Bessel potential space on the Laguerre hypergroup Taieb Ahmed | Ahmed Advances in Difference Equations 2011 2011 4 http content 2011 1 4 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Bessel potential space on the Laguerre hypergroup Taieb Ahmed Correspondence taiebahmed@ Faculty of Sciences of Tunis Department of Mathematics University of Tunis II 1060 Tunis Tunisia SpringerOpen0 Abstract In this article we define the fractional differentiation Ds of order s s 0 induced by the Laguerre operator L and associated with respect to the Haar measure dma. We obtain a characterization of the Bessel potential space Lp K using Ds and different equivalent norms. Keywords Heat-diffusion Poisson semigroups Fractional power Riesz potential Fractional differentiation 1 Introduction During the second half of the twentieth century until the 1990s the Continuous Time Random Walk CTRW method was practically the only tool available to describe subdiffusive and or superdiffusive phenomena associated with complex systems for many groups of research. The main reason behind the usefulness of fractional derivatives have been until this moment the close link that exists between fractional models and the so called Jump stochastic models such as the CTRW or those of the multiple trapping type. Note that fractional operators also provide a method for reflecting the memory properties and non-locality of many anomalous processes. In any case at the moment it is not clear what is the best fractional time derivative or the spatial fractional derivative to be used in the different models. Fractional calculus deals with the study of so-called fractional order integral and derivative operators over real or complex domains and their applications. Since 1990 there has been a spectacular increase in the use of fractional models to simulate the dynamics of many different anomalous processes especially those involving ultraslow diffusion. We hereby propose a few examples of fields where the
