A Remark on the Inequalities of Bernstein - Markov Type in Exponential Orlicz and Lorentz Spaces

We describe the Lorentz space $L(p, r), 0 < r < p, p > 1$, in terms of Orlicz type classes of functions L . As a consequence of this result it follows that Stein's characterization of the real functions on $R^n$ that are differentiable at almost all the points in $R^n$, is equivalent to the earlier characterization of those functions given by A. P. Calderon.

In this paper the necessary and sufficient conditions were given for Orlicz-Lorentz function space endowed with the Orlicz norm having non-squareness and local uniform non-squareness.

In this paper, based on concepts of convex sets and convex functions, we introduce new concepts of functions, Young functions, strong Young functions and Orlicz functions by relaxing definitions of functions, Young functions, strong Young functions, Orlicz functions respectively. We also give concepts of Orlicz spaces, weak Orlicz spaces, Orlicz Sobolev spaces, weak Orlicz Sobolev spaces, Orlicz Morrey spaces and weak Orlicz Morrey spaces, Orlicz Lorentz spaces and weak Orlic...

We generalize the well-known inequality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces.

Nikolskii type inequalities for entire functions of exponential type on Rn for the Lorentz Zygmund spaces are obtained. Some new limiting cases are examined. Application to Besov type spaces of logarithmic smoothness is given.

We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.

We obtain Marcinkiewicz--ygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal assumptions on the structural properties of these spaces. Our main results show that the Bernstein inequality in a general quasi-Banach function lattice $X$ implies Marcinkiewicz-Zygmund type estimates in $X$. We present a general approach to obtain MZ inequalities not only for polynomials but for other function classes including entire functions of exponential type, spli...

A new characterization of the exponential type Orlicz spaces generated by the functions $\exp(|x|^p)-1$ ($p\ge 1$) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with the Luxemburg norms. On the example of the Hoeffding inequality we present some application of these norms in a probabilistic context.

Let W: R to (0,1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm ||fW|| Linfinity(R) . The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorem...

Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Masty\l o, Maligranda, and Kami\'nska. In this paper, we consider the problem of comparing the Orlicz-Lorentz norms, and establish necessary and sufficient conditions for them to be equivalent. As a corollary, we give necessary and sufficient conditions for a Lorentz-Sharpley space to be equivalent to an Orlicz space, extending results o...