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The authors consider operators of the form L=\\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \\mathbb{R}^{p} where X_{0},X_{1},\\ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \\gamma for L and provide growth estimates for \\gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \\gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of \\gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,\\alpha} estimates on this solution.

In this work we deal with linear second order partial differential operators of the following type:

Let

The authors consider the nonlinear equation -\\frac 1m=z+Sm with a parameter z in the complex upper half plane \\mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \\mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \\mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\\in \\mathbb H, including the vicinity of the singularities.

In this paper, we find a set of structural equations for hyperbolic Ricci solitons admitting 2-conformal vector fields, which extends similar results for Ricci solitons. As a result of these equations, we obtain an integral formula for the case when the underlying manifold is compact, indicating that a nontrivial compact hyperbolic Ricci soliton with 2-conformal potential vector field is isometric to Euclidean sphere. Also, it will be shown that such manifolds either have constant scalar curvature or their associated vector fields are conformal. Furthermore, we use the Hodge- de Rham decomposition theorem to establish a link with 2-conformal vector fields associated with a hyperbolic Ricci soliton.

It is a remarkable fact that among the known examples of quadratic semicomplete vector fields on , it is always possible to find linear coordinates where the corresponding vector field has all—or “almost all”—coefficients in the real numbers. Indeed, the coefficients are very often integral. The space of quadratic vector fields on , up to linear equivalence, is a complex 9‐dimensional family. The main result of this work establishes that the degree of freedom in determining the coefficients of a semicomplete vector field (under very mild generic assumptions) is at most 3. In other words, there are 3 parameters from which all remaining parameters are determined. Moreover, if these 3 parameters are real, then so is the vector field. We start by considering a generic quadratic vector field on that is homogeneous and is not a multiple of the radial vector field. The first step in our work will be to construct a canonical form for the induced vector field X on . This canonical form will be invariant under the action of a specific group of symmetries. When n = 3, we then push further our approach by studying the singularities not lying on the exceptional divisor but at the hyperplane at infinity Δ≅. In this setting, the dynamics of the foliation turn out to be quite simple while the singularities tend to be degenerated. The advantage is that we can deal with degenerated singularities with the technique of successive blowups. This leads to simple expressions for the eigenvalues directly in terms of the coefficients of X .

In this paper, we initiate the study of shape vector fields on the hypersurfaces of a Riemannian manifold. We use a shape vector field on a compact hypersurface of a Euclidean space to obtain a characterization of round spheres. We also find a condition, under which a shape vector field that is on a compact hypersurface of a Euclidean space is a Killing vector field.

We study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums of squares of vector fields in terms of the Poisson–Treves stratification. The FBI transform is used. We prove hypoanalyticity for several classes of sums of squares and show that our method, though not general, includes almost every known hypoanalyticity result. Examples are discussed.

In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein?s field equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consider the geometry of two classes of sequential warped product space-time models which are sequential generalized Robertson-Walker space-times and sequential standard static space-times. nema

In this paper, we study 2−conformal vector fields on Riemannian manifolds. Some relations between 2−conformal vector fields, conformal vector fields, Killing and 2−Killing vector fields have been obtained. Also, relation between monotone vector fields and 2−conformal vector fields is investigated. We give the characterization of 2−conformal vector fields on standard Euclidean spaces. Finally, examples of such vector fields on some Riemannian manifolds is presented.