The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedevtransform

Semyon Yakubovich

doi:10.3934/cpaa.2011.10.745

Article Contents

2011, Volume 10, Issue 2: 745-760. Doi: 10.3934/cpaa.2011.10.745

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The heat kernel and Heisenberg inequalities related to the Kontorovich-Lebedev transform

Semyon Yakubovich^1,

1.
Department of Mathematics, Faculty of Sciences, University of Porto, Campo Alegre st., 687, Porto, 4169-007

Received: May 31, 2010

Revised: September 30, 2010

Published: February 2011

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Abstract

We introduce a notion of the heat kernel related to the familiar Kontorovich-Lebedev transform. We study differential and semigroup properties of this kernel and construct fundamental solutions of a generalized diffusion equation. An integral transformation with the heat kernel is considered. By using the Plancherel $L_2$-theory for the Kontorovich-Lebedev transform and norm estimates for its convolution we establish analogs of the classical Heisenberg inequality and uncertainty principle for this transformation. The proof is also based on the norm inequalities for the Mellin transform of the heat kernel.

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Mathematics Subject Classification: Primary: 44A15; Secondary: 44A05, 44A35.

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