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Complex Analysis and Operator Theory

, Volume 12, Issue 3, pp 669–681 | Cite as

The Kontorovich–Lebedev Transform and Sobolev Type Space

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Abstract

In this paper boundedness of convolution operator in \(L^p(\mathbb {R}_+; ~xdx)\) is given and continuity of pseudo-differential operator \(\mathcal {P}_a\) associated with the Kontorovich–Lebedev transform (KL-transform) from \(H(\mathbb {R}_+)\) into itself is discussed. The Kontorovich–Lebedev potential (KL-potential) \(\mathcal {P}_a^s\) is defined on \(H(\mathbb {R}_+)\) and then it is extended to the space of distributions also some of its properties are studied. A Sobolev type space \(W^{s,p}(\mathbb {R}_+)\) associated with the KL-transform is discussed and proved as a Banach space. Moreover, it is shown that the KL-potential is an isometry of \(W^{s,p}\). An \(L^p\)-boundedness for the KL-potential is obtained and at the end applications of the KL-potential in pseudo-differential equation are discussed.

Keywords

Kontorovich–Lebedev transform Convolution Pseudo-differential operator Banach space 

Mathematics Subject Classification

44A20 35S05 46F12 46E35 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

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