# Distributional fractional powers of the Laplacean. Riesz potentials

Celso Martínez; Miguel Sanzi; Francisco Periago

Studia Mathematica (1999)

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

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topMartínez, 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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