×

Convolution estimates for some measures on curves. (English) Zbl 0613.43002

The author conjectures that if \(\lambda\) is a nice measure supported on a nice k-dimensional surface in \({\mathbb{R}}^ n\), then \(\lambda *L^ p\subset L^ q\) precisely when (1/p,1/q) is in the closed triangle in \({\mathbb{R}}^ 2\) with vertices (0,0), (1,1) and (n/(2n-k), (n-k)/(2n-k)). The paper gives partial results about this conjecture.
Reviewer: K.Saka

MSC:

43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A05 Measures on groups and semigroups, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Christ, Convolution estimates for Cantor-Lebesgue measures, preprint. · Zbl 0644.42011
[2] I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Translated by Eugene Saletan, Academic Press, New York-London, 1964. · Zbl 0115.33101
[3] Lars Hörmander, Estimates for translation invariant operators in \?^{\?} spaces, Acta Math. 104 (1960), 93 – 140. · Zbl 0093.11402 · doi:10.1007/BF02547187
[4] Walter Littman, \?^{\?}-\?^{\?}-estimates for singular integral operators arising from hyperbolic equations, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 479 – 481.
[5] E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159 – 172. · Zbl 0083.34301
[6] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[7] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.