On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L¹(R²) for positive times is entirely determined by the trace of the vorticity at t=0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa & Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in R² is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed for arbitrary data in a function space that is large enough to contain the initial data of some self-similar solutions.     

Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

Graded Modules Over the q-Analog Virasoro-Like Algebra

In this paper, we deal with the classification of the irreducible Z-graded and Z ²-graded modules with finite dimensional homogeneous subspaces for the q analog Virasoro-like algebra L. We first prove that a Z-graded L-module must be a uniformly bounded module or a generalized highest weight module. Then we show that an irreducible generalized highest weight Z-graded module with finite dimensional homogeneous subspaces must be a highest (or lowest) weight module and give a necessary and sufficient condition for such a module with finite dimensional homogeneous subspaces. We use the Z-graded modules to construct a class of Z ²-graded irreducible generalized highest weight modules with finite dimensional homogeneous subspaces. Finally, we classify the Z ²-graded L-modules. We first prove that a Z ²-graded module must be either a uniformly bounded module or a generalized highest weight module. Then we prove that an irreducible nontrivial Z ²-graded module with finite dimensional homogeneous subspaces must be isomorphic to a module constructed as above. As a consequence, we also classify the irreducible Z-graded modules and the irreducible Z ²-graded modules with finite dimensional homogeneous subspaces and center acting nontrivial. Supported by the National Science Foundation of China (No 10671160), the China Postdoctoral Science Foundation (No. 20060390693), the Specialized Research fund for the Doctoral Program of Higher Education (No.20060384002), and the New Century Talents Supported Program from the Education Department of Fujian Province.     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

Automorphisms of hyperbolic domains inR n

The main result of this paper is a fixed-point theorem for projective automorphisms of a bounded strongly convex domain inR ⁿ . Several corollaries and applications are derived, especially on the dimension of the full automorphism group in the smooth case.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

A Characterization of Group Generators on Hilbert Spaces and the H ∞ -calculus

Abstract. We give necessary and sufficient conditions for an operator A on a Hilbert space to have a bounded H ^∈ fty -calculus on a vertical strip symmetric to the imaginary axis. From this, a characterization of group generators on Hilbert spaces is obtained yielding recent results of Liu and Zwart as corollaries.     

Operatori pseudo-differenziali inR n e applicazioni

Summary In §1 we study a class of pseudo-differential operators inR ⁿ. In §3 the results obtained in §§1, 2 are applied to study of an elliptic boundary value problem in the exterior of a bounded domain ofR ⁿ for differential operators whose coefficients have a polynomial growth to infinity. Entrata in Redazione il 23 marzo 1972.     

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into R ^n+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L ^n/2 Ricci curvature norm in R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.     

Stability and Boundary Behavior of the Kobayashi Metrics

Uniform stability and localization results for the higher order and singular Kobayashi metrics are established. As an application we obtain the non-tangential weighted limits of these metrics in an h-extendible boundary point of a bounded domain in C ⁿ. This revised version was published online in August 2006 with corrections to the Cover Date.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.     

Existence of bounded positive solutions of quasi-linear elliptic equations

Existence of bounded positive solutions of a class of quasi-linear elliptic equation is obtained in exterior domain of R ^N , N ≥ 1. Firstly, by using fixed point theory, the existence theorem of a class of ordinary differential equation is established. Then, by constructing super-solution and sub-solution, the existence of bounded positive solutions of quasi-linear elliptic equation is given. The results of this article are new and extend previously known results.     

An optimal multivariate spline method of recovery of twice differentiable functions

A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R ^d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R ^d .     

The maximal conjugate and Hilbert operators on real Hardy spaces

Summary We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H¹ to L¹, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T² and R².     

A contractible Levi-flat hypersurface which is a determining set for pluriharmonic functions

We find a real analytic Levi-flat hypersurface in C² containing a bounded contractible domain which is a determining set for pluriharmonic functions.     

Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .