Kernel operators and their boundedness from weighted Sobolev space to weighted Lebesgue space
Kernel operators and their boundedness from weighted Sobolev space to weighted Lebesgue space
In this paper, for a wide class of integral operators, we consider the problem of their boundedness from aweighted Sobolev space to a weighted Lebesgue space. The crucial step in the proof of the main result is to use theequivalence of the basic inequality and certain Hardy-type inequality, so we first state and prove this equivalence.
___
- [1] Eveson SP, Stepanov VD, Ushakova EP. A duality principle in weighted Sobolev spaces on the real line. Math Nachr
2015; 288: 877-897 doi: 10.1002/mana.201400019.
- [2] Kufner A, Maligranda L, Persson LE. The Hardy Inequality. About its History and Some Related Results. Pilsen,
Czech Republic: Vydavatelsky Servis, 2007.
- [3] Kufner A, Persson LE. Weighted Inequalities of Hardy type. London, UK: World Scientific, 2003.
- [4] Oinarov R. Boundedness and compactness of Volterra type integral operators. Siberian Math J 2007; 48: 884-896.
- [5] Oinarov R. Boundedness of integral operators from weighted Sobolev space to weighted Lebesgue space. Complex
Var Elliptic 2011; 56: 1021-1038.
- [6] Oinarov R. Boundedness of integral operators in weighted Sobolev spaces. Izv Math 2014; 78: 836-853.
- [7] Oinarov R. On weighted norm inequalities with three weights. J London Math Soc 1993; 48: 103-116.
- [8] Oinarov R. Boundedness and compactness in weighted Lebesgue spaces of integral operators with variable integration
limits. Siberian Math J 2011; 52: 1042-1055.
- [9] Prokhorov DV, Stepanov VD, Ushakova EP. Hardy-Steklov integral operators. Part I. Proc Steklov Inst Math 2018;
300: 1-111.
- [10] Stepanov VD, Ushakova EP. Kernel operators with variable intervals of integration in Lebesgue spaces and applications.
Math Inequal Appl 2010; 13: 449-510.