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Distributional fractional powers of the Laplacean. Riesz potentials

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Martínez C., Sanzi M., Periago F.

Studia Mathematica

1999

tom 135

nr 3

253-271

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.

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

100%

Martínez C., Redondo A., Sanz M.

Studia Mathematica

2011

tom 202

nr 2

145-164

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p}(ℝ)$-spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative operator. This identification allows us to give a unified view of the theory and provides some elegant proofs of some well-known results on the fractional integrals of Weyl.

Ograniczanie wyników

2 Studia Mathematica

2 Martínez C.

1 Periago F.

1 Redondo A.

1 Sanz M.

1 Sanzi M.

1 2011

1 1999

Wyniki wyszukiwania

Distributional fractional powers of the Laplacean. Riesz potentials

Suitable domains to define fractional integrals of Weyl via fractional powers of operators