SOME GENERALIZED DIFFERENCE SEQUENCE SPACES OF INVARIANT MEANS DEFINED BY ORLICZ FUNCTIONS

We define the sequence spaces [w θ ,M, p,u,∆] which are defined by combining the concepts of Orlicz functions, invariant means, and lacunary convergence. We also study some inclusion relations and linearity properties of the above-mentioned spaces. These are generalizations of those defined and studied by Savaş and Rhoades in 2002 and some others before

Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangementinvariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused

Approximation results for a general class of Kantorovich type operators

We introduce and study a family of integral operators in the Kantorovich sense for functions acting on locally compact topological groups. We obtain convergence results for the above operators with respect to the pointwise and uniform convergence and in the setting of Orlicz spaces with respect to the modular convergence. Moreover, we show how our theory applies to several classes of integral and discrete operators, as the sampling, convolution and Mellin type operators in the Kantorovich sense, thus obtaining a simultaneous approach for discrete and integral operators. Further, we derive our general convergence results for particular cases of Orlicz spaces, as $L^p-$spaces, interpolation spaces and exponential spaces. Finally we construct some concrete example of our operators and we show some graphical representations.Comment: 23 pages, 5 figure

New Martingale Inequalities and Applications to Fourier Analysis

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\varphi:\ \Omega\times[0,\infty)\to[0,\infty)$be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space$L^{\varphi}(\Omega)$. Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on$L^{\varphi}(\Omega)$. As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for$L^{\varphi}(\Omega)$, which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces$H_{\varphi}^s(\Omega)$,$P_{\varphi}(\Omega)$,$Q_{\varphi}(\Omega)$,$H_{\varphi}^S(\Omega)$and$H_{\varphi}^M(\Omega)$. From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on$H_{\varphi}^S(\Omega)$and$H_{\varphi}^M(\Omega)$, the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak--Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fej\'er operator is bounded from$H_{\varphi}[0,1)$to$L^{\varphi}[0,1)$, which further implies some convergence results of the Fej\'er means; these results are new even for the weighted Hardy spaces.Comment: 58 pages; Submitte

A crossed product approach to Orlicz spaces

We show how the known theory of noncommutative Orlicz spaces for semifinite von Neumann algebras equipped with an fns trace, may be recovered using crossed product techniques. Then using this as a template, we construct analogues of such spaces for type III algebras. The constructed spaces naturally dovetail with and closely mimic the behaviour of Haagerup $L^p$-spaces. We then define a modified $K$-method of interpolation which seems to better fit the present context, and give a formal prescription for using this method to define what may be regarded as type III Riesz-Fischer spaces.Comment: 39 pages, typos removed, presentation streamlined, non-essential results remove

Abstract Ces\`aro Spaces. I. Duality

We study abstract Ces\`aro spaces $CX$, which may be regarded as generalizations of Ces\`aro sequence spaces $ces_p$ and Ces\`aro function spaces $Ces_p(I)$ on $I = [0,1]$ or $I = [0,\infty)$, and also as the description of optimal domain from which Ces\`aro operator acts to $X$. We find the dual of such spaces in a very general situation. What is however even more important, we do it in the simplest possible way. Our proofs are more elementary than the known ones for $ces_p$ and $Ces_p(I)$. This is the point how our paper should be seen, i.e. not as generalization of known results, but rather like grasping and exhibiting the general nature of the problem, which is not so easy visible in the previous publications. Our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite interval.Comment: 20 page

A class of sequence spaces defined by ll-fractional difference operator

In this paper, we generalize the fractional order difference operator using $l$- Pochhammer symbol and define $l$- fractional difference operator. The $l$- fractional difference operator is further used to introduce a class of difference sequence spaces. Some topological properties and duals of the newly defined spaces are studied.Comment: 9 page

An affine Orlicz Polya-Szego principle

The author established the affine Orlicz Polya-Szego principle for log-concave functions and conjectured that the principle can be extended to the general Orlicz Sobolev functions. In this paper, we confirm this conjecture completely. An affine Orlicz Polya-Szego principle, which includes all the previous affine Polya-Szego principles as special cases, is formulated and proved. As a consequence, an Orlicz-Petty projection inequality for star bodies is established

Sobolev inequalities in arbitrary domains

A theory of Sobolev inequalities in arbitrary open sets of Euclidean space is established. Boundary regularity of domains is replaced with information on boundary traces of trial functions and of their derivatives up to some explicit minimal order. The relevant Sobolev inequalities involve constants independent of the geometry of the domain, and exhibit the same critical exponents as in the classical inequalities on regular domains. Our approach relies upon new representation formulas for Sobolev functions, and on ensuing pointwise estimates which hold in any open set

Maximal Ergodic Inequalities for Banach Function Spaces

We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of $\sigma$-compact locally compact Hausdorff groups acting measure-preservingly on $\sigma$-finite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on different Banach function spaces, and how the properties of these function spaces influence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages.Comment: 46 pages. The former Lemma 4.7 and Theorem 4.8 (which had a small gap in the proof) is replaced by Theorem 4.7. This change affects the latter part of section