Fredholmness of multipliers on Hardy–Sobolev spaces

This paper discusses the Fredholmness of multipliers on Hardy–Sobolev Spaces and obtains an index formula for the multipliers with some special symbols. Our results show that Hardy–Sobolev spaces have richer properties than classical holomorphic function spaces, and the behavior of the operators on these spaces is complex. Some methods of Hardy or Bergman spaces fail in the case of the Hardy–Sobolev space.     

Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

Sharp Sobolev inequalities in Lorentz spaces for a mean oscillation

We exhibit the optimal constant for Sobolev inequalities in Lorentz spaces for a mean oscillation, and its relation with a boundedness of the Hardy–Littlewood maximal operator in Sobolev spaces. Some applications to a scale invariant form of Hardy?s inequality in a limiting case are also considered.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces

We show the interpolation inequalities for derivatives in variable exponent Lebesgue–Sobolev spaces by applying the boundedness of the Hardy–Littlewood maximal operator on L^p(⋅)$L^{p(\cdot )}$.     

Shooting with degree theory: Analysis of some weighted poly-harmonic systems

In this paper, the author establishes the existence of positive entire solutions to a general class of semilinear poly-harmonic systems, which includes equations and systems of the weighted Hardy–Littlewood–Sobolev type. The novel method used implements the classical shooting method enhanced by topological degree theory. The key steps of the method are to first construct a target map which aims the shooting method and the non-degeneracy conditions guarantee the continuity of this map. With the continuity of the target map, a topological argument is used to show the existence of zeros of the target map. The existence of zeros of the map along with a non-existence theorem for the corresponding Navier boundary value problem imply the existence of positive solutions for the class of poly-harmonic systems.     

An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

We prove the existence of a weak solution of the Dirichlet problem for a class of elliptic partial differential systems in separable Orlicz–Sobolev spaces.     

Ill-posedness of nonlocal Burgers equations

Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [J.K. Hunter, Nonlinear surface waves, in: Current Progress in Hyberbolic Systems: Riemann Problems and Computations, Brunswick, ME, 1988, in: Contemp. Math., vol. 100, Amer. Math. Soc., 1989, pp. 185–202], and more recently by Benzoni-Gavage and Rosini [S. Benzoni-Gavage, M. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (3–4) (2009) 1463–1484], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [S. Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (3–4) (2009) 303–320] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature.     

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

The existence of martingale solutions of the hydrodynamic-type equations in 3D possibly unbounded domains is proved. The construction of the solution is based on the Faedo–Galerkin approximation. To overcome the difficulty related to the lack of the compactness of Sobolev embeddings in the case of unbounded domain we use certain Fréchet space. Besides, we use compactness and tightness criteria in some nonmetrizable spaces and a version of the Skorohod theorem in non-metric spaces. The general framework is applied to the stochastic Navier–Stokes, magneto-hydrodynamic (MHD) and the Boussinesq equations.     

Gevrey regularity for the supercritical quasi-geostrophic equation

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.     

Multiple positive solutions for Robin problem involving critical weighted Hardy–Sobolev exponents with boundary singularities

This paper deals with the existence of positive solutions for Robin elliptic problems involving critical weighted Hardy–Sobolev exponents with boundary singularities. Using the Caffarelli–Kohn–Nirenberg inequalities and variational methods, we prove the existence and multiplicity of positive solutions.     

Solutions for the quasilinear elliptic problems involving critical Hardy–Sobolev exponents

In this article, we study the quasilinear elliptic problem involving critical Hardy–Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign–changing solutions to the problem.     

The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces

We study the initial value problem associated to the dispersion generalized Benjamin–Ono equation. Our aim is to establish persistence properties of the solution flow in weighted Sobolev spaces and to deduce from them some sharp unique continuation properties of solutions to this equation. In particular, we shall establish optimal decay rate for the solutions of this model.     

An upper gradient approach to weakly differentiable cochains

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheim?s theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskela?s concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with p-integrable upper gradient in n-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey–Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.     

An approach of majorization in spaces with a curved geometry

The Hardy–Littlewood–Pólya majorization theorem is extended to the framework of some spaces with a curved geometry (such as the global NPC spaces and the Wasserstein spaces). We also discuss the connection between our concept of majorization and the subject of Schur convexity. Several applications are included.     

Sobolev type embeddings in the limiting case

We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W _n ^k/k as a special case. We deal with generalized Sobolev spaces W _A ^k , where instead of requiring the functions and their derivatives to be in L_n/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to L_n/k. We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator. In memory of Gene Fabes. Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund.     

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.     

Decay for nonlinear Klein-Gordon equations

We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. In these spaces of fractional order we obtain a weighted Sobolev embedding and a nonlinear estimate. Using these, we establish the decay estimate of the solution for large time provided the power of nonlinearity is greater than a critical value.     

New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces.