On inequalities for integral operators

Lφ Integral Inequalities for Maximal Operators

Sufficient (almost necessary) conditions are given on the weight functions u(·), v(·) for Φ−1 2 [ ∫ Rn Φ2 ( C2(Msf)(x) ) u(x)dx ] ≤ Φ−1 1 [ C1 ∫ Rn Φ1(|f(x)|)v(x)dx ] to hold when Φ1, Φ2 are φ-functions with subadditive Φ1Φ 2 , and Ms (0 ≤ s < n), is the usual fractional maximal operator. §

Some sharp inequalities for multilinear integral operators

In this paper, some sharp inequalities for certain multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained. As an application, we obtain the (L p , L q)-norm inequalities and Morrey spaces boundedness for the multilinear operators.

Weighted norm inequalities for singular integral operators

Inequalities for Norms of Some Integral Operators

Let (A(a)u)(x) := a 0 (1 − xt) −1 u(t) dt, 0 < a < 1. Properties of the operators A(a) as a → 1 are studied. It is proved that A := A(1) is a bounded, positive self-adjoint operator in H = L 2 [0, 1], ||A|| ≤ π, while A : C(0, 1) → C(0, 1) is unbounded.

On weighted inequalities for certain fractional integral operators

and Dn denotes the derivative operator ∂/∂x1, . . . ,∂xn. The operators in (1.1) provide multidimensional generalizations to the well-known one-dimensional Riemann-Liouville andWeyl fractional integral operators defined in [5] (see also [1]). The paper [7] considers several formulas and interesting properties of (1.1). By invoking the Gauss hypergeometric function 2F1(α,β;γ;x), the following ge...