# On multilinear fractional integrals

Studia Mathematica (1992)

• Volume: 102, Issue: 1, page 49-56
• ISSN: 0039-3223

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## Abstract

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In ${ℝ}^{n}$, we prove ${L}^{p₁}×...×{L}^{{p}_{K}}$ boundedness for the multilinear fractional integrals ${I}_{\alpha }\left(f₁,...,{f}_{K}\right)\left(x\right)=ʃf₁\left(x-\theta ₁y\right)...{f}_{K}\left(x-{\theta }_{K}y\right){|y|}^{\alpha -n}dy$ where the ${\theta }_{j}$’s are nonzero and distinct. We also prove multilinear versions of two inequalities for fractional integrals and a multilinear Lebesgue differentiation theorem.

## How to cite

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Grafakos, Loukas. "On multilinear fractional integrals." Studia Mathematica 102.1 (1992): 49-56. <http://eudml.org/doc/215913>.

@article{Grafakos1992,
abstract = {In $ℝ^n$, we prove $L^\{p₁\} ×...× L^\{p_\{K\}\}$ boundedness for the multilinear fractional integrals $I_α(f₁,...,f_K)(x) = ʃ f₁(x-θ₁ y)...f_K(x-θ_K y)|y|^\{α-n\} dy$ where the $θ_j$’s are nonzero and distinct. We also prove multilinear versions of two inequalities for fractional integrals and a multilinear Lebesgue differentiation theorem.},
author = {Grafakos, Loukas},
journal = {Studia Mathematica},
keywords = {Hardy-Littlewood maximal function; multilinear fractional integrals; inequalities; multilinear Lebesgue differentiation theorem},
language = {eng},
number = {1},
pages = {49-56},
title = {On multilinear fractional integrals},
url = {http://eudml.org/doc/215913},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Grafakos, Loukas
TI - On multilinear fractional integrals
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 1
SP - 49
EP - 56
AB - In $ℝ^n$, we prove $L^{p₁} ×...× L^{p_{K}}$ boundedness for the multilinear fractional integrals $I_α(f₁,...,f_K)(x) = ʃ f₁(x-θ₁ y)...f_K(x-θ_K y)|y|^{α-n} dy$ where the $θ_j$’s are nonzero and distinct. We also prove multilinear versions of two inequalities for fractional integrals and a multilinear Lebesgue differentiation theorem.
LA - eng
KW - Hardy-Littlewood maximal function; multilinear fractional integrals; inequalities; multilinear Lebesgue differentiation theorem
UR - http://eudml.org/doc/215913
ER -

## References

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9. [ST] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
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