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.
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.