Riesz transforms for Dunkl Hermite expansions

In the present paper, we establish that Riesz transforms for Dunkl Hermite expansions introduced by Nowak and Stempak are singular integral operators with Hörmander's type condition. We prove that they are bounded on L^p(R^d,d_μκ)$L^p({\mathbb{R}}^d,d{\mu }_{\kappa })$ for 1<p<∞$1<p<\infty$ and from L¹(R^d,d_μκ)$L^1({\mathbb{R}}^d,d{\mu }_{\kappa })$ into L^1,∞(R^d,d_μκ)$L^{1,\infty }({\mathbb{R}}^d,d{\mu }_{\kappa })$.     

Multi-tiling and Riesz bases

Let S   be a bounded, Riemann measurable set in R^d${\mathbb{R}}^d$, and Λ be a lattice. By a theorem of Fuglede, if S   tiles R^d${\mathbb{R}}^d$ with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles  R^d${\mathbb{R}}^d$ with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyer?s quasicrystals.     

The wave equation for the Bessel Laplacian

We study radial solutions of the Cauchy problem for the wave equation in the multidimensional unit ball B^d$B^d$, d≥1$d\ge 1$. In this case, the operator that appears is the Bessel Laplacian and the solution u(t,x)$u(t,x)$ is given in terms of a Fourier–Bessel expansion. We prove that, for initial L^p$L^p$ data, the series converges in the L²$L^2$ norm. The analysis of a particular operator, the adjoint of the Riesz transform for Fourier–Bessel series, is needed for our purposes, and may be of independent interest. As applications, certain L^p−L²$L^p-L^2$ estimates for the solution of the heat equation and the extension problem for the fractional Bessel Laplacian are obtained.     

The cutoff phenomenon for Ehrenfest chains

We consider families of Ehrenfest chains and provide a simple criterion on the L^p$L^p$-cutoff and the L^p$L^p$-precutoff with specified initial states for 1≤p<∞$1\le p<\infty$. For the family with an L^p$L^p$-cutoff, a cutoff time is described and a possible window is given. For the family without an L^p$L^p$-precutoff, the exact order of the L^p$L^p$-mixing time is determined. The result is consistent with the well-known conjecture on cutoffs of Markov chains proposed by Peres in 2004, which says that a cutoff exists if and only if the multiplication of the spectral gap and the mixing time tends to infinity.     

Stability estimates for the Radon transform with restricted data and applications

In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the L^p$L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.     

The natural Banach space for version independent risk measures

Risk measures, or coherent measures of risk, are often considered on the space L^∞$L^{\infty }$, and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure–the Average Value-at-Risk–are well defined on the larger space L¹$L^1$ and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some L^p$L^p$ space. But in many situations this is possibly unnatural, because any L^p$L^p$ with p>_p0$p>p_0$, say, is suitable to define the spectral risk measure as well. In addition to that, risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is?     

Uncertainty principles in finitely generated shift-invariant spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in L²(R^d)$L^2({\mathbb{R}}^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in L²(R)$L^2(\mathbb{R})$ whose generator and its translates form a Riesz basis.     

Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G  . In particular, we show that the operators _Tα:f?|⋅|^−αL^−α/2f$T_{\alpha }:f?{|\cdot |}^{-\alpha }{L}^{-\alpha /2}f$, where |⋅|$|\cdot |$ is a homogeneous norm, 0<α<Q/p$0<\alpha <Q/p$, and L   is the sub-Laplacian, are bounded on the Lebesgue space L^p(G)$L^p(G)$. As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the L^p$L^p$ norm of a function f   to the L^q$L^q$ norm of |⋅|^βf${|\cdot |}^{\beta }f$ and the L^r$L^r$ norm of L^δ/2f$L^{\delta /2}f$.     

Weighted estimates for commutators of some singular integrals related to Schrödinger operators

Let L=−Δ+V$L=-\mathrm{\Delta }+V$ be a Schrödinger operator with a non-negative potential V satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to L such as the Riesz transforms and fractional integrals with the new BMO functions introduced in Bongioanni et al. (2011) [1] on the weighted spaces L^p(w)$L^p(w)$ where w belongs to the new classes of weights introduced by Bongioanni et al. (2011) [2].     

L -decay rates of planar viscous rarefaction wave for scalar conservation law with degenerate viscosity in n -dimensions

We investigate the Cauchy problem and the initial-boundary value problem for multi-dimensional conservation laws with degenerate viscosity in the whole space and in the half-space respectively. We give the optimal decay estimates in the W^1,p(1≤p≤∞)$W^{1,p}(1\le p\le \infty )$ norm for the perturbation from the planar viscous rarefaction wave. The analysis based on the new L^p$L^p$-energy method and L¹$L^1$-estimates.     

Finite element approximation to nonlinear coupled thermal problem

A nonlinear coupled elliptic system modelling a large class of engineering problems was discussed in [A.F.D. Loula, J. Zhu, Finite element analysis of a coupled nonlinear system, Comp. Appl. Math. 20 (3) (2001) 321–339; J. Zhu, A.F.D. Loula, Mixed finite element analysis of a thermally nonlinear coupled problem, Numer. Methods Partial Differential Equations 22 (1) (2006) 180–196]. The convergence analysis of iterative finite element approximation to the solution was done under an assumption of ‘small’ solution or source data which guarantees the uniqueness of the nonlinear coupled system. Generally, a nonlinear system may have multiple solutions. In this work, the regularity of the weak solutions is further studied. The nonlinear finite element approximations to the nonsingular solutions are then proposed and analyzed. Finally, the optimal order error estimates in H¹$H^1$-norm and L²$L^2$-norm as well as in W^1,p$W^{1,p}$-norm and L^p$L^p$-norm are obtained.     

Dirichlet operators: A priori estimates and uniqueness problems II

We study some basic properties of the Dirichlet operator in weighted L^p$L^p$ spaces. We work under rather general assumptions on weights that destroy minimal local Sobolev regularity of the form domain.     

Gradient estimates in Orlicz spaces for quasilinear elliptic equation

In this paper we generalize classical L^q$L^q$, q≥p$q\ge p$, estimates of the gradient to the Orlicz space for weak solutions of quasilinear elliptic equations of p-Laplacian type.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

p-Laplacian boundary value problems on exterior regions

This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?R^N$U?{\mathbb{R}}^N$ with dimension N?3$N?3$. Existence-uniqueness results when p∈(1,N)$p\in (1,N)$ are provided in a space E^1,p(U)$E^{1,p}(U)$ of functions that contains W^1,p(U)$W^{1,p}(U)$. Functions in E^1,p(U)$E^{1,p}(U)$ are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an L^p$L^p$-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.     

Korn type inequalities in Orlicz spaces

We exhibit balance conditions between a Young function A and a Young function B   for a Korn type inequality to hold between the L^B$L^B$ norm of the gradient of vector-valued functions and the L^A$L^A$ norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in L^p$L^p$, with 1<p<∞$1<p<\infty$, and an Orlicz version involving a Young function A   satisfying both the _Δ2${\mathrm{\Delta }}_2$ and the _∇2${\mathrm{\nabla }}_2$ condition.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.