## Fractional integrals on weighted $H^ p$ and $L^ p$ spaces

HTML articles powered by AMS MathViewer

- by Jan-Olov Strömberg and Richard L. Wheeden
- Trans. Amer. Math. Soc.
**287**(1985), 293-321 - DOI: https://doi.org/10.1090/S0002-9947-1985-0766221-X
- PDF | Request permission

## Abstract:

We study the two weight function problem $\parallel {I_\alpha }f{\parallel _{H_u^q}} \leqslant c\parallel f{\parallel _{H_v^p}},0 < p \leqslant q < \infty$ , for fractional integrals on Hardy spaces. If $u$ and $v$ satisfy the doubling condition and $0 < p \leqslant 1$, we obtain a necessary and sufficient condition for the norm inequality to hold. If $1 < p < \infty$ we obtain a necessary condition and a sufficient condition, and show these are the same under various additional conditions on $u$ and $v$. We also consider the corresponding problem for $L_u^q$ and $L_v^p$, and obtain a necessary and sufficient condition in some cases.## References

- David R. Adams,
*A trace inequality for generalized potentials*, Studia Math.**48**(1973), 99–105. MR**336316**, DOI 10.4064/sm-48-1-99-105 - Ernst Adams,
*On the identification of weighted Hardy spaces*, Indiana Univ. Math. J.**32**(1983), no. 4, 477–489. MR**703279**, DOI 10.1512/iumj.1983.32.32034 - R. R. Coifman and C. Fefferman,
*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241–250. MR**358205**, DOI 10.4064/sm-51-3-241-250 - Björn E. J. Dahlberg,
*Regularity properties of Riesz potentials*, Indiana Univ. Math. J.**28**(1979), no. 2, 257–268. MR**523103**, DOI 10.1512/iumj.1979.28.28018 - C. Fefferman and E. M. Stein,
*$H^{p}$ spaces of several variables*, Acta Math.**129**(1972), no. 3-4, 137–193. MR**447953**, DOI 10.1007/BF02392215 - I. M. Gel′fand and G. E. Shilov,
*Generalized functions. Vol. 2. Spaces of fundamental and generalized functions*, Academic Press, New York-London, 1968. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR**0230128** - Kurt Hansson,
*Imbedding theorems of Sobolev type in potential theory*, Math. Scand.**45**(1979), no. 1, 77–102. MR**567435**, DOI 10.7146/math.scand.a-11827
R. Kerman and E. Sawyer, - Steven G. Krantz,
*Fractional integration on Hardy spaces*, Studia Math.**73**(1982), no. 2, 87–94. MR**667967**, DOI 10.4064/sm-73-2-87-94 - Roberto A. Macías and Carlos Segovia,
*Weighted norm inequalities for parabolic fractional integrals*, Studia Math.**61**(1977), no. 3, 279–291. MR**492116**, DOI 10.4064/sm-61-3-279-291 - Benjamin Muckenhoupt,
*Weighted norm inequalities for the Hardy maximal function*, Trans. Amer. Math. Soc.**165**(1972), 207–226. MR**293384**, DOI 10.1090/S0002-9947-1972-0293384-6 - Benjamin Muckenhoupt and Richard Wheeden,
*Weighted norm inequalities for fractional integrals*, Trans. Amer. Math. Soc.**192**(1974), 261–274. MR**340523**, DOI 10.1090/S0002-9947-1974-0340523-6 - Eric T. Sawyer,
*Weighted norm inequalities for fractional maximal operators*, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 283–309. MR**670111** - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095**
J.-O. Strömberg and A. Torchinsky,

*Weighted norm inequalities of trace type for potential operators*, preprint.

*Weighted Hardy spaces*(to appear). J.-O. Strömberg and R. L. Wheeden,

*Relations between*$H_u^p$

*and*$L_u^p$

*with polynomial weights*, Trans. Amer. Math. Soc.

**270**(1982), 439-467. —,

*Kernel estimates for fractional integrals with polynomial weights*(to appear).

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**287**(1985), 293-321 - MSC: Primary 42B30; Secondary 26A33, 47G05
- DOI: https://doi.org/10.1090/S0002-9947-1985-0766221-X
- MathSciNet review: 766221