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

In this article, we will define the Orlicz space and the Orlicz-Sobolev space, and develop their topological properties. We will also examine their applications to partial differential equations (PDEs), with an emphasis on the use of certain variational methods.

It is known that for $C^{\infty}$ determining sets Markov's property is equivalent to Bernstein's property. The purpose of this paper is to prove an analogous result in the case of compact subsets of algebraic varieties.

In this paper we give sharp $L_p$ Markov type inequalities for derivatives of multivariate polynomials for a wide family of UPC domains.

Certain inequalities between the values of the modular and the norm in the Orlicz spaces are established. These inequalities are applied then to the theory of solvability of nonlinear integral equations of Hammerstein type.

In this article, we study a number of properties of the K\"othe duals $\mathcal{M}_{\varphi,w}$ of Orlicz-Lorentz spaces. An explicit description of the order-continuous subspace of $\mathcal{M}_{\varphi,w}$ is provided. Moreover, the separability of these spaces is characterized by the growth condition $\Delta_2$. Consequently, the K\"othe dual space $\mathcal{M}_{\varphi,w}$ has the Radon-Nikod\'ym property if and only if the N-function at infinity $\varphi$ satisfies the a...

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.

We formulate and discuss a conjecture which would extend a classical inequality of Bernstein.

In the article we generalize the Marcinkiewicz sampling theorem in the context of Orlicz spaces. We establish conditions under which sampling theorem holds in terms of restricted submultiplicativity and supermultiplicativity of an $N$-function $\varphi$, boundedness of the Hilbert transform and Matuszewska-Orlicz indices. In addition we give a new criterion for boundedness of Hilbert transform on Orlicz space.

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}\gamma_k)} \leq C_p (\mathrm{deg}\, P)^{\frac{1}{2}+\frac{1}{\pi}\arctan\left(\frac{|p-2|}{2\sqrt{p-1}}\right)}\|P\|_{L^{p}(\mathbb{R}^{k}, \mathrm{d}\gamma_k)}, $$ where $\mathrm{d}\gamma_k$ is the sta...

In this paper we consider a generalized conditional-type Holder- inequality and investigate some classic properties of multiplication conditional expectation type operators on Orlicz-spaces.