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New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator M [resp. I], 0 ≤ s < n, [resp. 0 < s < n] sends the weighted Lebesgue space L(v(x)dx) into L(u(x)dx), 1 < p < ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

Necessary conditions and sufficient conditions are derived in order that Bessel potential operator $J_{s,\lambda }$ is bounded from the weighted Lebesgue spaces $L_v^p=L^p({ℝ}^n,v\left(x\right)dx)$ into $L_u^q$ when $1<p\le q<\infty$.