} \Vert + \Vert Q_{d}T_{ \alpha ,\beta } \Vert . $$

(4.8)

Further, we assume that the right-hand side of (4.8) tends to zero as $c\rightarrow 0$ and $d\rightarrow \infty$. Then the operator $T_{\alpha ,\beta }$ is compact from $L_{p,w}$ to $L_{q,v}$ as the uniform limit of compact operators.

Based on Theorem 3.1, we obtain

$$\begin{aligned} \Vert P_{c}T_{\alpha ,\beta }P_{c}f \Vert _{q,v}&= \biggl( \int _{0}^{ \infty }P_{c}v(x) \bigl\vert T_{\alpha ,\beta }P_{c}f(x) \bigr\vert ^{q}\,dx \biggr)^{ \frac{1}{q}} \\ &\ll \sup_{a< z< c}A_{\alpha ,\beta }(z) \Vert f \Vert _{p,w}. \end{aligned}$$

Therefore, $\|P_{c}T_{\alpha ,\beta }P_{c}\|\ll \sup_{a< z< c}A_{\alpha , \beta }(z)$.

Whence and from $\lim_{z\rightarrow 0^{+}}A_{\alpha ,\beta }(z)=0$ it follows that

$$\begin{aligned}& \begin{gathered} \lim_{c\rightarrow 0^{+}} \Vert P_{c}T_{\alpha ,\beta }P_{c} \Vert =0; \\ \begin{aligned} \Vert P_{cd}T_{\alpha ,\beta }P_{c}f \Vert _{q,v}&= \biggl( \int _{0}^{ \infty }P_{cd}v(x) \bigl\vert T_{\alpha ,\beta }P_{c}f(x) \bigr\vert ^{q}\,dx \biggr)^{ \frac{1}{q}} \\ &\ll A_{\alpha ,\beta }(c,d) \Vert f \Vert _{p,w}, \end{aligned} \end{gathered} \end{aligned}$$

(4.9)

where

$$\begin{aligned} A_{\alpha ,\beta }(c,d)&=\sup_{z>0} \biggl( \int _{z}^{ \infty }P_{cd}v(x)W^{q(\alpha -\beta -1)}(x) \,dx \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{z}W^{p'\beta }(s)P_{c}u^{p'}(s)w(s) \,ds \biggr)^{ \frac{1}{p'}} \\ &= \sup_{0< z< d}A_{\alpha ,\beta }(c,d)=A_{\alpha ,\beta }(c). \end{aligned}$$

Therefore $\|P_{cd}T_{\alpha ,\beta }P_{c}\|\ll A_{\alpha ,\beta }(c)$ and whence

$$ \lim_{c\rightarrow 0^{+}} \Vert P_{cd}T_{\alpha ,\beta }P_{c} \Vert =0 $$

(4.10)

holds.

Similarly, we have

$$\begin{aligned} \Vert Q_{d}T_{\alpha ,\beta }f \Vert _{q,v}&= \biggl( \int _{0}^{\infty }Q_{d}v(x) \bigl\vert T_{\alpha ,\beta }f(x) \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \\ &\ll \sup_{z>d}A_{\alpha ,\beta }(z) \Vert f \Vert _{p,w}. \end{aligned}$$

and $\|Q_{d}T_{\alpha ,\beta }\|\ll \sup_{z>d}A_{\alpha ,\beta }(z)$.

From this and from $\lim_{z\rightarrow \infty }A_{\alpha ,\beta }(z)=0$, we obtain

$$ \lim_{d\rightarrow \infty } \Vert Q_{d}T_{\alpha ,\beta } \Vert =0; $$

(4.11)

From (4.8), (4.9), (4.10), and (4.11) it follows that the operator $T_{\alpha ,\beta }$ is compact from $L_{p,w}$ to $L_{q,v}$. Theorem 4.1 is completely proved. □

5 Consequences

When $W(x)=x$ the operator $T_{\alpha ,\beta }$ has the form

$$ J_{\alpha ,\beta }f(x)= \int _{0}^{x} \frac{ (\ln \frac{x}{x-s} )^{\beta }}{(x-s)^{1-\alpha }}u(s)f(s)\,ds, \quad x>0. $$

Note that the operator

$$ Jf(x)= \int _{0}^{x}\ln \frac{x}{x-s}f(s) \frac{ds}{s} $$

is called [12] the infinitesimal order fractional integration operator.

From Theorems 3.1 and 4.1, as a consequence, we have the following.

Corollary 5.1

Let $0<\alpha <1$, $\frac{1}{\alpha }< p\leq q<\infty$, and $\beta \geq 0$. Let the function u be nonincreasing on I. Then the operator $J_{\alpha , \beta }$ is bounded from $L_{p}$ to $L_{q,v}$ if and only if $A_{\alpha ,\beta }=\sup_{z>0}A_{\alpha ,\beta }(z)<\infty$, where

$$ A_{\alpha ,\beta }(z)= \biggl( \int _{z}^{\infty }v(x)x^{q( \alpha -\beta -1)}(x)\,dx \biggr)^{\frac{1}{q}} \biggl( \int _{0}^{z}s^{ \beta p'}u^{p'}(s) \,ds \biggr)^{\frac{1}{p'}},$$

wherein $\|J_{\alpha , \beta }\|\approx A_{\alpha ,\beta }$, where $\|J_{\alpha , \beta }\|$ is the norm of the operator $J_{\alpha , \beta }$ from $L_{p}$ to $L_{q,v}$.

Corollary 5.2

Let $0<\alpha <1$, $\frac{1}{\alpha }< p\leq q<\infty$, and $\beta \geq 0$. Let the function u be nonincreasing on I. Then the operator $J_{\alpha , \beta }$ is compact from $L_{p}$ to $L_{q,v}$ if and only if $A_{\alpha ,\beta }<\infty$ and

$$ \lim_{z\rightarrow 0^{+}}A_{\alpha ,\beta }(z)=\lim_{z \rightarrow \infty }A_{\alpha ,\beta }(z)=0. $$

Note that the boundedness and compactness of the operator

$$ J'_{\alpha }f(x)= \int _{0}^{x}(x-s)^{\alpha -1} \biggl(\ln \frac{\gamma }{x-s} \biggr)^{\beta }f(s)\,ds $$

from $L_{p}(0,a)$ to $L_{q,v}(0,a)$ or from $L_{p,v}(0,a)$ to $L_{q}(0,a)$ were established in [13], where $0< a\leq \gamma <\infty$, $\alpha >\frac{1}{p}$.