• Bulletin of the Korean Mathematical Society
• /
• v.55 no.6
• /
• pp.1791-1809
• /
• 2018

Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and $T_q(q{\in}(1,{\infty}))$ be the vector-valued operator defined by $T_qf(x)=({\sum}_{k=1}^{\infty}{\mid}T\;f_k(x){\mid}^q)^{1/q}$. In this paper, by proving certain weak type endpoint estimate of L log L type for the grand maximal operator of T, the author establishes some quantitative weighted bounds for $T_q$ and the corresponding vector-valued maximal singular integral operator.

• Honam Mathematical Journal
• /
• v.36 no.2
• /
• pp.455-465
• /
• 2014

In recent years, diverse inequalities involving a variety of fractional integral operators have been developed by many authors. In this sequel, here, we aim at establishing certain new inequalities involving pathway type fractional integral operator by following the same lines, recently, used by Choi and Agarwal [7]. Relevant connections of the results presented here with those earlier ones are also pointed out.

• Journal of the Korean Mathematical Society
• /
• v.38 no.1
• /
• pp.163-176
• /
• 2001

We examine Orlicz-type integral inequalities for operators and obtain as a corollary a characterization of such inequalities for the Hardy-Littlewood maximal operator extending the well-known L(sup)p-norm inequalities.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.2
• /
• pp.207-213
• /
• 2010

In this paper, we define the q-extension of the Hardy-Littlewood-type maximal operator related to q-Volkenborn integral. By the meaning of the extension of q-Volkenborn integral, we obtain the boundedness of the q-extension of the Hardy-Littlewood-type maximal operator in the p-adic integer ring.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.1
• /
• pp.1-10
• /
• 2008

We prove a sharp inequality for some multilinear operator related to Marcinkiewicz integral operator. As application, we obtain the weighted norm inequality and L log L type estimate for the multilinear operator.

• Korean Journal of Mathematics
• /
• v.29 no.2
• /
• pp.227-237
• /
• 2021

Opial inequality and its consequences are useful in establishing existence and uniqueness of solutions of initial and boundary value problems for differential and difference equations. In this paper we analyze Opial-type inequalities for convex functions. We have studied different versions of these inequalities for a generalized integral operator. Further difference of Opial-type inequalities are utilized to obtain generalized mean value theorems, which further produce various interesting derivations for fractional and conformable integral operators.

• East Asian mathematical journal
• /
• v.30 no.3
• /
• pp.283-291
• /
• 2014

A remarkably large number of inequalities involving the fractional integral operators have been investigated in the literature by many authors. Very recently, Baleanu et al. [2] gave certain interesting fractional integral inequalities involving the Gauss hypergeometric functions. Using the same fractional integral operator, in this paper, we present some (presumably) new fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Saigo, Erd$\acute{e}$lyi-Kober and Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

• Kyungpook Mathematical Journal
• /
• v.50 no.2
• /
• pp.297-306
• /
• 2010

In this paper, the integral operator is used. We give a new Hilbert-type integral inequality, whose kernel is the homogeneous form with degree - $\lambda$ and with three pairs of conjugate exponents and the best constant factor and its reverse form are also derived. It is shown that the results of this paper represent an extension as well as some improvements of the earlier results.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.1
• /
• pp.181-193
• /
• 2016

During the past two decades or so, fractional integral inequalities have proved to be one of the most powerful and far-reaching tools for the development of many branches of pure and applied mathematics. Very recently, many authors have presented some generalized inequalities involving the fractional integral operators. Here, using the pathway fractional integral operator, we give some presumably new and potentially useful fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

• Communications of the Korean Mathematical Society
• /
• v.21 no.3
• /
• pp.451-464
• /
• 2006

In this paper, a sharp inequality for some multilineal singular integral operators are obtained. As the applications, we get the weighted $L^p(p>1)$ norm inequalities and L log L type estimate for the multilinear operators.