Local solvability for a class of evolution equations

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L² and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg-Treves condition (ψ) which is suitable to our study.     

Removable sets for homogeneous linear partial differential equations in Carnot groups

Let L be a homogeneous left-invariant differential operator on a Carnot group. Assume that both L and L^t are hypoelliptic. We study the removable sets for L-solutions. We give precise conditions in terms of the Carnot- Caratheodory Hausdorff dimension for the removability for L-solutions under several auxiliary integrability or regularity hypotheses. In some cases, our criteria are sharp on the level of the relevant Hausdorff measure. One of the main ingredients in our proof is the use of novel local self-similar tilings in Carnot groups.     

On the nodal sets of toral eigenfunctions

We study the nodal sets of eigenfunctions of the Laplacian on the standard d-dimensional flat torus. The question we address is: Can a fixed hypersurface lie on the nodal sets of eigenfunctions with arbitrarily large eigenvalue? In dimension two, we show that this happens only for segments of closed geodesics. In higher dimensions, certain cylindrical sets do lie on nodal sets corresponding to arbitrarily large eigenvalues. Our main result is that this cannot happen for hypersurfaces with nonzero Gauss-Kronecker curvature. In dimension two, the result follows from a uniform lower bound for the L ²-norm of the restriction of eigenfunctions to the curve, proved in an earlier paper (Bourgain and Rudnick in C. R. Math. 347(21?C22):1249?C1253, 2009). In high dimensions we currently do not have this bound. Instead, we make use of the real-analytic nature of the flat torus to study variations on this bound for restrictions of eigenfunctions to suitable submanifolds in the complex domain. In all of our results, we need an arithmetic ingredient concerning the cluster structure of lattice points on the sphere. We also present an independent proof for the two-dimensional case relying on the ??abc-theorem?? in function fields.     

Abelian actions with globally hypoelliptic leafwise Laplacian and rigidity

In this paper, we prove several results concerning smooth R^k actions on a smooth compact manifold with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and their cohomology is often finite dimensional, even trivial. Further, we consider a class of examples of R² actions on two-step nilmanifolds that have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.     

Randverteilungen von Nullösungen hypoelliptischer Differentialgleichungen

In this paper a distributional boundary value is defined for solutions f (defined on ?ⁿ⁺¹\?ⁿ) of a partially hypoelliptic differential operator (on ?ⁿ⁺¹)with constant coefficients. Then the following is equivalent:

1. f has a distributional boundary value.
2. f can be continued to ?ⁿ⁺¹ as a distribution. For hypoelliptic operators this is equivalent to:
3. f ist a locally slowly growing function. A topology is given on this function space, that makes the boundary value mapping a topological homomorphism.

     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

On the left side derivative of the Hausdorff dimension of the Julia sets for z 2 + c at c = −3/4

Let d(c) denote the Hausdorff dimension of the Julia set J _c of the polynomial f _c (z) = z ² + c. The function c ? d(c) is real-analytic on the interval (?5/4,?3/4), which is included in the 1/2 bulb of the Mandelbrot set. The number c = ?3/4 is the parameter at which, going from the right, the attracting fixed point bifurcates to an orbit of period two. Recently [13], we studied d′(c) when c ? ?3/4 from the right. Here, under numerically verified assumption d(?3/4) < 4/3, we will show that there exists K _?3 ^4/? > 0 such that d′(c)(?3/4 ? c)^{?3d(?3/4)/2+2} → ?K _?3 ^4/? , when c tends to ?3/4 from the left. In particular we obtain d′(c) ? ?∞. This case is much harder than that considered in [13].     

On cohomology of finitely generated function groups

Let Γ be a non-elementary finitely generated Kleinian group with region of discontinuity Ω. Letq be an integer,q ≥ 2. The group Λ acts on the right on the vector space Π_2q?2 of polynomials of degree less than or equal to 2q ? 2 via Eichler action. We note by A_qq(Ω, Λ) the space of cusp forms for Λ of weight (?2q) and the space of parabolic cohomology classes by PH¹ (Λ, Π2q?2). Bers introduced an anti-linear map $$\beta _q^* :A^q \left( {\Omega ,\Gamma } \right) - - - \to PH^1 \left( {\Gamma ,\Omega _{2q - 2} } \right)$$ . We try to determine the class of Kleinian groups for which the Bers map is surjective. We show that the Bers map is surjective for geometrically finite function groups. We also obtain a characterization of geometrically finite function groups. As an application, we reprove theorems of Maskit on inequalities involving the dimension of the space of cusp forms supported on an invariant component and the dimension of the space of cusp forms supported on the other components for finitely generated function groups. We also show all these inequalities are equalities for geometrically finiteB-groups.     

Jacobi Maaß forms

In this paper, we give a new definition for the space of non-holomorphic Jacobi Maaß forms (denoted by J _k,m ^nh ) of weight k∈? and index m∈? as eigenfunctions of a degree three differential operator \(\mathcal{C}^{k,m}\). We show that the three main examples of Jacobi forms known in the literature: holomorphic, skew-holomorphic and real-analytic Eisenstein series, are contained in J _k,m ^nh . We construct new examples of cuspidal Jacobi Maaß forms F _f of weight k∈2? and index 1 from weight k?1/2 Maaß forms f with respect to Γ₀(4) and show that the map f ? F _f is Hecke equivariant. We also show that the above map is compatible with the well-known representation theory of the Jacobi group. In addition, we show that all of J _k,m ^nh can be “essentially” obtained from scalar or vector valued half integer weight Maaß forms.     

On Analytic Interpolation Manifolds in Boundaries of Weakly Pseudoconvex Domains

Let Ω be a bounded, weakly pseudoconvex domain in C ⁿ , n ≤ 2, with real-analytic boundary. A real-analytic submanifold M ? ?Ω is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to (Ω¯). We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of ?Ω.     

Global Morrey estimates for a class of Ornstein-Uhlenbeck operators

In this paper, we consider a class of hypoelliptic Ornstein-Uhlenbeck operators in ? ^N given by $\mathcal{A} = \sum\limits_{i,j = 1}^{p_0 } {a_{ij} \partial _{x_i x_j }^2 + } \sum\limits_{i,j = 1}^N {b_{ij} x_i \partial _x } ,$ where (a _ij ), (b _ij ) are N × N constant matrices, and (a _ij ) is symmetric and positive semidefinite. We deduce global Morrey estimates forA from similar estimates of its evolution operator L on a strip domain S = ? ^N × [?1, 1].     

Smoothness of solutions of almost hypoelliptic equations

The paper studies a class of almost hypoelliptic equations P(D)U = ? in a strip. It is proved that for \(\mathcal{H}\) great enough and for δ > 0 small enough all solutions of this equation, which are square summable with the weight e ^?δ|x| and for which \(D_2^{\alpha _2 } U\), where α ₂ = 0, …, \(ord_{\alpha _2 } P\), are infinitely differentiable in x ₁ functions, provided D ₁ ^j ? ∈ L ₂(\(\Omega _\mathcal{H} \)) for any j.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

On Formal Maps Between Generic Submanifolds in Complex Space

Let H:(M,p)→(M ^′,p ^′) be a formal mapping between two germs of real-analytic generic submanifolds in ? ^N with nonvanishing Jacobian. Assuming M to be minimal at p and M ^′ holomorphically nondegenerate at p ^′, we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M ^′ are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank.     

Cauchy-Riemann Automorphism Groups that cannot be projectively realized

LetM ? ?ⁿ be a real-analytic, nonspherical hypersurface passing through the origin and having nondegenerate Levi form. Let Aut₀ M be the stability group of 0. Whenn = 12 an example is constructed for which Aut₀ M cannot be linearized.     

New extension phenomena for solutions of tangential Cauchy–Riemann equations

In our recent work, we showed that C^∞ CR-diffeomorphisms of real-analytic Levi-nonflat hypersurfaces in ?² are not analytic in general. This result raised again the question on the nature of CR-maps of a real-analytic hypersurfaces.

In this paper, we give a complete picture of what CR-maps actually are. First, we discover an analytic continuation phenomenon for CR-diffeomorphisms which we call the sectorial analyticity property. It appears to be the optimal regularity property for CR-diffeomorphisms in general. We emphasize that such type of extension never appeared previously in the literature. Second, we introduce the class of Fuchsian type hypersurfaces and prove that (infinitesimal generators of) CR-automorphisms of a Fuchsian type hypersurface are still analytic. In particular, this solves a problem formulated earlier by Shafikov and the first author.

Finally, we prove a regularity result for formal CR-automorphisms of Fuchsian type hypersurfaces.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

A note on the cubical dimension of new classes of binary trees

The cubical dimension of a graph G is the smallest dimension of a hypercube into which G is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with 2 ⁿ vertices, n ? 1, is n. The 2-rooted complete binary tree of depth n is obtained from two copies of the complete binary tree of depth n by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.     

Pfister forms and their applications

This note investigates the properties of the 2ⁿ-dimensional quadratic forms ?_i=2ⁿ 〈1, a_i〉, called n-fold Pfister forms. Utilizing these properties, various applications are made to k-theory of fields, and field invariants. In particular, the set of orderings of a field and the maximum dimension of anisotropic forms with everywhere zero signature are investigated. Details will appear elsewhere.