Weighted norm inequalities for the Fourier transform
Author:
Benjamin Muckenhoupt
Journal:
Trans. Amer. Math. Soc. 276 (1983), 729742
MSC:
Primary 42A38; Secondary 26D15, 42B10, 44A15
MathSciNet review:
688974
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given and satisfying , sufficient conditions on nonnegative pairs of functions are given to imply where denotes the Fourier transform of , and is independent of . For the case the sufficient condition is that for all positive , where and are positive and independent of . For the condition is more complicated but also is invariant under rearrangements of and . In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of and . Examples are given to show that the sufficient condition is not necessary for a pair if the norm inequality is assumed only for that pair.
 [1]
Néstor
E. Aguilera and Eleonor
O. Harboure, On the search for weighted norm inequalities for the
Fourier transform, Pacific J. Math. 104 (1983),
no. 1, 1–14. MR 683723
(85e:42007)
 [2]
Björn
E. J. Dahlberg, Regularity properties of Riesz potentials,
Indiana Univ. Math. J. 28 (1979), no. 2,
257–268. MR
523103 (80g:31004), http://dx.doi.org/10.1512/iumj.1979.28.28018
 [3]
Peter Knopf and Karl Rudnick, Weighted norm inequalities for the Fourier transform (preprint).
 [4]
Benjamin
Muckenhoupt, Weighted norm inequalities for classical
operators, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure
Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure
Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979,
pp. 69–83. MR 545240
(80i:42015)
 [5]
Y.
Sagher, Real interpolation with weights, Indiana Univ. Math.
J. 30 (1981), no. 1, 113–121. MR 600037
(82e:46045), http://dx.doi.org/10.1512/iumj.1981.30.30010
 [6]
Elias
M. Stein, Interpolation of linear
operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. MR 0082586
(18,575d), http://dx.doi.org/10.1090/S00029947195600825860
 [7]
Elias
M. Stein, Singular integrals and differentiability properties of
functions, Princeton Mathematical Series, No. 30, Princeton University
Press, Princeton, N.J., 1970. MR 0290095
(44 #7280)
 [8]
Elias
M. Stein and Guido
Weiss, Introduction to Fourier analysis on Euclidean spaces,
Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical
Series, No. 32. MR 0304972
(46 #4102)
 [1]
 N. E. Aguilera and E. O. Harboure, On the search of weighted norm inequalities for the Fourier transform (to appear). MR 683723 (85e:42007)
 [2]
 B. Dahlberg, Regularity properties of Riesz potentials, Indiana Univ. Math. J. 28 (1979), 257268. MR 523103 (80g:31004)
 [3]
 Peter Knopf and Karl Rudnick, Weighted norm inequalities for the Fourier transform (preprint).
 [4]
 B. Muckenhoupt, Weighted norm inequalities for classical operators, Proc. Sympos. Pure Math., vol. 35, part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 6983. MR 545240 (80i:42015)
 [5]
 Y. Sagher, Real interpolation with weights, Indiana Univ. Math. J. 30 (1981), 113121. MR 600037 (82e:46045)
 [6]
 E. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482492. MR 0082586 (18:575d)
 [7]
 , Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
 [8]
 E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
42A38,
26D15,
42B10,
44A15
Retrieve articles in all journals
with MSC:
42A38,
26D15,
42B10,
44A15
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719830688974X
PII:
S 00029947(1983)0688974X
Article copyright:
© Copyright 1983
American Mathematical Society
