Integral operators and weighted amalgams

C. Carton-Lebrun; H. Heinig; S. Hofmann

Studia Mathematica (1994)

  • Volume: 109, Issue: 2, page 133-157
  • ISSN: 0039-3223

Abstract

top
For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ℓ q ̅ ( L v p ̅ ) into ℓ q ( L u p ) . For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted L p -spaces. Amalgams of the form ℓ q ( L w p ) , 1 < p,q < ∞ , q ≠ p, w ∈ A p , are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

How to cite

top

Carton-Lebrun, C., Heinig, H., and Hofmann, S.. "Integral operators and weighted amalgams." Studia Mathematica 109.2 (1994): 133-157. <http://eudml.org/doc/216065>.

@article{Carton1994,
abstract = {For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from $ℓ^\{q̅\}(L^\{p̅\}_\{v\})$ into $ℓ^\{q\}(L^\{p\}_\{u\})$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form $ℓ^\{q\}(L^\{p\}_\{w\})$, 1 < p,q < ∞ , q ≠ p, $w ∈ A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.},
author = {Carton-Lebrun, C., Heinig, H., Hofmann, S.},
journal = {Studia Mathematica},
keywords = {amalgam spaces; weights; $A_p$ weights; Hardy operator; Hardy-Littlewood maximal operator; weighted amalgam inequalities; integral operators; weighted amalgam spaces},
language = {eng},
number = {2},
pages = {133-157},
title = {Integral operators and weighted amalgams},
url = {http://eudml.org/doc/216065},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Carton-Lebrun, C.
AU - Heinig, H.
AU - Hofmann, S.
TI - Integral operators and weighted amalgams
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 133
EP - 157
AB - For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from $ℓ^{q̅}(L^{p̅}_{v})$ into $ℓ^{q}(L^{p}_{u})$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form $ℓ^{q}(L^{p}_{w})$, 1 < p,q < ∞ , q ≠ p, $w ∈ A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.
LA - eng
KW - amalgam spaces; weights; $A_p$ weights; Hardy operator; Hardy-Littlewood maximal operator; weighted amalgam inequalities; integral operators; weighted amalgam spaces
UR - http://eudml.org/doc/216065
ER -

References

top
  1. [1] K. F. Andersen and H. P. Heinig, Weighted norm inequalities for certain integral operators, SIAM J. Math. Anal. 14 (4) (1983), 834-844. Zbl0527.26010
  2. [2] J. J. Benedetto, H. P. Heinig and R. Johnson, Weighted Hardy spaces and the Laplace transform II, Math. Nachr. 132 (1987), 29-55. Zbl0626.44002
  3. [3] G. Bennett, Some elementary inequalities III, Quart. J. Math. Oxford Ser. (2) 42 (1991), 149-174. Zbl0751.26007
  4. [4] J. S. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408. Zbl0402.26006
  5. [5] A. P. Calderón, Inequalities for the maximal function relative to a metric, Studia Math. 57 (1978), 297-306. Zbl0341.44007
  6. [6] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. Zbl0222.26019
  7. [7] J. J. F. Fournier and J. Stewart, Amalgams of L p and ℓ q , Bull. Amer. Math. Soc. 13 (1985), 1-21. 
  8. [8] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, 1985. 
  9. [9] F. Holland, Harmonic analysis on amalgams of L p and ℓ q , J. London Math. Soc. (2) 10 (1975), 295-305. 
  10. [10] B. Jawerth, Weighted inequalities for maximal operators : linearization, localization and factorization, Amer. J. Math. 108 (1986), 361-414. Zbl0608.42012
  11. [11] V. G. Maz'ya, Sobolev Spaces, Springer, Berlin, 1985. 
  12. [12] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. Zbl0236.26016
  13. [13] B. Opic and A. Kufner, Hardy Type Inequalities, Pitman Res. Notes Math. 219, Longman, 1990. 
  14. [14] G. Sinnamon, Spaces defined by their level function and their dual, preprint. 
  15. [15] R. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977. 

NotesEmbed ?

top

You must be logged in to post comments.