Some integral and maximal operators related to starlike sets
Chanillo, Sagun ; Watson, David ; Wheeden, Richard
Studia Mathematica, Tome 104 (1993), p. 223-255 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:216031 
@article{bwmeta1.element.bwnjournal-article-smv107i3p223bwm,
     author = {Sagun Chanillo and David Watson and Richard Wheeden},
     title = {Some integral and maximal operators related to starlike sets},
     journal = {Studia Mathematica},
     volume = {104},
     year = {1993},
     pages = {223-255},
     zbl = {0809.42008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv107i3p223bwm}
}

Chanillo, Sagun; Watson, David; Wheeden, Richard. Some integral and maximal operators related to starlike sets. Studia Mathematica, Tome 104 (1993) pp. 223-255. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv107i3p223bwm/

[00000] [Ca] C. P. Calderón, Differentiation through starlike sets in ℝm, Studia Math. 48 (1973), 1-13.

[00001] [Ch] M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. 128 (1988), 19-42. | Zbl 0666.47027

[00002] [ChR] M. Christ and J. L. Rubio de Francia, Weak type (1, 1) bounds for rough operators, II, Invent. Math. 93 (1988), 225-237. | Zbl 0695.47052

[00003] [Cor] A. Córdoba, Maximal functions, covering lemmas and Fourier multipliers, in: Proc. Sympos. Pure Math. 35, Part 1, Amer. Math. Soc., 1979, 29-50.

[00004] [F] R. Fefferman, A theory of entropy in Fourier analysis, Adv. in Math. 30 (1978), 171-201. | Zbl 0441.42019

[00005] [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.

[00006] [M] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-227. | Zbl 0236.26016

[00007] [P] C. Perez, Two weighted inequalities for potential and fractional type maximal operators, Indiana Univ. Math. J., to appear. | Zbl 0809.42007

[00008] [Sa] E. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545. | Zbl 0665.42023

[00009] [SaWh] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. | Zbl 0783.42011

[00010] [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.

[00011] [StWe] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. | Zbl 0182.10801

[00012] [W1] D. Watson, Vector-valued inequalities, factorization, and extrapolation for a family of rough operators, J. Funct. Anal., to appear.

[00013] [W2] D. Watson, A₁ weights and weak type (1,1) estimates for rough operators, to appear.