Given a weight in , and a Young function , we consider the weighted modular ( ( )) ( ) and the resulting weighted Orlicz space ( ). For a Young function ∆ ( ) we present a necessary and sufficient conditions in order that ( ) = ( ) up to the equivalence of norms. We find a necessary and sufficient condition for in order that there exists an unbounded weight such that the above equality of space...

We prove higher integrability of the spatial gradient weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The need be neither degenerate nor singular. Our result generalized version one J. Kinnunen and Lewis [Duke Math. 102 (2000), no. 2, 253--271] for $p$-Laplace systems.

In this paper, we study two classes of Kirchhoff-type problems set on a double-phase framework. That is, the functional space where finding solutions coincides with Musielak–Orlicz–Sobolev $$W^{1,{\mathcal {H}}}_0(\Omega )$$ , modular function $${\mathcal {H}}$$ related to so-called operator. Via variational approach, provide existence and multiplicity results.

We show that the canonical embedding C(K)→ LΦ(μ) has Gaussian cotype p, where μ is a Radon probabilty measure on K, and Φ is an Orlicz function equivalent to t(log t) p 2 for large t. * * * * * * In [6], I showed that the Gaussian cotype 2 constant of the canonical embedding l ∞ → L2,1 is bounded by log logN . Talagrand [9] showed that this embedding does not have uniformly bounded cotype 2 con...

This chapter is a revised version of tutorial lecture that I presented at the École de Physique des Houches on July 26–31 2020. Topics include: Non-parametric Information Geometry, Statistical bundle, exponential Orlicz spaces, and Gaussian Orlicz-Sobolev spaces.

A result is found which is similar to BDG-inequalities, but in the framework of exponential (non moderate) Orlicz spaces. A special class of such spaces is introduced and its properties are discussed with respect to probability measures, whose densities are connected by an exponential model. Acknowledgement: Thanks are due to Prof. M. Mania (Georgian Academy of Sciences) for the discussions and...

The current article discusses the existence of monotonic discontinuous solutions for general nonlinear quadratic Volterra-Urysohn functional-integral equations in Orlicz spaces E? when function ? satisfies ?2-condition. case Lebesgue Lp (p > 1) are also examined. We use arguments measure noncompactness with Darbo fixed point theorem to prove our results.

It can be checked that this is a valid norm (on the set of random variables for which the left side of the above display is finite). Of special interest to us will be the Orlicz norms corresponding to the functions {ψp : p ≥ 1} where ψp(x) = exp(xp) − 1. Lemma 8.1 of Kosorok (2008) provides a necessary and sufficient condition for the ψp Orlicz norm to be finite in terms of the tail-behavior of...

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group Γ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum Λ ⊆ Γ , the norm of the Hilbert transform and th...

Our concern in this paper lies with imbeddings of general spaces of Besov and Lizorkin-Triebel type with dominating mixed derivatives in the first critical case. We employ multivariate exponential Orlicz and Lorentz-Orlicz spaces in the role of targets. We study basic properties of the target spaces, in particular, we compare them with usual exponential spaces, showing that in this case the mul...