Quasi-banded operators,convolutions with almost periodic or quasi-continuous data,and their approximations

We study the stability and Fredholm property of the finite sections of quasi-banded operators acting on L^p$L^p$ spaces over the real line. This family is significantly larger than the set of band-dominated operators, but still permits to derive criteria for the stability and results on the splitting property, as well as an index formula in the form as it is known for the classical cases. In particular, this class covers convolution type operators with semi-almost periodic and quasi-continuous symbols, and operators of multiplication by slowly oscillating, almost periodic or even more general coefficients.     

Preserving closedness of operators under summation

We give a sufficient condition for the sum of two closed operators to be closed. In particular, we study the sum of two sectorial operators with the sum of their sectoriality angles greater than π  . We show that if one of the operators admits bounded H^∞$H^{\infty }$-calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality, but weaker than the bounded imaginary powers property in the case of UMD spaces, then the sum is closed. We apply the result to the abstract parabolic problem and give a sufficient condition for L^p$L^p$-maximal regularity.     

Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds

In this Note, we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of invariant operator and its full symbol associated with an elliptic operator are introduced. Some applications to the study of r  -nuclearity on L^p$L^p$ spaces are also obtained.     

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L   be a non-negative self-adjoint operator acting on L²(X)$L^2(X)$ where X is a space of homogeneous type. Assume that L   generates a holomorphic semigroup e^−tL$e^{-tL}$ which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.     

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.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

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$L^{\delta /2}f$.     

Riesz transform and Riesz potentials for Dunkl transform

Analogues of Riesz potentials and Riesz transforms are defined and studied for the Dunkl transform associated with a family of weight functions that are invariant under a reflection group. The L^p$L^p$ boundedness of these operators is established in certain cases.     

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)$L^{1,\infty }({\mathbb{R}}^d,d{\mu }_{\kappa })$.     

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.     

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.     

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.     

Logarithmic bump conditions and the two-weight boundedness of Calderón–Zygmund operators

We prove that if a pair of weights (u,v)$(u,v)$ satisfies a sharp _Ap$A_p$-bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map L^p(v)$L^p(v)$ into L^p(u)$L^p(u)$ with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón–Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the _A2$A_2$ conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele.     

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?     

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.     

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.     

Cauchy integrals,Calderón projectors,and Toeplitz operators on uniformly rectifiable domains

We develop properties of Cauchy integrals associated to a general class of first-order elliptic systems of differential operators D on a bounded, uniformly rectifiable (UR) domain Ω in a Riemannian manifold M. We show that associated to such Cauchy integrals are analogues of Hardy spaces of functions on Ω annihilated by D  , and we produce projections, of Calderón type, onto subspaces of L^p(∂Ω)$L^p(\partial \Omega )$ consisting of boundary values of elements of such Hardy spaces. We consider Toeplitz operators associated to such projections and study their index properties. Of particular interest is a “cobordism argument,” which often enables one to identify the index of a Toeplitz operator on a rough UR domain with that of one on a smoothly bounded domain.     

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.