Sobolev Lorentz Spaces Research Articles

A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the Cauchy problem for the H^{s} -critical inhomogeneous nonlinear Schrödinger (INLS) equation iu_{t} +\Delta u=\lvert{x}\rvert^{-b}f(u),\quad u(0)=u_{0} \in H^{s} (\mathbb{R}^{n}), where n\in \mathbb{N} , 0\le s<\smash{\frac{n}{2}} , 0<b<\min\{2, n-s, 1+\smash{\frac{n-2s}{2}}\} and f(u) is a nonlinear function that behaves like \lambda \lvert{u}\rvert^{\sigma}u with \lambda \in \mathbb{C} and \sigma=\smash{\frac{4-2b}{n-2s}} . First, we establish the local well-posedness as well as the small data global well-posedness in H^{s}(\mathbb{R}^{n}) for the H^{s} -critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev–Lorentz spaces. Next, we obtain some standard continuous dependence results for the H^{s} -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the H^{s} -critical INLS equation improve the ones of Aloui–Tayachi [Discrete Contin. Dyn. Syst. 41 (2021), 5409–5437] and An–Kim [Evol. Equ. Control Theory 12 (2023), 1039–1055] by extending the validity of s and b . Based on the local well-posedness in H^{1}(\mathbb{R}^{n}) , we finally establish the blow-up criteria for H^{1} -solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Sobolev-Lorentz spaces with an application to the inhomogeneous biharmonic NLS equation

Sharp Adams type inequalities in Lorentz-Sobole space

<abstract><p>This article addresses several sharp weighted Adams type inequalities in Lorentz-Sobolev spaces by using symmetry, rearrangement and the Riesz representation formula. In particular, the sharpness of these inequalities were also obtained by constructing a proper test sequence.</p></abstract>

Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

EMBEDDING THEOREM OF THE WEIGHTED SOBOLEV–LORENTZ SPACES

AbstractWeight criteria for embedding of the weighted Sobolev–Lorentz spaces to the weighted Besov–Lorentz spaces built upon certain mixed norms and iterated rearrangement are investigated. This gives an improvement of some known Sobolev embedding. We achieve the result based on different norm inequalities for the weighted Besov–Lorentz spaces defined in some mixed norms.

The sharp higher-order Lorentz–Poincaré and Lorentz–Sobolev inequalities in the hyperbolic spaces

In this paper, we study the sharp Poincaré inequality and the Sobolev inequalities in the higher-order Lorentz–Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in Nguyen VH (J Math Anal Appl, 490(1):124197, 2020) to the higher-order derivatives and seem to be new in the context of the Lorentz–Sobolev spaces defined in the hyperbolic spaces.

Removability of zero modular capacity sets

We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory. We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients. We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values. As a key tool, we establish a modular Poincare inequality. With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions. Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets. The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak–Orlicz–Sobolev spaces, Lorentz–Sobolev spaces, variable exponent Sobolev spaces.

The sharp Sobolev type inequalities in the Lorentz–Sobolev spaces in the hyperbolic spaces

Let W1Lp,q(Hn), 1≤q,p<∞ denote the Lorentz–Sobolev spaces of order one in the hyperbolic spaces Hn. Our aim in this paper is three-fold. First of all, we establish a sharp Poincaré inequality in W1Lp,q(Hn) with 1≤q≤p which generalizes the result in [41] to the setting of Lorentz–Sobolev spaces. Second, we prove several sharp Poincaré–Sobolev type inequalities in W1Lp,q(Hn) with 1≤q≤p<n which generalize the results in [45] to the setting of Lorentz–Sobolev spaces. Finally, we provide the improved Moser–Trudinger type inequalities in W1Ln,q(Hn) in the critical case p=n with 1≤q≤n which generalize the results in [43] and improve the results in [57]. In the proof of the main results, we shall prove a Pólya–Szegö type principle in W1Lp,q(Hn) with 1≤q≤p which maybe is of independent interest.

Measures of Non-Compactness and Sobolev–Lorentz Spaces

We show that the measure of non-compactness of the limiting embedding of Sobolev–Lorentz spaces is equal to the norm. This is a consequence of our general theorem for arbitrary Banach spaces.

Sobolev embeddings in Orlicz and Lorentz spaces with measures

Embedding theorems for Orlicz-Sobolev spaces into Orlicz or Orlicz-Lorentz spaces in domains in Rn, endowed with a Frostman measure, are offered. Parallel embeddings for Lorentz-Sobolev spaces into (generalized) Lorentz spaces are also established. The relevant embeddings yield an optimal target norm, in the relevant classes, whenever it does exist. In particular, various results available in the literature are improved by special choices of the function spaces and of the measure.

On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev–Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local $$L^2$$ -mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scaling of the single-power NLS-equation. Our results extend and complement those of Weissler (Adv Differ Equ 6(4):419–440, 2001), particularly because we are working in the larger setting of Sobolev-weak- $$L^p$$ spaces and considering non-local terms. The two nonlinearities of power-type and the generality of the non-local terms allow us to cover in a unified way a large number of dispersive equations and systems.

Banach algebras of weakly differentiable functions

The question is addressed of when a Sobolev type space, built upon a general rearrangement-invariant norm, on an n-dimensional domain, is a Banach algebra under pointwise multiplication of functions. A sharp balance condition among the order of the Sobolev space, the strength of the norm, and the (ir)regularity of the domain is provided for the relevant Sobolev space to be a Banach algebra. The regularity of the domain is described in terms of its isoperimetric function. Related results on the boundedness of the multiplication operator into lower-order Sobolev type spaces are also established. The special cases of Orlicz-Sobolev and Lorentz-Sobolev spaces are discussed in detail. New results for classical Sobolev spaces on possibly irregular domains follow as well.

On a Hamiltonian System with Critical Exponential Growth

We are interested in finding nontrivial solutions for a Hamiltonian elliptic system in dimension two involving a potential function which can be coercive and nonlinearities that have maximal growth with respect to the Trudinger–Moser inequality. To establish the existence of solutions, we use variational methods combined with Trudinger–Moser type inequalities in Lorentz–Sobolev spaces and a finite-dimensional approximation.

Equivalence of Sharp Trudinger-Moser Inequalities in Lorentz-Sobolev Spaces

The critical and subcritical Trudinger-Moser inequalities in Lorentz Sobolev space have been studied by Cassani and Tarsi (Asymptot. Anal. 64(1-2):29–51, 2009), Lu and Tang (Adv. Nonlinear Stud. 16(3):581–601, 2016). In this paper, we will prove that these critical and subcritical Trudinger-Moser inequalities are actually equivalent and thus extend those equivalence results of Lam et al. (Rev. Mat. Iberoam 33(4):1219–1246, 2017) into Lorentz Sobolev spaces.

Equivalent and attained version of Hardy's inequality in [formula omitted

We investigate connections between Hardy's inequality in the whole space Rn and embedding inequalities for Sobolev–Lorentz spaces. In particular, we complete previous results due to [1, Alvino] and [28, Talenti] by establishing optimal embedding inequalities for the Sobolev–Lorentz quasinorm ‖∇⋅‖p,q also in the range p<q, which remained essentially open since [1].Attainability of the best embedding constants is also studied, as well as the limiting case when q=∞. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev–Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called “ghost” extremal functions of [6, Brezis–Vazquez], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised in [6].

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

AbstractWe will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system urn:x-wiley:0025584X:media:mana201700215:mana201700215-math-0001where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.

Absolute Continuity of Sobolev Functions and Moduli of Families of Hypersurfaces Related to Lorentz Spaces

We study traces of functions in Sobolev–Lorentz spaces on Lipschitz hypersurfaces. Some aspects concerning the (n − 1)-absolute continuity are studied.

Sharp Sobolev type embeddings on the entire Euclidean space

A comprehensive approach to Sobolev type embeddings, involving arbitrary rearrangement-invariant norms on the entire Euclidean space \begin{document}${\mathbb R^n}$\end{document} , is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in our analysis is a new reduction principle for the relevant embeddings, showing their equivalence to a couple of considerably simpler one-dimensional inequalities. Applications to the classes of the Orlicz-Sobolev and the Lorentz-Sobolev spaces are also presented. These contributions fill in a gap in the existing literature, where sharp results in such a general setting are only available for domains of finite measure.

Nonlinear Elliptic Equations with Maximal Growth Range

En este trabajo nos interesa estudiar la existencia de soluciones débiles no triviales para una clase de ecuaciones elípticas no lineales definidas en un dominio limitado en dimensión dos, donde las no linealidades poseen un rango de crecimiento exponencial máximo motivado por las desigualdades de Trudinger-Moser en espacios de Lorentz-Sobolev. Para estudiar la solubilidad se utiliza un enfoque variacional. Más específicamente, usamos el teorema del paso de montaña junto con desigualdades de tipo Trudinger-Moser.

On weak continuity of the Moser functional in Lorentz-Sobolev spaces

Let $B(R)\subset \mathbb{R}^n $, $n\in \mathbb{N} $, $n\geq 2$, be an open ball. By a result from~\cite {AdT}, the Moser functional with the borderline exponent from the Moser-Trudinger inequality fails to be sequentially weakly continuous on the set of radial functions from the unit ball in $W_0^{1,n}(B(R))$, only in the exceptional case of sequences acting like a~concentrating Moser sequence. We extend this result into the Lorentz-Sobolev space $W_0^1L^{n,q}(B(R))$, with $q\in (1,n]$, equipped with the norm $$ ||\nabla u||_{n,q}:= ||t^{1/n-1/q}|\nabla u|^*(t)||_{L^q((0,|B(R)|))}. $$ We also consider the case of a nontrivial weak limit and the corresponding Moser functional with the borderline exponent from the concentration-compactness alternative.