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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Weighted norm estimates for the Fourier transform with a pair of weights

Authors: Jan-Olov Strömberg and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 318 (1990), 355-372
MSC: Primary 42B10; Secondary 42B30
MathSciNet review: 1002924
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Abstract: We prove weighted norm inequalities of the form

$\displaystyle {\left\Vert {\hat f} \right\Vert _{L_u^q}} \leq C{\left\Vert f \right\Vert _{H_\upsilon ^p}},\quad 0 < p \leq q < \infty ,$

for the Fourier transform on $ {{\mathbf{R}}^n}$. For some weight functions $ \upsilon $, the Hardy space $ H_\upsilon ^p$ on the right can be replaced by $ L_\upsilon ^p$. The proof depends on making an atomic decomposition of $ f$ and using cancellation properties of the atoms.

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  • [A] E. Adams, On the identification of weighted Hardy spaces, Indiana Univ. Math. J. 32 (1983), 477-489. MR 703279 (85g:42024)
  • [B-H] J. J. Benedetto and H. Heinig, Fourier transform inequalities with measure weights, Adv. in Math. (to appear). MR 1196988 (93m:42004)
  • [B-H-J] J. J. Benedetto, H. Heinig and R. Johnson, Fourier inequalities with $ {A_p}$ weights, General Inequalities, $ 5$ (W. Walter, ed.), Internat. Ser. Numer. Math. 80 (1987), 217-232. MR 1018148 (90k:42015)
  • [H1] H. Heinig, Weighted norm inequalities for classes of operators, Indiana Univ. Math. J. 33 (1984), 573-582. MR 749315 (86c:42016)
  • [H2] H. Heinig, Fourier operators on weighted Hardy spaces, Proc. Cambridge Philos. Soc. 101 (1987), 113-121. MR 877705 (88c:42031)
  • [J-Sam] W. B. Jurkat and G. Sampson, On rearrangement and weight inequalities for the Fourier transform, Indiana Univ. Math. J. 33 (1984), 257-267. MR 733899 (85k:42040)
  • [M] B. Muckenhoupt, Weighted norm inequalities for the Fourier transform, Trans. Amer. Math. Soc. 176 (1983), 729-742. MR 688974 (84m:42019)
  • [Sad-W] C. Sadosky and R. L. Wheeden, Some weighted norm inequalities for the Fourier transform of functions with vanishing moments, Trans. Amer. Math. Soc. 300 (1987), 521-533. MR 876464 (88c:42027)
  • [St] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482-492. MR 0082586 (18:575d)
  • [Str-T] J.-O. Stràmberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., vol. 1381, Springer, 1989. MR 1011673 (90j:42053)
  • [Str-W] J.-O. Stràmberg and R. L. Wheeden, Relations between $ H_u^p$ and $ L_u^p$ with polynomial weights, Trans. Amer. Math. Soc. 270 (1982), 439-467.

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