Curvature estimates for minimal hypersurfaces in singular spaces

We here prove pointwise curvature estimates for minimal hypersurfaces in singular spaces, using the integral curvature estimate and a generalized Simons inequality which were established recently. A further basic ingredient is a new Sobolev type inequality for stationary hypersurfaces.Oblatum 19-IX-1994This research was supported by the Deutsche Forschungsgemeinschaft through a Heisenberg award. Part of the work described here was carried out during visits to Dipartimento Matematica Applicata, University of Firenze and Department of Mathematics, Stanford University. The author would like to express his gratitude to both institutions for their kind hospitality and support. Also it is a pleasure to thank the referee for his useful comments concerning the formulation of the main theorem.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

Quantitative gradient estimates for harmonic maps into singular spaces

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space (X, d_X) with curvature bounded above by a constant κ (κ ? 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.

     

Schauder estimates for equationswith fractional derivatives

The equation where and are fractional derivatives of order and is studied. It is shown that if , , and are Hölder-continuous and , then there is a solution such that and are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (). Finally the solution of () with is studied.     

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.     

Endpoint estimates for multilinear fractional singular integral operators on some Hardy spaces

In this paper, the endpoint estimates for some multilinear operators related to certain fractional singular integral operators on some Hardy and Herz-type Hardy spaces are obtained.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

VMO spaces not having Schauder basis

ПустьA _n (n=0, 1, 2, ...) — последо вательность δ-алгебр, порождаемых двоичны ми отрезками [k2^?n , (k+1)2^?n ) (0≦k<2 ⁿ ). Замыкание множества двоичных с тупенчатых функций по норме      

Uniform Schauder estimates for regularized hypoelliptic equations

In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: in , where Δ is the Laplace operator, m < n, and the limit operator is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter ϵ, of solution of the approximated equation L _ϵ u = f, using a modification of the lifting technique of Rothschild and Stein. These estimates can be used in particular while studying regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.      

Isomorphic Schauder decompositions in certain Banach spaces

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ?_ψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.     

Weighted Schauder estimates for Evolution Stokes Problem

Abstract We prove the solvability of the evolution Stokes problem in bounded or exterior domain Ω∈ Rⁿ under the assumption that the initial value of the velocity vector field v₀ belongs to C^s(Ω), (in particular, it can be only continuous). The solution is obtained in some weighted H?lder spaces. This result makes it possible to prove the local solvability of a nonlinear problem under the same assumption concerning v₀. Keywords: Stokes equations, Weighted norms Mathematics Subject Classification (2000): 35Q30, 76D03     

Partial Schauder estimates for second-order elliptic and parabolic equations

We establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving H?lder semi-norms not with respect to all, but only with respect to some of the independent variables.     

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by