Volume 53, Issue 1
O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and some Applications

J. Math. Study, 53 (2020), pp. 90-124.

Published online: 2020-03

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• Abstract

In this paper we prove an O'Neil inequality for the convolution operator ($G$-convolution) associated with the Gegenbauer differential operator $G_{\lambda}$. By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the $G$% -convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the $G$-fractional maximal and $G$-fractional integral operators from the spaces $L_{p,\lambda}$ to $L_{q,\lambda }$ and from the spaces $L_{1,\lambda }$ to the weak spaces $WL_{p,\lambda}$.

• Keywords

Gegenbauer differential operator, G-convolution, O'Neil inequality, G-fractional integral, G-fractional maximal function.

42B20, 42B25, 42B35, 47G10, 47B37

vagif@guliyev.com (Vagif S. Guliyev)

elmanibrahimov@yahoo.com (E.J. Ibrahimov)

elifnurekincioglu@gmail.com (S.E. Ekincioglu)

• BibTex
• RIS
• TXT
@Article{JMS-53-90, author = {Guliyev , Vagif S. and Ibrahimov , E.J. and Ekincioglu , S.E. and Jafarova , S. Ar.}, title = {O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and some Applications}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {1}, pages = {90--124}, abstract = {

In this paper we prove an O'Neil inequality for the convolution operator ($G$-convolution) associated with the Gegenbauer differential operator $G_{\lambda}$. By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the $G$% -convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the $G$-fractional maximal and $G$-fractional integral operators from the spaces $L_{p,\lambda}$ to $L_{q,\lambda }$ and from the spaces $L_{1,\lambda }$ to the weak spaces $WL_{p,\lambda}$.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n1.20.05}, url = {http://global-sci.org/intro/article_detail/jms/15209.html} }
TY - JOUR T1 - O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and some Applications AU - Guliyev , Vagif S. AU - Ibrahimov , E.J. AU - Ekincioglu , S.E. AU - Jafarova , S. Ar. JO - Journal of Mathematical Study VL - 1 SP - 90 EP - 124 PY - 2020 DA - 2020/03 SN - 53 DO - http://doi.org/10.4208/jms.v53n1.20.05 UR - https://global-sci.org/intro/article_detail/jms/15209.html KW - Gegenbauer differential operator, G-convolution, O'Neil inequality, G-fractional integral, G-fractional maximal function. AB -

In this paper we prove an O'Neil inequality for the convolution operator ($G$-convolution) associated with the Gegenbauer differential operator $G_{\lambda}$. By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the $G$% -convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the $G$-fractional maximal and $G$-fractional integral operators from the spaces $L_{p,\lambda}$ to $L_{q,\lambda }$ and from the spaces $L_{1,\lambda }$ to the weak spaces $WL_{p,\lambda}$.

Vagif S. Guliyev, E.J. Ibrahimov, S.E. Ekincioglu & S. Ar. Jafarova. (2020). O'Neil Inequality for Convolutions Associated with Gegenbauer Differential Operator and some Applications. Journal of Mathematical Study. 53 (1). 90-124. doi:10.4208/jms.v53n1.20.05
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