On the Multi-Dimensional Duality Principle of Sawyer Type

A multi-dimensional version of the duality principle of Sawyer type [1] is obtained whenever the corresponding weight satisfies some doubling property. Received June 27, 2000, Accepted September 14, 2000     

Characterization of a Two-Weighted Vector-Valued Inequality for Fractional Maximal Operators

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M _s is bounded from the weighted Lebesgue spaces L ^p(l ^r, vdx) into L ^q(l ^r, udx) whenever 0 s < n, 1 < p, r < , and 1 q < .     

On the boundedness of classical operators on weighted lorentz spaces

Conditions on weightsu(·),v(·) are given so that a classical operatorT sends the weighted Lorentz spaceL ^Lrs (vdx) intoL ^pq (udx). HereT is either a fractional maximal operatorM _α or a fractional integral operatorI _α or a Calderón-Zygmund operator. A characterization of this boundedness is obtained forM _α andI _α when the weights have some usual properties and max(r, s) ≤ min(p, q).     

Two-Dimensional Discrete Hardy Inequalities

A sufficient condition on nonnegative double-sequences

Two-Weight Norm Inequality for Calderón-Zygmund Operators

Sufficient conditions on weight functions u(·) and v(·) are given so that any Calderón-Zygmund operator is bounded from the weighted Lebesgue space Lvp into Lup.     

Weighted LΦ ntergral inequalities for maximal operators

Sufficient (almost necessary) conditions are given on the weight funotiousu(·),v(·) for $$\Phi _2^{ - 1} \left[ {\int\limits_{\mathbb{R}^n } {\Phi _2 (C_2 (M_s f)(x))u(x)dx} } \right] \leqslant \Phi _1^{ - 1} \left[ {C_1 \int\limits_{\mathbb{R}^n } {\Phi _1 (|f(x)|)} v(x)dx} \right]$$ to hold when Φ₁, Φ₂ are ?-functions with subadditive Φ₁Φ ₂ ^?1 , andM _s (0≤s<n), is the usual fractional maximal operator.