Some integral and maximal operators related to starlike sets

Sagun Chanillo; David Watson; Richard Wheeden

Studia Mathematica (1993)

  • Volume: 107, Issue: 3, page 223-255
  • ISSN: 0039-3223

Abstract

top
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.

How to cite

top

Chanillo, Sagun, Watson, David, and Wheeden, Richard. "Some integral and maximal operators related to starlike sets." Studia Mathematica 107.3 (1993): 223-255. <http://eudml.org/doc/216031>.

@article{Chanillo1993,
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.},
author = {Chanillo, Sagun, Watson, David, Wheeden, Richard},
journal = {Studia Mathematica},
keywords = {two-weight norm inequalities; fractional integrals; fractional maximal functions},
language = {eng},
number = {3},
pages = {223-255},
title = {Some integral and maximal operators related to starlike sets},
url = {http://eudml.org/doc/216031},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Chanillo, Sagun
AU - Watson, David
AU - Wheeden, Richard
TI - Some integral and maximal operators related to starlike sets
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 3
SP - 223
EP - 255
AB - 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.
LA - eng
KW - two-weight norm inequalities; fractional integrals; fractional maximal functions
UR - http://eudml.org/doc/216031
ER -

References

top
  1. [Ca] C. P. Calderón, Differentiation through starlike sets in ℝ m , 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. Zbl0666.47027
  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. Zbl0695.47052
  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. Zbl0441.42019
  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. Zbl0236.26016
  8. [P] C. Perez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J., to appear. Zbl0809.42007
  9. [Sa] E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545. Zbl0665.42023
  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. Zbl0783.42011
  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. Zbl0182.10801
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.