# Distributional fractional powers of the Laplacean. Riesz potentials

Studia Mathematica (1999)

• Volume: 135, Issue: 3, page 253-271
• ISSN: 0039-3223

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## Abstract

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For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, $\left({\left(-\Delta \right)}^{\alpha }u,\varphi \right)=\left(u,{\left(-\Delta \right)}^{\alpha }\varphi \right)$, 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.

## How to cite

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Martínez, Celso, Sanzi, Miguel, and Periago, Francisco. "Distributional fractional powers of the Laplacean. Riesz potentials." Studia Mathematica 135.3 (1999): 253-271. <http://eudml.org/doc/216654>.

@article{Martínez1999,
abstract = {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.},
author = {Martínez, Celso, Sanzi, Miguel, Periago, Francisco},
journal = {Studia Mathematica},
keywords = {fractional powers; Laplacean operator; Riesz potentials; singular integrals; fractional power of Laplacian; Riesz potential; distributional space},
language = {eng},
number = {3},
pages = {253-271},
title = {Distributional fractional powers of the Laplacean. Riesz potentials},
url = {http://eudml.org/doc/216654},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Martínez, Celso
AU - Sanzi, Miguel
AU - Periago, Francisco
TI - Distributional fractional powers of the Laplacean. Riesz potentials
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 3
SP - 253
EP - 271
AB - 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.
LA - eng
KW - fractional powers; Laplacean operator; Riesz potentials; singular integrals; fractional power of Laplacian; Riesz potential; distributional space
UR - http://eudml.org/doc/216654
ER -

## References

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15. [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. Zbl0790.35023
16. [16] H. H. Schaefer, Topological Vector Spaces, Springer, New York, 1971.
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20. [20] K. Yosida, Functional Analysis, Springer, New York, 1980.

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