The Hardy–Littlewood–Sobolev theorem for Riesz potential generated by Gegenbauer operator

Boundedness criteria for the fractional integral and fractional maximal operator on Morrey spaces generated by the Gegenbauer differential operator

“…Consider the Gegenbauer‐Riesz potential [18] $$$$ {I}_G^{\alpha }f(chx)=\frac{1}{\Gamma \left(\frac{\alpha }{2}\right)}\int_0^{\infty}\left(\int_0^{\infty }{r}^{\frac{\alpha }{2}-1}\kern0.1em {h}_r(cht)\kern0.1em dr\right){A}_{cht}^{\lambda }f(chx)\kern0.1em s{h}^{2\lambda }t\kern0.1em dt, $$$$ where $$$$ {h}_r(cht)=\int_1^{\infty }{e}^{-\nu \left(\nu +2\lambda \right)}{P}_{\nu}^{\lambda }(cht)\kern0.1em {\left({\nu}^2-1\right)}^{\lambda -\frac{1}{2}}\kern0.1em d\nu, \kern6.05pt 0<\alpha <2\lambda +1, $$$$ and …”

“…In [18, Sec. 4], it was shown that, for $$$ \alpha \ge 2\lambda +1 $$$ and for $$$ f\in {L}_{p,\lambda}\left({\mathrm{\mathbb{R}}}_{+}\right), $$$ $$$ {I}_G^{\alpha } $$$ does not exist.…”

“…Remark In [18, Corollary 3.1] proven that $$$$ \left|{I}_G^{\alpha }f(chx)\right|\lesssim \int_0^{\infty}\frac{\left|{A}_{chy}^{\lambda }f(chx)\right|}{(shy)^{2\lambda +1-\alpha }}\kern0.1em s{h}^{2\lambda }y\kern0.1em dy, $$$$ for $$$ 0<\alpha <2\lambda +1. $$$ From last inequality, we have …”

Boundedness criteria for the commutators of fractional integral and fractional maximal operators on Morrey spaces generated by the Gegenbauer differential operator

“…Let $$$ {H}_r=\left(0,r\right)\subset {\mathrm{\mathbb{R}}}_{+} $$$. Below, we'll need the following relation [16, Lemma 2.3]: $$$$ {\left|{H}_r\right|}_{\lambda }=\int_0^rs{h}^{2\lambda } xdx\approx {\left( sh\frac{r}{2}\right)}^{\gamma }, $$$$ where $$$ 0<\lambda <\frac{1}{2} $$$ and $$$$ \gamma ={\gamma}_{\lambda }(r)=\left\{\begin{array}{cc}2\lambda +1,& 0<r<2,\\ {}4\lambda, & 2\le r<\infty .\end{array}\right. $$$$ By analogy with (), we introduce the following definition.…”

O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and Some Applications

“…The Gegenbauer differential operator was introduced in [5]. For the properties of the Gegenbauer differential operator, we refer to [3,4,[10][11][12].…”