On Hardy Inequality in Variable Lebesgue Spaces with Mixed Norm (original) (raw)

A Characterization of a Two-Weight Inequality for Discrete Two-Dimensional Hardy Operators

Zeitschrift für Analysis und ihre Anwendungen, 1999

We establish necessary and sufficient conditions on a weight pair (v, w) governing the boundedness of the Riesz potential operator I α defined on a homogeneous group G from L p dec,r (w, G) to L q (v, G), where L p dec,r (w, G) is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels I α 1 ,α 2 defined on a product of two homogeneous groups G 1 × G 2. In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and G 1 × G 2 , respectively. MSC: 42B20; 42B25

Weighted weak-type inequalities for generalized Hardy operators

Journal of Inequalities and Applications, 2006

We characterize the pairs of weights (v,w) for which the Hardy-Steklov-type operator T f (x) = g(x) h(x) s(x) K(x, y) f (y)dy applies L p (v) into weak-L q (w), q < p, assuming certain monotonicity conditions on g, s, h, and K.

Conditions for boundedness into Hardy spaces

Mathematische Nachrichten, 2019

We obtain the boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman–Meyer class, sums of products of linear Calderón–Zygmund operators and combinations of these two types.

Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces

Real Analysis Exchange

In this paper the boundedness of the weighted iterated Hardy-type operators T u,b and T * u,b involving suprema from weighted Lebesgue space L p (v) into weighted Cesàro function spaces Ces q (w, a) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator R u from L p (v) into Ces q (w, a) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator P u,b from L p (v) into Ces q (w, a) on the cone of monotone nonincreasing functions. Under additional condition on u and b, we are able to characterize the boundedness of weighted iterated Hardy-type operator T u,b involving suprema from L p (v) into Ces q (w, a) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function M γ from Λ p (v) into Γ q (w). 2010 Mathematics Subject Classification. 46E30, 26D10, 42B25, 42B35. Key words and phrases. weighted iterated Hardy operators involving suprema, Cesàro function spaces, fractional maximal functions, classical Lorentz spaces. L p (w, I) = { f ∈ M(I) : f p,w,I < ∞} and it is equipped with the quasi-norm • p,w,I. When I = (0, ∞), we write L p (w) instead of L p (w, (0, ∞)). We adopt the following usual conventions.