Tangential characterizations of Hardy and mixed-norm spaces of harmonic functions on the real hyperbolic ball

Summary The Hardy and mixed-norm spaces of harmonic functions on the real hyperbolic ball are characterized in terms of the tangential gradient.     

The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains

We introduce certain Sobolev-Besov spaces which are particularly well adapted for measuring the smoothness of data and solutions of mixed boundary value problems in Lipschitz domains. In particular, these are used to obtain sharp well-posedness results for the Poisson problem for the Laplacian with mixed boundary conditions on bounded Lipschitz domains which satisfy a suitable geometric condition introduced by R.Brown in (1994). In this context, we obtain results which generalize those by D.Jerison and C.Kenig (1995) as well as E.Fabes, O.Mendez and M.Mitrea (1998). Applications to Hodge theory and the regularity of Green operators are also presented.

     

区域Hardy空间的原子分解和对偶空间

研究D-C chang等人引进的五个区域Hardy空间，刻划这些空间的原子分解和对偶空间，揭示了这些空间的内在联系。     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

多线性分数次奇异积分在弱 Hardy 空间的Lipschitz 估计

讨论了一类具有粗糙核多线性分数次奇异积分算子在弱 Hardy 空间的性质,通过原子分解,得到了这类算子在弱Hardy空间的有界性.     

交换子在加权Herz型Hardy空间上的有界性

主要讨论由Lipschitz函数b与广义C-Z算子T生成的交换子[b,T]在加权Herz型Hardy空间上的有界性,证明了[6,T]从HKq1^α,p（w1,w2^q1）到HKq2^α,p（w1,w2^q2）的有界性．     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Real interpolation method,Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.     

Abstract Hardy-Sobolev spaces and interpolation

The purpose of this work is to describe an abstract theory of Hardy-Sobolev spaces on doubling Riemannian manifolds via an atomic decomposition. We study the real interpolation of these spaces with Sobolev spaces and finally give applications to Riesz inequalities.     

A rough hypersingular integral operator with an oscillating factor

We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|^−n−α(α>0) at the origin, an oscillating factor ei|y|^−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

A sharper uncertainty principle

The work strengthens the result established by L. Cohen on uncertainty principle involving phase derivative. We propose stronger uncertainty principles not only in the classical setting for Fourier transform, but also for self-adjoint operators. We also deduce the conditions that give rise to the equal relation of the uncertainty principle. Examples are provided to show that the new uncertainty principle is truly sharper than the existing ones in literature.     

The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains

We study the regularity of the Navier-Stokes equations in arbitrary Lipschitz domains.     

A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials

Let be the solution operator for in , Tr on , where is a bounded domain in . B. E. J. Dahlberg proved that for a bounded Lipschitz domain maps boundedly into weak- and that there exists such that is bounded for . In this paper, we generalize this result by addressing two aspects. First we are also able to treat the solution operator corresponding to Neumann boundary conditions and, second, we prove mapping properties for these operators acting on Sobolev (rather than Lebesgue) spaces.

     

An <Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>-Theory of SPDEs on Lipschitz Domains

Stochastic partial differential equations are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular, it can be negative and fractional. It is allowed that the coefficients of the equations blow up near the boundary.     

Scattering properties for a pair of Schrödinger type operators on cylindrical domains

Strong asymptotic completeness is shown for a pair of Schrödinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre’s method within the context of pseudo-selfadjoint operators.     

Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].