Convolution Estimates and Generalized de Leeuw Theorems for Multipliers of Weak Type (1,1)

Asmar, Nakhlé ; Berkson, Earl ; Gillespie, T. A.

Canadian journal of mathematics, Tome 47 (1995) no. 2, pp. 225-245

Résumé

In the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

Détail

DOI : 10.4153/CJM-1995-011-x

Mots-clés : 43A32

Bibliographie

[1] 1. Asmar, N., Berkson, E. and Bourgain, J., Restrictions from ℝn to ℤn of weak type (1,1) multipliers, Studia Math. 108(1994), 291–299. Google Scholar

[2] 2. Asmar, N., Berkson, E. and Gillespie, T.A., Transference of strong type maximal inequalities by separationpreserving representations, Amer. J. Math. 113(1991), 47–74. Google Scholar

[3] 3. Asmar, N., Berkson, E. and Gillespie, T.A., Distributional control and generalized analyticity, Integral Equations and Operator Theory 14(1991), 311–341. Google Scholar

[4] 4. Asmar, N., Berkson, E. and Gillespie, T.A., Maximal estimates on measure spaces for weak type mutlipliers, J. Geom. Anal., to appear. Google Scholar

[5] 5. Asmar, N., Berkson, E. and Gillespie, T.A., Maximal estimates on groups, subgroups, and the Bohr compactification, J. Funct. Anal., to appear. Google Scholar

[6] 6. Berkson, E., Gillespie, T.A. and R Muhly, S., LP-multiplier transference induced by representations in Hilbert space, Studia Math. 94(1989), 51–61. Google Scholar

[7] 7. de Leeuw, K., On Lp multipliers, Ann. of Math. 81(1965), 364–379. Google Scholar

[8] 8. Dowson, H.R., Spectral Theory of Linear Operators, London Math. Soc. Monographs 12, Academic Press, New York, 1978. Google Scholar

[9] 9. Edwards, R.E. and Gaudry, G.I., Littlewood-Paley and Multiplier Theory, Ergeb. Math. Grenzgeb. 90, Springer-Verlag, Berlin, 1977. Google Scholar

[10] 10. Hewitt, E. and Ross, K.A., Abstract Harmonic Analysis I, Grundlehren Math. Wiss. 115, Academic Press, New York, 1963. Google Scholar

[11] 11. Hewitt, E., Abstract Harmonic Analysis II, Grundlehren Math. Wiss. 152, Springer-Verlag, Berlin, 1970. Google Scholar

[12] 12. Saeki, S., Translation invariant operators on groups, Tôhoku Math. J. 22(1970), 409–419. Google Scholar

[13] 13. Sawyer, S., Maximal inequalities of weak type, Ann. of Math. 84(1966), 157–174. Google Scholar

[14] 14. Stein, E.M., On limits of sequences of operators, Ann. of Math. 74(1961), 140–170. Google Scholar

[15] 15. Zygmund, A., Trigonometric Series, 2nd edition, vol. I, Cambridge University Press, 1959. Google Scholar