A new equivalent condition for boundedness of Hardy-Volterra operator Eshimova M

A new equivalent condition for boundedness of Hardy-Volterra operator
Eshimova M

Let and be weight functions on i.e., are nonnegative measurable functions defined on Let introduce weighted Lebesgue space of all measurable functions on such that

and similarly Consider the following Hardy-Volterra operator

where is a kernel-nonnegative measurable function defined
In this paper we study boundedness of operator (1) with the so-called Oinarov’s kernel-that is continuous, non-negative and increasing in the first argument, decreasing in the second argument and there exists a constant , such that for all the inequality holds: see [1-4]. This problem is equivalent to the validity of the Hardy-type integral inequality:

where constant does not depend on the function and parameters . We give new necessary and sufficient conditions for satisfying of (2), i.e., the boundedness of the (1) in the case with the Oinarov’s kernel.
Let us make some notations concerning to our theorem:




Then our main result reads.
Theorem. Let and be Oinarov’s kernel. Then inequality (2) holds for all if and only if

Moreover, the best constant of the inequality-the least constant for which the inequality holds- satisfies


References
1. Kufner A., Persson L.-E., Weighted inequalities of Hardy type. World scientific: New Jersey, London, Singapore, Hong Kong, 2003.
2. Kufner A., Maligranda L., Persson L.-E., The Hardy inequality-about its history and some related results. Pilsen, 2007.
3. Stepanov V.D., The weighted Hardy’s inequality for nonincreasing functions, Transactions of the AMS, 338, 1(1993).
4. Kuliev K., Kulieva G., Eshimova M., Estimates for the norm of some Hardy type operators, Scientific journal of SamSU, 127, 3(2021).

The initial attempts in the investigation of Hardy-type integral inequalities for special kernels were done by F.J.Martin-Reyes, E.Sawyer, S.Bloom, R.Kerman, V.D.Stepanov, R.Oinarov , see[].

A variety of authors have studied several classes of such operators, for example S.Bloom, R.Kerman, F.J.Martin-Reyes, E.Sawyer, R.Oinarov, V. D. Stepanov and ets... There has been done lots of investigations and results in case . By contrast, in case there are little contributions which was studied by V.D.Stepanov, R. Oinarov, G. SINNAMON, H. P. HEINIG, L.-E. Persson, L.S. Arendarenko and others. Some necessary and sufficient conditions for the validity of (2) in case were obtained by V. D. Stepanov and E.P. Ushakova. The problem of finding two-sided estimates in the case remains open.