In this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two-weight norm inequalities for both operators on weighted Morrey spaces. K E Y W O R D S fractional integrals, fractional maximal operators, weighted Morrey spaces M S C ( 2 0 1 0 ) 42B20, 42B25 970

In this paper, we introduce a new class of weights, the A λ,∞ weights, which contains the classical A ∞ weights. We prove a mixed A p,q-A λ,∞ type estimate for fractional integral operators.

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We study the Fourier extension operator E on the truncated paraboloid ℙ 2 := { ( ω 1 , ω 2 , ω 1 2 + ω 2 2 ) : ω 1 2 + ω 2 2 ≤ 1 } ⊂ ℝ 3 . \mathbb{P}^{2}:=\bigl{\{}(\omega_{1},\omega_{2},\omega_{1}^{2}+\omega_{2}^{2})% :\omega_{1}^{2}+\omega_{2}^{2}\leq 1\bigr{\}}\subset\mathbb{R}^{3}. For a function f in the Morrey space L 2 , λ ⁢ ( B 2 ) {L^{2,\lambda}(B^{2})} , where B 2 {B^{2}} denotes the unit disk in ℝ 2 {\mathbb{R}^{2}} , we prove that for each R > 1 {R>1} and ε > 0 {\varepsilon>0} , ∥ E ⁢ f ∥ L p ⁢ ( B 3 ⁢ ( 0 , R ) ) ≲ ε , λ R ε ⁢ ∥ f ∥ L 2 , λ ⁢ ( B 2 ) \lVert Ef\rVert_{L^{p}(B^{3}(0,R))}\lesssim_{\varepsilon,\lambda}R^{% \varepsilon}\lVert f\rVert_{L^{2,\lambda}(B^{2})} holds for all p ≥ ( 6 ⁢ λ + 7 ) / ( 2 ⁢ λ + 2 ) {p\geq(6\lambda+7)/(2\lambda+2)} if 5 6 < λ ≤ 1 {\frac{5}{6}<\lambda\leq 1} and for all p ≥ 6 / ( 1 + λ ) {p\geq 6/(1+\lambda)} if 1 2 < λ ≤ 5 6 {\frac{1}{2}<\lambda\leq\frac{5}{6}} . The main tool used in our proof is the polynomial partitioning technique.