Weighted inequalities for Hardy-type operators on the cone of decreasing functions in an Orlicz space

We establish criteria for the validity of modular inequalities for the Hardy operator on the cone Ω of nonnegative decreasing functions from weighted Orlicz spaces with general weight. The result is based on the theorem on the reduction of modular inequalities for positively homogeneous operators on the cone Ω, which enables passing to modular inequalities for modified operators on the cone of all nonnegative functions from an Orlicz space. It is shown that, for the Hardy operator, the modified operator is a generalized Hardy operator. This enables us to establish explicit criteria for the validity of modular inequalities.     

Inequalities for Hardy-type operators on the cone of decreasing functions in a weighted Orlicz space

Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces.     

Boundedness of quasilinear integral operators on the cone of monotone functions

We study the problem of characterizing weighted inequalities on Lebesgue cones of monotone functions on the half-axis for one class of quasilinear integral operators.     

Boundedness of a class of quasilinear operators on the cone of monotone functions

Necessary and sufficient conditions for the weighted boundedness of a class of positive quasilinear integral two-kernel operators of iterative type on the real half-line are given.

     

Representation theorem for local operators in the space of continuous and monotone functions

We prove that every locally defined operator mapping the space of nondecreasing continuous functions into itself is a Nemytskii composition operator.     

On equivalent criteria for the boundedness of Hardy type operators on the cone of decreasing functions

We prove the equivalence of some three-weighted criteria for the boundedness of Hardy type operators on the cone of decreasing functions. These criteria are then applied to realize the direct comparisons of the results obtained in different forms in the papers of several authors (M. Goldman, G. Bennett and K.-G. Grosse-Erdmann, M. Carro, A. Gogatishvili, J. Martin, and L. Pick).     

The weighted inequalities for a certain class of quasilinear integral operators on the cone of monotone functions

We study the problem of characterizing the weighted inequalities on the Lebesgue cones of monotone functions on the semiaxis for a class of quasilinear integral operators.     

Two-sided estimates for matrix operators on the cone of monotone sequences

Weighted estimates for a class of non-negative lower triangular matrices have been established on the cone of monotone sequences.     

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.

     

Some properties of functions in Orlicz space

For functions in Orlicz space L _M ^* , we study the behavior of (t)dt, where x^* (t) is non-increasing and equimeasurable with ¦ x(t) ¦. We establish the existence of unbounded functions in L _M ^* that are not limits of bounded functions and for which . Moreover, we establish a new criterion for an N-function to belong to the class ₂ and a sufficiency test for a function to belong to Orlicz space.Translated from Matematicheskie Zametki, Vol. 3, No. 2, pp. 145–156, February, 1968.     

几类Kantorovich型插值算子在Orlicz空间内的逼近

分别讨论了以第二类Chebyshev多项式的零点、Jacobi多项式的零点、第一类Chebyshev多项式的零点为插值结点组的五类Kantorovich型插值算子在Orlicz空间内的逼近问题,得到了逼近阶的上界估计.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

On the boundary limits of monotone Sobolev functions in variable exponent Orlicz spaces

Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.     

The modular inequalities for a class of convolution operators on monotone functions

This paper is devoted to the study of modular inequality

where and is a class of Volterra convolution operators restricted to the monotone functions. When with and the kernel , our results will extend those for the Hardy operator on monotone functions on Lebesgue spaces.