Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces

Abstract

In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential {I_{G}^{\alpha}} , {0<\alpha<2\lambda+1} , is bounded from the G-Morrey space {L_{p,\lambda,\gamma}} to {L_{q,\lambda,\gamma}} if and only if \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1<p<\frac{2% \lambda+1-\gamma}{\alpha}. Also, we prove that the G-Riesz potential {I_{G}^{\alpha}} is bounded from the G-Morrey space {L_{1,\lambda,\gamma}} to the weak G-Morrey space {WL_{q,\lambda,\gamma}} if and only if 1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

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

“…4], it was shown that, for 𝛼 ≥ 2𝜆 + 1 and for 𝑓 ∈ L p,𝜆 (R + ) , I 𝛼 G does not exist. In [17] for the G-Riesz potential I 𝛼 G on G-Morrey space, the following theorem is proved by analogue of the Theorem 1.1.…”

“…Let $$$ {H}_r=\left(0,r\right)\subset {\mathrm{\mathbb{R}}}_{+}. $$$ Next, we need the following relation (see [16, 17]) $$$$ {\left|{H}_r\right|}_{\lambda }=\int_0^rs{h}^{2\lambda } tdt\approx {\left( sh\frac{r}{2}\right)}^{\gamma }, $$$$ where $$$$ \gamma ={\gamma}_{\lambda }(r)=\left\{\begin{array}{c}2\lambda +1,\kern3.0235pt \mathrm{if}\kern6.05pt 0<r<2,\\ {}4\lambda, \kern9.07pt \mathrm{if}\kern0.1em \hspace{6.05pt}2\le r<\infty, \end{array}\right. $$$$ and $$$ 0<\lambda <\frac{1}{2} $$$.…”

“…In [17] for the $$$ G $$$‐Riesz potential $$$ {I}_G^{\alpha } $$$ on $$$ G $$$‐Morrey space, the following theorem is proved by analogue of the Theorem 1.1.…”

“…According to Proposition 2.1, Definitions 2.1 and 2.2 are equivalent to $$$ G $$$‐Morrey space introduced in [17].…”

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].…”