JournalszaaVol. 15, No. 2pp. 309–328

Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator

  • Y. Rakotondratsimba

    Institut Polytechnique St. Louis, Cergy-Pontoise, France
Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator cover

Abstract

A sufficient condition is given on weight functions uu and vv on Rn\mathbb R^n for which the fractional maximal operator Ms(0s<n)M_s (0 ≤ s < n) defined by (Msf)(x)=supQxQsn1Qf(y)dy(M_sf)(x) = \mathrm {sup}_{Q \ni x} |Q|^{\frac{s}{n}–1} \int_Q | f (y) | dy or the fractional integral operator Is(0<s<n)I_s (0 < s < n) defined by (Is,f)(x)=Rnxysnf(y)dy(I_s,f)(x) = \int_{\mathbb R^n} | x - y |^{s–n} f(y)dy is bounded from Lp(Rn,vdx)L^p (\mathbb R^n, vdx) into Lq(Rn,udx)L^q(\mathbb R^n,udx) for 0<q<p0 < q < p with p>1p> 1, where QQ is a cube and n a non-negative integer.

Cite this article

Y. Rakotondratsimba, Weighted Inequalities for the Fractional Maximal Operator and the Fractional Integral Operator. Z. Anal. Anwend. 15 (1996), no. 2, pp. 309–328

DOI 10.4171/ZAA/702