1 Introduction
Let \(I=(a,b)\), \(0\leq a < b\leq\infty\). Let v and u be almost everywhere positive functions, which are locally integrable on the interval I.
Let \(0< p<\infty\) and \(\frac{1}{p}+\frac{1}{p'}=1\). Denote by \(L_{p,v}\equiv L_{p}(v,I)\) the set of all functions f measurable on I such that \(\|f\|_{p,v}:= (\int_{a}^{b}|f(x)|^{p}v(x)\,dx )^{\frac{1}{p}}<\infty\).
Let W be a non-negative, strictly increasing and locally absolutely continuous function on I. Suppose that \(\frac{dW(x)}{dx}=w(x)\), a.e. \(x\in I\).
We consider the Hardy type operator
\(T_{\alpha,\beta}\) defined by
$$ T_{\alpha, \beta}f(x):= \int_{a}^{x}\frac{u(s)W^{\beta}(s)f(s)w(s)\, ds}{ (W(x)-W(s) )^{1-\alpha}},\quad x\in I. $$
(1.1)
When
\(u\equiv1\) and
\(\beta=0\) the operator
\(T_{\alpha,\beta}\) is called the fractional integral operator of a function
f with respect to a function
W ([
1], p.248). When
\(u\equiv1\) and
\(W(x)=x\) the operator (
1.1) becomes the Riemann-Liouville operator
\(I_{\alpha}\) defined by
$$ I_{\alpha}f(x):= \int_{a}^{x}\frac{s^{\beta}f(s)\,ds}{(x-s)^{1-\alpha}}. $$
(1.2)
When
\(u\equiv1\) and
\(W(x)\equiv \ln\frac{x}{a}\),
\(a>0\), this operator is the Hadamard operator
\(\mathcal{H}_{\alpha}\) defined by
$$\mathcal{H}_{\alpha}f(x):= \int_{a}^{x}\frac{ (\ln \frac {s}{a} )^{\beta}f(s)\,ds}{s (\ln \frac{x}{s} )^{1-\alpha}}. $$
Moreover, when
\(u\equiv1\) and
\(W(x)=x^{\sigma}\),
\(\sigma>0\), we get the operator
\(E_{\alpha, \beta}\) of Erdelyi-Kober type ([
1], p.246) defined by
$$E_{\alpha, \beta}f(x):= \sigma \int_{a}^{x}\frac{f(s)s^{\sigma\beta +\sigma-1}\,ds}{ (x^{\sigma}-s^{\sigma} )^{1-\alpha}}. $$
There are a lot of works devoted to the mapping properties of the Riemann-Liouville operator
\(I_{\alpha}\). Two-weighted estimates of the operator
\(I_{\alpha}\) of the order
\(\alpha>1\) in weighted Lebesgue spaces were first obtained in the papers [
2] and [
3]. The singular case
\(0<\alpha<1\) was studied with different restrictions in [
4‐
9] and some others. The most general results among them are given in [
5] and [
9] under the assumption that one of the weight functions is increasing or decreasing.
In this work we investigate the problems of boundedness and compactness of the operator
\(T_{\alpha,\beta}\) defined by (
1.1) from
\(L_{p,w}\) to
\(L_{q,v}\) when
\(0<\alpha<1\). When
\(\alpha>1\) the results follow from the results in [
10].
The operator
\(T_{\alpha,\beta}\) was studied in [
11] and [
12] when
\(u\equiv1\),
\(\beta=0\) and
\(u\equiv1\),
\(\beta>-\frac{1}{p'}\), respectively.
Due to the non-negativity and monotone increase of the function W the limit \(\lim_{x\rightarrow a^{+}}W(x)\equiv W(a)\geq0\) exists.
We also consider the Hardy type operator
\(T_{\alpha, \beta}^{0}\) defined by
$$T_{\alpha, \beta}^{0}f(x):= \int_{a}^{x}\frac{u(s)W^{\beta }_{0}(s)f(s)w(s)\,ds}{ (W_{0}(x)-W_{0}(s) )^{1-\alpha}},\quad x\in I, $$
where
\(W_{0}(x)=W(x)-W(a)\).
Since we also suppose that \(\beta\geq0\), for \(f\geq0\) we have \(T_{\alpha, \beta}f(x)\approx T_{\alpha, \beta}^{0}f(x)+W(a)T_{\alpha, 0}^{0}f(x)\), where the equivalence constants do not depend on x and f. Therefore, without loss of generality, we can assume that \(W(a)=0\). For short writing we denote by \(\|K\|\) the norm of a linear operator K acting from one normalized space to another, since from the context we shall in each case clearly see which spaces the operator is acting between.
The paper is organized as follows: In order not to disturb our discussions later on some auxiliary statements are given in Section
2. The main results concerning the boundedness of operator
\(T_{\alpha, \beta}\), including the corresponding Hardy type inequalities, can be found in Section
3. The main results about the compactness are presented in Section
4. Moreover, in Section
5 some similar results for the dual operator
\(T_{\alpha, \beta}^{0}\) are given. Finally, Section
6 is reserved for some applications (both new and well-known results).
3 Boundedness of the operator \(T_{\alpha, \beta}\)
The main results in this section read as follows.
These two theorems can be reformulated as the following new information in the theory of Hardy type inequalities.
6 Applications
By applying our results in special cases we obtain both new and well-known results. Here we just consider the Riemann-Liouville, Erdelyi-Kober, and Hadamard operators mentioned in our introduction. We use the weight functions
ρ and
ω and consider these operators on the forms
\(\widetilde {I}_{\alpha}\),
\(\widetilde{E}_{\alpha, \gamma}\) and
\(\widetilde {\mathcal{H}}_{\alpha}\) defined by
$$\begin{aligned}& \widetilde{I}_{\alpha}f(x):= \rho(x) \bigl[I_{\alpha}(f\omega ) \bigr](x), \\& \widetilde{E}_{\alpha, \gamma}f(x):= \rho(x) \bigl[E_{\alpha, \gamma}(f\omega) \bigr](x), \\& \widetilde{\mathcal {H}}_{\alpha}f(x):= \rho(x) \bigl[\mathcal {H}_{\alpha}(f\omega ) \bigr](x), \end{aligned}$$
where
ρ and
ω are almost everywhere positive functions locally summable on
I with degrees
q and
\(p'\), respectively.
The action of the operator
\(T_{\alpha,\beta}\) from
\(L_{p,v}\) to
\(L_{q,w}\) is equivalent to the action of the operator
$$\widetilde{T}_{\alpha,\beta}f(x)=v^{\frac{1}{q}}(x) \int_{a} ^{x}\frac{u(s)W^{\beta}(s)w^{\frac{1}{p'}}(s)f(s)\,ds}{ (W(x)-W(s) )^{1-\alpha}} $$
from
\(L_{p}\) to
\(L_{q}\). Therefore, in the case
\(W(x)=x\) we have
\(\rho (x)=v^{\frac{1}{q}}(x)\),
\(\omega(x)=u(x)x^{\beta}\) and
$$\widetilde{I}_{\alpha}f(x)=\rho(x) \int_{a} ^{x}\frac{\omega (s)f(s)\,ds}{ (x-s )^{1-\alpha}}. $$
If
\(W(x)=x^{\sigma}\),
\(\sigma>0\), then
\(u(s)W^{\beta}(s)w^{\frac {1}{p'}}(s)=u(s)s^{\sigma\beta-\frac{\sigma-1}{p'}}=u(s)s^{\sigma \gamma+\sigma-1}\), where
\(\gamma=\beta-\frac{\sigma-1}{\sigma p}\). Consequently,
\(\rho (x)=v^{\frac{1}{q}}(x)\),
\(\omega(s)=u(s)\) and
$$\widetilde{E}_{\alpha, \gamma}f(x)=\rho(x) \int_{a} ^{x}\frac {\omega(s)s^{\sigma\gamma+\sigma-1}f(s)\,ds}{ (x^{\sigma }-s^{\sigma} )^{1-\alpha}}. $$
Now, we assume that
\(a>0\) and
\(W(x)=\ln\frac{x}{a}\). Then
\(u(s)W^{\beta}(s)w^{\frac{1}{p'}}(s)=u(s) (\ln\frac {s}{a} )^{\beta} (\frac{a}{s} )^{\frac{1}{p'}}= a^{\frac{1}{p'}}u(s)s^{\frac{1}{p}} (\ln\frac{s}{a} )^{\beta} \frac{1}{s}\). In this case
\(\rho(x)=v^{\frac{1}{q}}(x)\),
\(\omega(s)=u(s)s^{\frac {1}{p}} (\ln\frac{s}{a} )^{\beta}\) and
$$\widetilde{\mathcal{H}}_{\alpha}f(x)=\rho(x) \int_{a} ^{x}\frac {\omega(s)f(s)\,ds}{s (\ln\frac{x}{s} )^{1-\alpha}}. $$
Below we present statements for boundedness and compactness of the operators
\(\widetilde{I}_{\alpha}\),
\(\widetilde{E}_{\alpha, \gamma}\) and
\(\widetilde{\mathcal{H}}_{\alpha}\) from
\(L_{p}\) to
\(L_{q}\). These statements are consequences of Theorems
3.1,
3.2,
4.1, and
4.2.
We define
$$\begin{aligned}& A^{1}_{\alpha}(z):= \biggl( \int_{z} ^{b} \bigl(\rho(x)x^{\alpha -1} \bigr)^{q}\,dx \biggr)^{\frac{1}{q}} \biggl( \int_{a} ^{z}\omega ^{p'}(s)\,ds \biggr)^{\frac{1}{p'}}, \qquad A^{1}_{\alpha}:= \sup _{z\in I}A^{1}_{\alpha}(z), \\& B^{1}_{\alpha}:= \biggl( \int_{a}^{b} \biggl( \int_{z}^{b}\bigl|\rho (x)x^{\alpha-1}\bigr|^{q} \,dx \biggr)^{\frac{p}{p-q}} \biggl( \int _{a}^{z}\omega^{p'}(s)\,ds \biggr)^{\frac{p(q-1)}{p-q}}\omega ^{p'}(z)\,dz \biggr)^{\frac{p-q}{pq}}. \end{aligned}$$
We define
$$\begin{aligned}& A^{2}_{\alpha,\gamma}(z):= \biggl( \int_{z}^{b}\bigl\vert \rho(x)x^{\sigma (\alpha-1)} \bigr\vert ^{q}\,dx \biggr)^{\frac{1}{q}} \biggl( \int_{a}^{z} \bigl\vert \omega(s)s^{\sigma\gamma+\sigma-1} \bigr\vert ^{p'}\,ds \biggr)^{\frac{1}{p'}}, \\& A^{2}_{\alpha,\gamma}:= \sup_{z\in I}A^{2}_{\alpha,\gamma}(z), \\& B^{2}_{\alpha,\gamma}:= \biggl( \int_{a}^{b} \biggl( \int_{z}^{b}\bigl\vert \rho (x)x^{\sigma(\alpha-1)} \bigr\vert ^{q}\,dx \biggr)^{\frac{p}{p-q}} \\& \hphantom{B^{2}_{\alpha,\gamma}:={}}{}\times \biggl( \int_{a}^{z} \bigl\vert \omega(s)s^{\sigma\gamma+\sigma-1} \bigr\vert ^{p'}\,ds \biggr)^{\frac {p(p-1)}{p-q}}\bigl\vert \omega(z)z^{\sigma\gamma+\sigma-1}\bigr\vert ^{p'}\,dz \biggr)^{\frac{p-q}{pq}}. \end{aligned}$$
To formulate statements corresponding to the operator
\(\widetilde{\mathcal{H}}_{\alpha}\) we define
$$\begin{aligned}& A^{3}_{\alpha}(z):= \biggl( \int_{z} ^{b}\biggl\vert \rho(x) \biggl(\ln { \frac{x}{a}} \biggr)^{\alpha-1}\biggr\vert ^{q}\,dx \biggr)^{\frac {1}{q}} \biggl( \int_{a} ^{z}\omega^{p'}(s)\,ds \biggr)^{\frac {1}{p'}},\qquad A^{3}_{\alpha}:= \sup _{z\in I}A^{3}_{\alpha}(z), \\& B^{3}_{\alpha}:= \biggl( \int_{a}^{b} \biggl( \int_{z}^{b}\biggl\vert \rho (x) \biggl(\ln{ \frac{x}{a}} \biggr)^{\alpha-1}\biggr\vert ^{q} \,dx \biggr)^{\frac{p}{p-q}} \biggl( \int_{a}^{z}\omega^{p'}(s)\,ds \biggr)^{\frac {p(q-1)}{p-q}}\omega^{p'}(z)\,dz \biggr)^{\frac{p-q}{pq}}. \end{aligned}$$
Finally, we consider the operator
\(\widetilde{I}^{*}_{\alpha}g(s)=\rho(s)[I^{*}_{\alpha}(g\omega)](s)\),
\(s\in I\), acting from
\(L_{p}\) to
\(L_{q}\), where
\(I^{*}_{\alpha}\) is the Weyl operator
$$I^{*}_{\alpha}g(s)= \int^{b}_{s}{\frac{g(x)\,dx}{(x-s)^{1-\alpha}}}. $$
The action of the operator
\(K^{*}_{\alpha,\beta}\) from
\(L_{p,v}\) to
\(L_{q,w}\) is equivalent to the action of the operator
$$\widetilde{K}^{*}_{\alpha,\beta}g(s)=w^{\frac{1}{q}}(s)u(s)W^{\beta }(s) \int_{s}^{b}{\frac{v^{\frac{1}{p'}}(x)g(x)\, dx}{(W(x)-W(s))^{1-\alpha}}} $$
from
\(L_{p}\) to
\(L_{q}\). Therefore, when
\(W(x)=x\) we have
$$\begin{aligned}& \rho(s)=u(s)s^{\beta}, \qquad \omega(x)=v^{\frac{1}{p'}}(x), \\& \widetilde{I}^{*}_{\alpha}g(s)=\rho(s) \int^{b}_{s}{\frac{\omega (x)g(x)\,dx}{(x-s)^{1-\alpha}}}. \end{aligned}$$
We define
$$\begin{aligned}& A^{*}_{\alpha}(z):= \biggl( \int_{a}^{z} \rho^{q}(s)\,ds \biggr)^{\frac{1}{q}} \biggl( \int_{z}^{b}\bigl\vert \omega(x)x^{\alpha-1} \bigr\vert ^{p'}\,dx \biggr)^{\frac{1}{p'}},\qquad A^{*}_{\alpha}:= \sup_{z\in I}{A^{*}_{\alpha}(z)}, \\& B^{*}_{\alpha}:= \biggl( \int_{a}^{b} \biggl( \int_{z}^{b}\bigl\vert \omega(x)x^{\alpha -1} \bigr\vert ^{p'}\,dx \biggr)^{\frac{q(p-1)}{p-q}} \biggl( \int_{a}^{z} \rho^{q}(s)\,ds \biggr)^{\frac{q}{p-q}}\rho^{q}(z)\, dz \biggr)^{\frac{p-q}{pq}}. \end{aligned}$$
From Theorems
5.1 and
5.2 we have the following result.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.