Boundedness of fractional operators on $L^{p}$ spaces with different weights

Abstract:

Let ${T_\alpha }$ be either the fractional integral operator $\smallint f(y)|x - y{|^{\alpha - n}}\; dy$, or the fractional maximal operator $\sup \left \{ {{r^{\alpha - n}}{\smallint _{|x - y| < r}}|f(y)|dy: r > 0} \right \}$. Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $\upsilon$ (resp. $w$) such that ${(\smallint |{T_\alpha }f{|^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint |f{|^p}w\;dx)^{1/p}}$ holds. Weak type substitutes in limiting cases are considered.