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1999 | 135 | 3 | 253-271
Tytuł artykułu

Distributional fractional powers of the Laplacean. Riesz potentials

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Języki publikacji
EN
Abstrakty
EN
For different reasons it is very useful to have at one's disposal a duality formula for the fractional powers of the Laplacean, namely, $((-Δ)^α u,ϕ ) = (u,(-Δ)^α ϕ)$, α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean to obtain some properties of the Riesz potentials in a wide class of spaces which contains the $L^p$-spaces.
Twórcy
  • Departament de Matemàtica Aplicada, Universitat de València, 46100 Burjassot, València, Spain, martinel@uv.es
autor
  • Departament de Matemàtica Aplicada, Universitat de València, 46100 Burjassot, València, Spain, sanzma@uv.es
  • Departament de Matemàtica Aplicada, Universitat de València, 46100 Burjassot, València, Spain, periago@uv.es
Bibliografia
  • [1] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419-437.
  • [2] P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer, Berlin, 1967.
  • [3] G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z. 196 (1987), 189-201.
  • [4] H. O. Fattorini, The Cauchy Problem, Encyclopedia Math. Appl. 18, Addison-Wesley, 1983.
  • [5] S. Guerre, Some remarks on complex powers of( -Δ ) and UMD spaces, Illinois J. Math. 35 (1991), 401-407.
  • [6] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Providence, 1957.
  • [7] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad. 36 (1960), 94-96.
  • [8] H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285-346.
  • [9] H. Komatsu, Fractional powers of operators, II. Interpolation spaces, ibid. 21 (1967), 89-111.
  • [10] H. Komatsu, Fractional powers of operators, III. Negative powers, J. Math. Soc. Japan 21 (1969), 205-220.
  • [11] C. Martínez and M. Sanz, Fractional powers of non-densely defined operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), 443-454.
  • [12] C. Martínez and M. Sanz, Spectral mapping theorem for fractional powers in locally convex spaces, ibid. 24 (1997), 685-702.
  • [13] C. Martínez, M. Sanz and V. Calvo, Fractional powers of non-negative operators in Fréchet spaces, Internat. J. Math. Math. Sci. 12 (1989), 309-320.
  • [14] C. Martínez, M. Sanz and L. Marco, Fractional powers of operators, J. Math. Soc. Japan 40 (1988), 331-347.
  • [15] J. Prüss and H. Sohr, Imaginary powers of elliptic second order differential operators in $L^p$-spaces, Hiroshima Math. J. 23 (1993), 161-192.
  • [16] H. H. Schaefer, Topological Vector Spaces, Springer, New York, 1971.
  • [17] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970.
  • [18] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
  • [19] J. Watanabe, On some properties of fractional powers of linear operators, Proc. Japan Acad. 37 (1961), 273-275.
  • [20] K. Yosida, Functional Analysis, Springer, New York, 1980.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv135i3p253bwm
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