Weighted Hardy-Littlewood-Sobolev-type inequality for \(\psi\)-Riemann-Liouville fractional integrals (Q6100864)

The paper under review is a good piece of functional analysis devoted to prove the boundedness character of certain integral operators defined on weighted Lebesgue spaces. Given a finite real interval \([a,b]\), recall that a weight function on \([a,b]\) is a measurable function on \([a,b]\), positive and finite almost everywhere. For \(1\leq p <\infty\), denote by \(L_w^p [a,b]\) the weighted Lebesgue space associated to the weight \(w\) and endowed with the finite norm \(\left\|u\right\|_{L_w^p [a,b]}=\left(\int_a^b |u(x)|w(x)dx\right)^{\frac{1}{p}}\). As weight function, the authors consider the derivative function \(\psi'\in C[a,b]\), with \(\psi'\neq 0\), of an increasing and positive continuous function \(\psi\) defined on \([a,b]\). On the other hand, for \(\alpha\in (0,1)\) and \(\xi\in L^1[a,b]\), the authors consider the \(\psi\)-Riemman Liouville left-sided fractional integral of \(\xi\) by \(\mathbf{I}^{\alpha; \psi}_ {a+,x}\xi(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^x\psi'(t)\left(\psi(x)-\psi(t)\right)^{\alpha-1}\xi(t)dt,\) and the right-sided fractional integral operator \(\mathbf{I}^{\alpha; \psi}_ {x,b-}\) is similarly defined, taking limits between \(x\) and \(b\). Once the ingredients for a full understanding of the statement of main results have been presented, we proceed to state them. Previously, the authors prove, in the form of a lemma, that for \(\alpha\in (0,1), p\in [1,\infty),\) and \(u\in L^p_{\psi'}[a,b]\), it holds \(\mathbf{I}^{\alpha; \psi}_ {a+}u\in L^p_{\psi'}[a,b]\), with \( \left\|\mathbf{I}^{\alpha; \psi}_ {a+}u\right\|_{L^p_{\psi'}[a,b]}\leq \frac{\left(\psi(b)-\psi(a)\right)^{\alpha}}{\Gamma(\alpha +1)}\left\|u\right\|_{L^p_{\psi'}[a,b]}\). Theorem 1. Let \(\alpha\in (0,1)\) and \(1\leq p <\frac{1}{\alpha}\). Then, the operators \(\mathbf{I}^{\alpha; \psi}_ {a+}, \mathbf{I}^{\alpha; \psi}_ {b-}:L^p_{\psi'}[a,b]\rightarrow L^q_{\psi'}[a,b]\) are continuous for every \(q\in\left[1,\frac{p}{1-\alpha p}\right]\). Theorem 2. If \(\alpha=\frac{1}{p}\), then the operators \(\mathbf{I}^{\alpha; \psi}_ {a+}, \mathbf{I}^{\alpha; \psi}_ {b-}:L^p_{\psi'}[a,b]\rightarrow L^q_{\psi'}[a,b]\) are continuous for every \(q\in [1,\infty)\). Theorem 3. If \(\alpha\in \left(\frac{1}{p},1\right)\), then the operators \(\mathbf{I}^{\alpha; \psi}_ {a+}, \mathbf{I}^{\alpha; \psi}_ {b-}:L^p_{\psi'}[a,b]\rightarrow L^q_{\psi'}[a,b]\) are continuous for every \(p\leq q <\infty\). Moreover, when \(q=\infty\), we have that \(\mathbf{I}^{\alpha; \psi}_ {a+}, \mathbf{I}^{\alpha; \psi}_ {b-}:L^p_{\psi'}[a,b]\rightarrow L^{\infty} [a,b]\) are continuous. Theorem 4. If \(\alpha\in (0,1)\) and \(p=+\infty\), then the operators \(\mathbf{I}^{\alpha; \psi}_ {a+}, \mathbf{I}^{\alpha; \psi}_ {b-}:L^p[a,b]\rightarrow C[a,b]\cap L^{\infty}[a,b]\) are continuous. Apart from continuity, in the proofs of the previous theorems, the authors present bounds for all the integral inequalities. It is also worth noticing that the authors give a succint introduction to the subject of Hardy-Littlewood-Sobolev and Stein-Weiss inequalities; additionally, it can be stressed the connection of the main results obtained in the paper with other works in the literature, as the authors briefly point out in a subsection.