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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.

We introduce the notion of an almost conformal vector field, which generalizes conformal vector fields and recently introduced m-modified conformal vector fields on a Riemannian manifold. The definition of an almost conformal vector field ζ on an n-dimensional Riemannian manifold (N,g) requires two smooth functions σ and f called the potential and copotential and a skew symmetric tensor φ called the associated tensor of ζ. Many examples of almost conformal vector fields which are not conformal vector fields are provided. We find conditions using σ, f and φ under which an almost conformal vector field ζ on an n-dimensional compact Riemannian manifold (N,g) is either conformal or a Killing vector field. We also find conditions under which a compact Riemannian manifold (N,g) admitting an almost conformal vector field is isometric to the sphere S[sup.n](c). Finally, we find conditions under which an almost conformal vector field ζ on a noncompact Riemannian manifold (N,g) 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.

The purpose of the present paper is to study Ricci solitons on Sasaki–Kenmotsu manifolds. It is shown that if the characteristic vector fields and are recurrent torse-forming vector fields on the Sasaki–Kenmotsu metric as a Ricci soliton, then both and are concurrent and Killing vector fields. We classify and characterize a Sasaki–Kenmotsu manifold admitting holomorphically planar conformal vector field. Also, we prove that an field on a Sasaki–Kenmotsu manifold is solenoidal. Moreover, in a Sasaki–Kenmotsu manifold admitting field with , the field is the eigen vector of the Ricci operator with eigen value

We consider piecewise smooth vector fields (PSVF) defined in open sets M⊆Rn with switching manifold being a smooth surface Σ . We assume that M\\Σ contains exactly two connected regions, namely Σ+ and Σ- . Then, the PSVF are given by pairs X=(X+,X-) , with X=X+ in Σ+ and X=X- in Σ-. A regularization of X is a 1-parameter family of smooth vector fields Xε , ε>0, satisfying that Xε converges pointwise to X on M\\Σ , when ε→0 . Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization Xε . While the linear regularization requires that for every ε>0 the regularized field Xε is in the convex combination of X+ and X- , the nonlinear regularization requires only that Xε is in a continuous combination of X+ and X- . We prove that, for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R2 and in R3 .

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 .