Musielak-Orlicz Function Research Articles

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Lower Local Uniform Monotonicity in F-Normed Musielak–Orlicz Spaces

Lower strict monotonicity points and lower local uniform monotonicity points are considered in the case of Musielak–Orlicz function spaces LΦ endowed with the Mazur–Orlicz F-norm. The findings outlined in this study extend the scope of geometric characteristics observed in F-normed Orlicz spaces, as well as monotonicity properties within specific F-normed lattices. They are suitable for the Orlicz spaces of ordered continuous elements, specifically in relation to the Mazur–Orlicz F-norm. In addition, in this paper presents results that can be used to derive certain monotonicity properties in F-normed Musielak–Orlicz spaces.

Density of smooth functions in Musielak–Orlicz–Sobolev spaces Wk,Φ(Ω)$W^{k,\Phi }(\Omega)$

AbstractWe consider here Musielak–Orlicz–Sobolev (MOS) spaces , where is an open subset of , and is a Musielak–Orlicz function. The main outcomes consist of the results on density of the space of compactly supported smooth functions in . One section is devoted to compare the various conditions on appearing in the literature in the context of maximal operator and density theorems in MOS spaces. The assumptions on we apply here are substantially weaker than in the earlier papers on the topics of approximation by smooth functions. One of the reasons is that in the process of proving density theorems, we do not use the fact that the Hardy–Littlewood maximal operator on Musielak–Orlicz space is bounded, a standard tool employed in density results for different types of Sobolev spaces. We show in particular that under some regularity assumptions on , (A1) and that are not sufficient for the maximal operator to be bounded, the space of is dense in . In the case of variable exponent Sobolev space , we obtain the similar density result under the assumption that , , , , satisfies the log‐Hölder condition and the exponent function is essentially bounded.

On the convexity coefficient of Musielak–Orlicz function spaces equipped with the Orlicz norm

AbstractIn Hudzik and Landes, the convexity coefficient of Musielak–Orlicz function spaces over a non‐atomic measure space equipped with the Luxemburg norm is computed whenever the Musielak–Orlicz functions are strictly convex see [6]. In this paper, we extend this result to the case of Musielak–Orlicz spaces equipped with the Orlicz norm. Also, a characterization of uniformly convex Musielak–Orlicz function spaces as well as k‐uniformly convex Musielak–Orlicz spaces equipped with the Orlicz norm is given.

Martingale inequalities on Musielak–Orlicz Hardy spaces

AbstractGiven a probability space and a Musielak–Orlicz function , we investigate martingale inequalities in the framework of Musielak–Orlicz spaces by constructing atomic decompositions. Especially, the obtained results for Musielak–Orlicz functions with particular structure, including the variable Orlicz functions , , the variable double phase‐growth , and the perturbed variable exponent function are also new. Hence, we handle the martingale inequality for the above functional in a universal way.

Entropy and renormalized solutions for a nonlinear elliptic problem in Musielak-Orlicz spaces

In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak-Orlicz functions and the right-hand side from the space L1(Ω). In nonreflexive Musielak-Orlicz-Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.

Equivalence of Entropy and Renormalized Solutions of a Nonlinear Elliptic Problem in Musielak–Orlicz Spaces

We consider second-order elliptic equations with nonlinearities determined by Musielak–Orlicz functions and with right-hand side in the space. In the Musielak–Orlicz–Sobolev spaces, we establish some properties and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with Lipschitz boundary. In addition, the equivalence and sign-definiteness of the entropy and renormalized solutions is proved.

Duals of generalized Orlicz Hilbert sequence spaces and matrix transformations

In this paper, wedefine the sequence spaces hc (?(m)v,M,u,p), h0 (?(m) v,M,u,p) and h?(?(m)v,M,u,p) resulting from the infinite Hilbert matrix and the Musielak-Orlicz function. We give some topological properties and inclusion relations of these newly created spaces. We also identified ??, ?? and ??duals of the spaces. Finally, we tried to characterize some matrix transformations between these spaces.

Musielak–Orlicz Hardy space estimates for commutators of Calderón–Zygmund operators

AbstractLet and T be a δ‐Calderón–Zygmund operator. When and , it is well‐known (see the work by Harboure, Segovia, and Torrea [Illinois J. Math. 41 (1997), no. 4, 676–700]) that the commutator is not bounded from the Hardy space into the Lebesgue space if b is not a constant function. Let φ be a Musielak–Orlicz function satisfying that, for any , belongs to the Muckenhoupt weight class with the critical weight exponent and is an Orlicz function with the critical lower type . In this paper, we find a proper subspace of such that, if then is bounded from the Musielak–Orlicz Hardy space into the Musielak–Orlicz space . Conversely, if and the commutators of the classical Riesz transforms are bounded from into , then . Our results generalize some recent results by Huy and Ky [Vietnam J. Math. (2020). https://doi.org/10.1007/s10013‐020‐00406‐2] and Liang, Ky, and Yang [Proc. Amer. Math. Soc. 144 (2016), no. 12, 5171–5181].

On a Range of Exponents for Absence of Lavrentiev Phenomenon for Double Phase Functionals

For a class of functionals having the (p, q)-growth, we establish an improved range of exponents p, q for which the Lavrentiev phenomenon does not occur. The proof is based on a standard mollification argument and Young convolution inequality. Our contribution is two-fold. First, we observe that it is sufficient to regularise only bounded functions. Second, we exploit the L^{infty } bound on the function rather than the L^p estimate on the gradient. Our proof does not rely on the properties of minimizers to variational problems but it is rather a consequence of the underlying Musielak–Orlicz function spaces. Moreover, our method works for unbounded boundary data, the variable exponent functionals and vectorial problems. In addition, the result seems to be optimal for pleqq d.

Trudinger's inequality for Riesz potentials on Musielak-Orlicz spaces over metric measure spaces

We prove a generalized Trudinger's inequality for Riesz potentials of Musielak-Orlicz functions defined in a bounded metric measure space. As a corollary we obtain Trudinger's inequality for double phase functionals , where and satisfy log-Hölder conditions and is non-negative, bounded and Hölder continuous.

A flow approach to the Musielak-Orlicz-Gauss image problem

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure C˜G,Ψ,λ(Ω,⋅) of convex body Ω in Rn+1 containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

Some classes of sequence spaces defined by a Musielak-Orlicz function

Abstract In the present paper we introduce the sequence spaces c0{ℳ, Λ , p, q}, c{ℳ, Λ, p, q} and l ∞ {ℳ, Λ, p, q} defined by a Musielak-Orlicz function ℳ = (ℳk). We study some topological properties and prove some inclusion relations between these spaces.

Boundedness of fractional integral operators on Musielak–Orlicz Hardy spaces

AbstractLet and let be Musielak–Orlicz functions such that , are Orlicz functions and , are Muckenhoupt weights. In this paper, we give the necessary and sufficient condition for the boundedness of the fractional integral operator from the Musielak–Orlicz Hardy space into the Musielak–Orlicz Hardy space . We also give the necessary and sufficient condition for the boundedness of from into the Musielak–Orlicz space . Our results generalize the main results in [5, 20, 21].

The Interval Vectors of χ2 Sequence Space Defined by Musielak Orlicz Function

The Interval Vectors of χ2 Sequence Space Defined by Musielak Orlicz Function

Correction to: Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

Correction to: Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

Some sequence spaces over $n$-normed spaces defined by fractional difference operator and Musielak-Orlicz function

In the present paper we introduce some sequence spaces over $n$-normed spaces defined by fractional difference operator and Musielak-Orlicz function $\mathcal{M} = (\mathfrak{F}_i)$. We also study some topological properties and prove some inclusion relations between these spaces.

Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm

AbstractIt is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

Continuous imbedding in Musielak spaces with an application to anisotropic nonlinear Neumann problems

We prove a continuous embedding that allows us to obtain a boundary trace imbedding result for anisotropic Musielak-Orlicz spaces, which we then apply to obtain an existence result for Neumann problems with nonlinearities on the boundary associated to some anisotropic nonlinear elliptic equations in Musielak-Orlicz spaces constructed from Musielak-Orlicz functions on which and on their conjugates we do not assume the \(\Delta_2\)-condition. The uniqueness of weak solutions is also studied.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/26/abstr.html