Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T23:42:31.605Z Has data issue: false hasContentIssue false

On Weighted Norm Inequalities for Fractional and Singular Integrals

Published online by Cambridge University Press:  20 November 2018

T. Walsh*
Affiliation:
Princeton University, Princeton, New Jersey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a recent paper [12] Muckenhoupt and Wheeden have established necessary and sufficient conditions for the validity of norm inequalities of the form ‖ |x|αqC‖ |x|αƒp, where denotes a Calderón and Zygmund singular integral of ƒ or a fractional integral with variable kernel. The purpose of the present paper is to prove, by somewhat different methods, similar inequalities for more general weight functions.

In what follows, for p ≧ 1, p′ is the exponent conjugate to p, given by l/p + l/p′ = 1. Ω will always denote a locally integrable function on Rn which is homogeneous of degree 0, Ω will denote a measurable function on Rn × Rn such that for each xRn, Ω(x, .) is locally integrable and homogeneous of degree 0.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Benedek, A. and Panzone, R., The spaces Lp with mixed norm, Duke Math. J. 28 (1961), 301324.Google Scholar
2. Calderón, A. P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113190.Google Scholar
3. Calderon, A. P. and Zygmund, A., On singular integrals, Amer. J. Math. 78 (1956), 289309.Google Scholar
4. Chen, Y.-M., Theorems of asymptotic approximation, Math. Ann. 140 (1960), 360407.Google Scholar
5. Chen, Y.-M., Some asymptotic approximation methods I and II, Proc. London Math. Soc. 15 (1965), 323-345, and 16 (1966), 241263.Google Scholar
6. Halperin, I., Reflexivity in the Lx function spaces, Duke Math. J. 21 (1954), 205208.Google Scholar
7. Hunt, R. A., OnL(p,q) spaces, Enseignement Math. 12 (1966), 249276.Google Scholar
8. Krée, P., Surles multiplicateurs dans FLV avec poids, Ann. Inst. Fourier (Grenoble) 16 (1966), 91121.Google Scholar
9. Muckenhoupt, B., On certain singular integrals, Pacific J. Math. 10 (1960), 239261. lQm 1 Hardy's inequality with weights (to appear). IX., Weighted norm inequalities for the Hardy maximal function (to appear).Google Scholar
12. Muckenhoupt, B. and Wheeden, R. L., Weighted norm inequalities for singular and fractional integrals (to appear in Trans. Amer. Math. Soc).Google Scholar
13. O'Neil, R. O., Integral transforms and tensor products on Orlicz spaces and L(p, q) spaces, J. Analyse Math. 21 (1968), 1276.Google Scholar
14. Neveu, J., Mathematical foundations of the calculus of probability (Holden-Day, San Francisco, 1965).Google Scholar
15. Stein, E. M., Interpolation of linear operators, Trans. Amer. Math. Soc. 88 (1956), 482492.Google Scholar
16. Stein, E. M., Note on singular integrals, Proc. Amer. Math. Soc. 8 (1957), 250254.Google Scholar
17. Stein, E. M. and Weiss, G., Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503514.Google Scholar
18. Strichartz, R. S., IP estimates for integral transforms, Trans. Amer. Math. Soc. 186 (1969), 3350.Google Scholar
19. Walsh, T., On Z> estimates for integral transforms, Trans. Amer. Math. Soc. 155 (1971), 195215.Google Scholar
20. Zygmund, A., Trigonometric series, Vol. II, 2nd ed. (Cambridge University Press, New York, 1959).Google Scholar