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Register a new userNonlocal characterizations of variable exponent Sobolev spaces
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We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
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Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
Let $L$ be a linear operator on $L^2(mathbb R^n)$ generating an analytic semigroup ${e^{-tL}}_{tge0}$ with kernels having pointwise upper bounds and $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(cdot)}(mathbb R^n)$, and the BMO-type space ${mathrm{BMO}}_{p(cdot),L}(mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(cdot)}(mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(cdot)}(mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$.
We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.
In this article we introduce Variable exponent Fock spaces and study some of their basic properties such as the boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality.
For a given Lipschitz domain $Omega$, it is a classical result that the trace space of $W^{1,p}(Omega)$ is $W^{1-1/p,p}(partialOmega)$, namely any $W^{1,p}(Omega)$ function has a well-defined $W^{1-1/p,p}(partialOmega)$ trace on its codimension-1 boundary $partialOmega$ and any $W^{1-1/p,p}(partialOmega)$ function on $partialOmega$ can be extended to a $W^{1,p}(Omega)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $mathbb{R}^dbackslashOmega$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical $W^{1-1/p,p}(partialOmega)$ space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.
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