On multilinear fractional integrals

Loukas Grafakos

Studia Mathematica (1992)

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

Abstract

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In ℝ n , we prove L p ₁ × . . . × L p K boundedness for the multilinear fractional integrals I α ( f ₁ , . . . , f K ) ( x ) = ʃ f ₁ ( x - θ ₁ y ) . . . f K ( x - θ K y ) | y | α - n d y 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.

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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  1. [A] D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398. Zbl0672.31008
  2. [CM] S.-Y. A. Chang and D. E. Marshall, On a sharp inequality concerning the Dirichlet integral, Amer. J. Math. 107 (1985), 1015-1033. Zbl0578.30010
  3. [CG] R. R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators I, Rev. Mat. Iberoamericana, to appear. Zbl0785.47025
  4. [G] L. Grafakos, Hardy space estimates for multilinear operators II, Rev. Mat. Iberoamericana, to appear. Zbl0785.47026
  5. [HE] L. I. Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505-510. Zbl0283.26003
  6. [HMT] J. A. Hempel, G. R. Morris and N. S. Trudinger, On the sharpness of a limiting case of the Sobolev embedding theorem, Bull. Austral. Math. Soc. 3 (1970), 369-373. Zbl0205.12801
  7. [HI] I. I. Hirschman Jr., A convexity theorem for certain groups of transformations, J. Analyse Math. 2 (1953), 209-218. Zbl0052.06302
  8. [M] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. Zbl0203.43701
  9. [ST] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
  10. [SWE] E. M. Stein and G. Weiss, An Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. 
  11. [STR] R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841-842. Zbl0241.46028
  12. [T] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483. Zbl0163.36402

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