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1
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Distributional fractional powers of the Laplacean. Riesz potentials

100%
Martínez C., Sanzi M., Periago F.
Studia Mathematica
|
1999
|
tom 135
|
nr 3
253-271
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.
2
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On approach regions for the conjugate Poisson integral and singular integrals

100%
Ferrando S., Jones R. L., Reinhold K.
Studia Mathematica
|
1996
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tom 120
|
nr 2
169-182
EN
Let ũ denote the conjugate Poisson integral of a function $f ∈ L^{p}(ℝ)$. We give conditions on a region Ω so that $lim_{(v,ε)→(0,0)}_{(v,ε)∈Ω} ũ(x+v,ε) = Hf(x)$, the Hilbert transform of f at x, for a.e. x. We also consider more general Calderón-Zygmund singular integrals and give conditions on a set Ω so that $sup_{(v,r)∈Ω} |ʃ_{|t|>r} k(x+v-t)f(t)dt|$ is a bounded operator on $L^p$, 1 < p < ∞, and is weak (1,1).
3
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On local properties of functions and singular integrals in terms of the mean oscillation

100%
Rzaev R., Aliyeva L.
Open Mathematics
|
2008
|
tom 6
|
nr 4
595-609
EN
This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L p-sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.
4
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Ondelettes et poids de Muckenhoupt

100%
Gilles Lemarié-Rieusset P.
Studia Mathematica
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1994
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tom 108
|
nr 2
127-147
EN
We study, for a basis of Hölderian compactly supported wavelets, the boundedness and convergence of the associated projectors $P_j$ on the space $L^p(dμ)$ for some p in ]1,∞[ and some nonnegative Borel measure μ on ℝ. We show that the convergence properties are related to the $A_p$ criterion of Muckenhoupt.
5
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Weighted estimates for commutators of linear operators

100%
Alvarez J., Bagby R. J., Kurtz D. S., Pérez C.
Studia Mathematica
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1993
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tom 104
|
nr 2
195-209
EN
We study boundedness properties of commutators of general linear operators with real-valued BMO functions on weighted $L^p$ spaces. We then derive applications to particular important operators, such as Calderón-Zygmund type operators, pseudo-differential operators, multipliers, rough singular integrals and maximal type operators.
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