We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $ρ$ the weight function, we compare the space $W^{1,p}(X,d,ρm)$ (which always concides with the closure $H^{1,p}(X,d,ρm)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_ρ(X,d,m)$ and $H^{1,p}_ρ(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,ρm)=H^{1,p}_ρ(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $ρ$ that ensure the equality $W^{1,p}_ρ(X,d,m)=H^{1,p}_ρ(X,d,m)$.
In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities, that is Sobolev type inequalities where different derivatives have different weight functions. The inequalities we are dealing with, are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider here Sobolev inequalities on finite cylinders, the weights being different powers of the distance function from the top and the bottom of the cylinder…
In the setting of the Euclidean space equipped with an arbitrary Radon measure, we prove the equivalence between several notions of function of bounded variation present in the literature. We also study the relation between various definitions of 1-Sobolev function.
We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces.
In this paper we study weighted Hardy-Sobolev spaces of vector valued functions analytic on double-napped cones of the complex plane. We introduce these spaces as a tool for complex scaling of linear ordinary differential equations with dilation analytic unbounded operator coefficients. As examples we consider boundary value problems in cylindrical domains and domains with quasicylindrical ends.
In this paper we give connections between mappings which generate bounded composition operators on Sobolev spaces and $Q$-mappings. On this base we obtain measure distortion properties $Q$-homeomorphisms. Using the composition operators on Sobolev spaces we obtain weighted Sobolev inequalities with special weights which are Jacobians of $Q$-mappings.
In this study, we consider weighted stochastic field exponent function spaces $L_{\vartheta }^{p(.,.)}\left( D\times Ω\right) $ and $W_{\vartheta }^{k,p(.,.)}\left( D\times Ω\right) $. Also, we investigate some basic properties and embeddings of these spaces. Finally, we present an application of these spaces to the stochastic partial differential equations with stochastic field growth.
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( Ω,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(.)$-Laplacian equation \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in Ω\\ f=0 & x\in \partial Ω\end{array} \right. \end{equati…
We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and rad…
New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.
We study the weak regularity of mappings inverse to weighted Sobolev homeomorphisms $φ:Ω\to\widetildeΩ$, where $Ω$ and $\widetildeΩ$ are domains in $\mathbb R^n$. Using the weak regularity of inverse mappings we obtain the composition duality property of composition operators on weighted Sobolev spaces.
In this paper we study connections between composition operators on Sobolev spaces and mappings defined by $p$-moduli inequalities ($p$-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on corresponding Sobolev spaces and inverse, composition operators on Sobolev spaces imply weighted moduli inequalities.
In this paper we develop elements of the global calculus of Fourier integral operators in $R^n$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev $L^2$ estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of smoothing estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.
In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant.
We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.
We present weighted Sobolev spaces and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $Ω\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.
In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{p,q}, are also obtained along the way.
We prove weighted inequalities between the Gagliardo and Sobolev seminorms and also between the Marcinkiewicz quasi-norm and the Sobolev seminorm. With $A_1$ weights we improve earlier results of Bourgain, Brezis, and Mironescu.
We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.
We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polinomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.
Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
We provide a family of global weighted Sobolev inequalities and Hardy inequalities on PI spaces with possibly non-maximal volume growth. Our results apply notably to non-trivial Ahlfors regular spaces like Laakso spaces and Kleiner-Schioppa spaces.
We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…
We consider the Benjamin-Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a consequence, unconditional uniqueness for the Benjamin-Ono equation is proved.
Let $Ω$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $Ω$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $Ω$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.
We study persistence properties of solutions to some canonical dispersive models, namely the semi-linear Schrödinger equation, the $k$-generalized Korteweg-de Vries equation and the Benjamin-Ono equation, in weighted Sobolev spaces $H^s(\R^n)\cap L^2(|x|^ldx),\;s,\,l>0$
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up…
We consider a scattering map that arises in the $\bar \partial $-approach to the scattering theory for the Davey-Stewartson II equation and show that the map is an invertible map between certain weighted $L^2$ Sobolev spaces.
We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel similar to $x\mapsto |x|^{-d-sp}$. One may consider the results to be a~version of the Meyers--Serrin theorem.
We study the initial value problem associated to the dispersion generalized Benjamin-Ono equation. Our aim is to establish well posedness results 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.
We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of re…
We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.
We study the well-posedness in weighted Sobolev spaces, for the initial value problem (IVP) associated with the dissipative Benjamin-Ono (dBO) equation. We establish persistence properties of the solution flow in the weighted Sobolev spaces $\z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\geq r>0$. We also prove some unique continuation properties in these spaces. In particular, such results of unique continuation show that our results of well posedness are sharp.
We generalize the Abstract Interpolation Lemma proved by the authors in [2]. Using this extension, we show in a more general context, the persistence property for the generalized Korteweg-de Vries equation, see (1.2), in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear dispersive models, for instance the multidimensional nonlinear Schrodinger equation.
Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
School of Science and the Environment, Grenfell Campus, Memorial University of Newfoundland（纪念大学格伦菲分校科学与环境学院）；Department of Mathematics, University of Connecticut（康涅狄格大学数学系）
We prove the existence of time periodic solution to the 3D Ginzburg-Landau equation in weighted Sobolev spaces. We consider the cubic Ginzburg-Landau equation with an external force $g$ satisfying the oddness condition $g(-x,t)=-g(x,t)$. The existence of the periodic solution is proved for small time-periodic external force. The stability of the time periodic solution is also considered.
In this note, we prove global weighted Sobolev inequalities on non-compact CD(0,N) spaces satisfying a suitable growth condition, extending to possibly non-smooth and non-Riemannian structures a previous result by V. Minerbe stated for Riemannian manifolds with non-negative Ricci curvature. We use this result in the context of RCD(0,N) spaces to get a uniform bound of the corresponding weighted heat kernel via a weighted Nash inequality.
We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces. Our method of proof is based on the contraction mapping principle and it mainly relies on the well-posedness results recently obtained for this equation in the Sobolev spaces H^s(\R^2) and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered
In this paper we present a weighted $L_p$-theory of parabolic systems on a half space. The leading coefficients are assumed to be only measurable in $t$ and have small bounded mean oscillations (BMO) with respect to $x$, and the lower order coefficients are allowed to blow up near the boundary.
We study the initial value problem associated to a higher dimensional version of the Benjamin-Ono equation. Our purpose is to establish local well-posedness results in weighted Sobolev spaces and to determinate according to them some sharp unique continuation properties of the solution flow. In consequence, optimal decay rate for this model is determined. A key ingredient is the deduction of a new commutator estimate involving Riesz transforms.
The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through $\overline{\partial}$-derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces, where the weight satisfies some specific conditions. Our proof relies on a classical bound for the derivatives of fundamental solution to parabolic equations.
We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces $Z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\in\R, \,s\geq 1$ and $s\geq r$. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demostrate that our persistence properties are sharp.
The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations.
