Narrow Results

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen–Cahn type given by

In this paper, we study the local regularity of very weak solution of the elliptic equation D_j(a_ij(x)D_iu) = f - D_ig_i. Using the bootstrap argument and the difference quotient method, we obtain that if , and with 1 < p < ∞, then . Furthermore, we consider the higher regularity of u.

This paper is devoted to the description of the lack of compactness of the Sobolev space in the Orlicz space . The approach that we adopt to establish this characterization is in the spirit of the one adopted in the case of into the Orlicz space in [H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal. 260 (2011) 208–252].

We consider a parabolic system in divergence form with measurable coefficients in a cylindrical space–time domain with nonsmooth base. The associated nonhomogeneous term is assumed to belong to a suitable weighted Orlicz space. Under possibly optimal assumptions on the coefficients and minimal geometric requirements on the boundary of the underlying domain, we generalize the Calderón–Zygmund theorem for such systems by essentially proving that the spatial gradient of the weak solution gains the same weighted Orlicz integrability as the nonhomogeneous term.

This paper is devoted to the qualitative study of the nonlinear Schrödinger equation with exponential growth, where the Orlicz norm plays a crucial role. The approach we adopted in this paper which is based on profile decompositions consists of comparing the evolution of oscillations and concentration effects displayed by sequences of solutions to 2D linear and nonlinear Schrödinger equations associated to the same sequence of Cauchy data, up to small remainder terms both in Strichartz and Orlicz norms. The analysis we conducted in this work emphasizes the correlation between the nonlinear effect highlighted in the behavior of the solutions to the 2D nonlinear Schrödinger equation and the $1$-oscillating component of the sequence of the Cauchy data.

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of $\delta$-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of $\delta$-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

In this paper, we aim to provide a general paradigm for dealing with the sampling and random sampling problem in a reproducing kernel subspace of Orlicz space $L^{\varphi }({ℝ}^n)$. We consider the function space $V$ as the image of an idempotent integral operator on $L^{\varphi }({ℝ}^n)$, where the integral kernel satisfies certain off-diagonal decay and regularity conditions. The model example of such reproducing kernel subspace of $L^{\varphi }({ℝ}^n)$ includes the finitely generated shift-invariant space and signal space with a finite rate of innovation. We show that a signal in $V$ can be stably reconstructed from its samples at distinct points separated by a sufficiently small gap. Next, we deduce that the random sampling inequality holds with a high probability for the class of functions in $V$ concentrated on a cube $C_R={[-\frac{R}{2},\frac{R}{2}]}^n$, when the samples collected at i.i.d. random points are drawn on $C_R$ of order ${𝒪}(R^{n}log{R}^n)$.

The motivation of this paper is to introduce Henstock–Orlicz space with non-absolute integrable functions. We prove that $C_0^{\infty }$ is dense in the Henstock–Orlicz space, which is not dense in the classical Orlicz space.

Suppose X is a metric measure space. As a generalization of the p-modulus and the p-capacities in X, the Ψ-modulus and the Ψ-capacities are defined by means of the Luxemburg norm in the Orlicz space L^Ψ(X) with a general Yang function Ψ, instead of the ordinary norm in L^p(X). The relations between the Ψ-modulus and Ψ-capacities are discussed and some related equalities are established.

General aspects of the domination and factorization properties of multilinear operators from quasi-Banach lattices to quasi-Banach spaces are presented.

Using a modern approach that allows a full generality, we analyze some classical and new results, showing when certain vector valued norm inequalities provide factorization through some well-known Banach function lattices, as for instance Orlicz spaces. For example, we show that our framework allows to prove a multilinear variant of the Grothendieck factorization theoremfor products of C(K)-spaces.