Download PDF - Some integral and maximal operators related to starlike setsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Some integral and maximal operators related to starlike sets

Authors 1, 2, 1

Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101, U.S.A.
  2. Department of Mathematics, Rutgers University, Camden, New Jersey 08102, U.S.A.

Abstract

We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product spaces. We consider both weak type and strong type results, and we show that the conditions imposed on the weight functions are fairly sharp.

Bibliography

  1. [Ca] C. P. Calderón, Differentiation through starlike sets in ℝm, Studia Math. 48 (1973), 1-13.
  2. [Ch] M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988), 19-42.
  3. [ChR] M. Christ and J. L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), 225-237.
  4. [Cor] A. Córdoba, Maximal functions, covering lemmas and Fourier multipliers, in: Proc. Sympos. Pure Math. 35, Part 1, Amer. Math. Soc., 1979, 29-50.
  5. [F] R. Fefferman, A theory of entropy in Fourier analysis, Adv. in Math. 30 (1978), 171-201.
  6. [GK] M. Gabidzashvili and V. Kokilashvili, Two weight weak type inequalities for fractional-type integrals, preprint, No. 45, Math. Inst. Czech. Acad. Sci., Prague, 1989.
  7. [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-227.
  8. [P] C. Perez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J., to appear.
  9. [Sa] E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545.
  10. [SaWh] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874.
  11. [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
  12. [StWe] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54.
  13. [W1] D. Watson, Vector-valued inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. Anal., to appear.
  14. [W2] D. Watson, A₁ weights and weak type (1,1) estimates for rough operators, to appear.
Studia Mathematica
1993/107/3
Export to PBN, Opens in new tab
Pages:
223-255
Main language of publication
English
Received
1991-11-29
Published
1993
Exact and natural sciences