Abstract
We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C 0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.
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Research partially supported by Project MTM2010-16679, DGI-FEDER, of the MCYTS, Spain, and Project E-64, D.G. Aragón, Spain. The third author was partially supported by grant MTM2011-28149-C02-01 from Spanish Government.
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Galé, J.E., Miana, P.J. & Stinga, P.R. Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equ. 13, 343–368 (2013). https://doi.org/10.1007/s00028-013-0182-6
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DOI: https://doi.org/10.1007/s00028-013-0182-6