We study the analytic and Gevrey regularity for a “sum of squares” operator closely connected to the Schrödinger equation with minimal coupling. We however assume that the (magnetic) vector potential has some degree of homogeneity and that the Hörmander bracket condition is satisfied. It is shown that the local analytic/Gevrey regularity of the solution is related to the multiplicities of the zeroes of the Lie bracket of the vector fields.

We prove a minimal Gevrey regularity theorem for Hormander's sum of squares type operators (), improving the result of Derridj and Zuily []. The Gevrey index given here is optimal, in the sense that there are operators of this type that just attain that regularity and not any better.

We study the real analytic and Gevrey regularity of the solutions to a type of “sum of squares” model operator, see (1), in two variables and obtain a result in agreement with Treves conjecture.

We prove a Gevrey regularity theorem for the sum of squares of two vector fields in two variables with the assumption that the H & ouml;rmander bracket hypothesis is satisfied at length three. In Appendix A, a procedure is discussed for what attains to the possible optimality of the obtained Gevrey order.

We consider the operator in (1.1) and prove that it is analytic hypoelliptic. This operator is linked to a stationary Schrödinger equation with a magnetic field and an anharmonic type potential. It is also a sum of squares of vector fields exhibiting a symplectic characteristic variety. This aspect is discussed in the introduction.

For $ q $, $ a $ integers such that $ a \geq 1 $, $ 1 < q $, $ (x, y) \in U $, $ U $ a neighborhood of the origin in $ \mathbb{R}^{2} $, we consider the operator $$ D_{x}^{2} + x^{2(q-1)} D_{y}^{2} + y^{2a} D_{y}^{2} . $$ Slightly modifying the method of proof of \cite{monom} we can see that it is Gevrey $ s_{0} $ hypoelliptic, where $ s_{0}^{-1} = 1 - a^{-1} (q - 1) q^{-1} $. Here we show that this value is optimal, i.e. that there are solutions to $ P u = f $ with $ f $ more regular than $ G^{s_{0}} $ that are not better than Gevrey $ s_{0} $. The above operator reduces to the M\'etivier operator (\cite{metivier81}) when $ a = 1 $, $ q = 2 $. We give a description of the characteristic manifold of the operator and of its relation with the Treves conjecture on the real analytic regularity for sums of squares.

We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hörmander condition.

Analytic or Gevrey hypoellipticity is proved for a class of sums of squares of vector fields having a symplectic characteristic manifold of dimension 2 and arbitrary (even) codimension. We note that this class contains examples for which the Treves stratification seems to work as well as examples for which the Treves stratification does not identify properly the non symplectic stratum.

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

Objective: To determine the cumulative 30-day and 1-year mortality as well as personal independence after hip fracture in patients on hemodialysis. Design: Prospective, observational cohort study with matched controls. Setting: One teaching hospital, one metropolitan trauma center, one peripheral hospital. Patients and Intervention: Study group: a consecutive cohort of 64 patients with end-stage renal disease receiving chronic hemodialysis who had undergone surgery for a trochanteric or femoral neck fracture from June 2008 to November 2016. Control group: subjects without end-stage renal disease who underwent surgery for similar hip fractures. Main Outcome Measure: One-year mortality, activities of daily living, and ambulatory activity. Results: The 30-day and 1-year mortality rate in patients with a hip fracture undergoing hemodialysis was 25.0% and 57.8%, respectively. Hemodialysis was independently associated with increased 30-day (Hazard ratio 2.933; 95% confidence interval 1.270-6.770; P = 0.018) and 1-year (hazard ratio 2.535; 95% confidence interval, 1.494-4.299; P < 0.001) mortality compared with the matched controls. At the 1-year follow-up, loss of personal independence in comparison with the prefracture status was detected. Conclusions: Hemodialysis was associated with increased mortality after hip fracture. A worse prefracture functional status predicted the loss of functional independence at follow-up.

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying Hörmander’s condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity.

We consider a class of operators of the type sum of squares of real analytic vector fields satisfying the Hörmander bracket condition. The Poisson-Treves stratification is associated to the vector fields. We show that if the deepest stratum in the stratification, i.e., the stratum associated to the longest commutators, is symplectic, then the Gevrey regularity of the solution is better than the minimal Gevrey regularity given by the Derridj-Zuily theorem.

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. The Treves conjecture defines a stratification and states that an operator of this type is analytic hypoelliptic if and only if all the strata in the stratification are symplectic manifolds.Albano, Bove, and Mughetti (2016) produced an example where the operator has a single symplectic stratum, according to the conjecture, but is not analytic hypoelliptic.If the characteristic manifold has codimension 2 and if it consists of a single symplectic stratum, defined again according to the conjecture, it has been shown that the operator is analytic hypoelliptic.We show here that the above assertion is true only if the stratum is single, by producing an example with two symplectic strata which is not analytic hypoelliptic.

We consider an operator being a sum of squares of vector fields. It has the form, p, r is an element of N,P(x, D-x, D-y, D-t) = D-x(2) + x(2(p-1)) (D-y - x(r)D(t))(2).This type of operator is C-infinity hypoelliptic by Hormander's theorem, [18]. Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson-Treves stratification of P includes both symplectic and non-symplectic strata.In this paper we show that P is Gevrey (p+r)/p hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal. (C) 2016 Elsevier Inc. All rights reserved.

We study a class of third order hyperbolic operators $P$ in $G = \Omega \cap \{0 \leq t \leq T\},\: \Omega \subset \R^{n+1}$ with triple characteristics on $t = 0$. We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$.

Analytic and Gevrey hypo-ellipticity are studied for operators of the formP(x,y,D-x,D-y) = D-x(2) + Sigma(j=1)(P-j(X, Y)D-y)(2),in R-2. We assume that the vector fields D-x and p(j)(x,y)D-y satisfy Hormander's condition, that is, that they as well as their Poisson brackets generate a two-dimensional vector space. It is also assumed that the polynomials p(j) are quasi-homogeneous of degree m(j), that is, that p(j) (lambda x, lambda(theta)y) = lambda(mj) p(j) (x, y), for every positive number lambda. We prove that if the associated Poisson-Treves stratification is not symplectic, then P is Gevrey S hypo-elliptic for an s which can be explicitly computed. On the other hand, if the stratification is symplectic, then P is analytic hypo-elliptic.

We study a class of third-order effectively hyperbolic operators P in G = {(t, x): 0 <= t <= T,x epsilon U subset of R-n} with triple characteristics at p = (0, x(0),xi),xi epsilon R-n\ (0). V. Ivrii introduced the conjecture that every effectively hyperbolic operator is strongly hyperbolic, that is the Cauchy problem for P + Q is locally well posed for any lower-order terms Q. For operators with triple characteristics, this conjecture was established [3] in the case when the principal symbol of P admits a factorization as a product of two symbols of principal type. A strongly hyperbolic operator in G could have triple characteristics in G only for t = 0 or for t = T. The operators that we investigate have a principal symbol which in general is not factorizable and we prove that these operators are strongly hyperbolic if T is small enough. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.

In this paper we consider sums of squares of vector fields in R 2 \mathbb {R}^2 satisfying Hörmander’s condition and with polynomial, but non-(quasi-)homoge- neous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is \[ P = X 2 + Y 2 + ∑ j = 1 L Z j 2 , P=X^2+Y^2+\sum _{j=1}^{L}Z_j^2, \] with \[ X = D x , Y = a 0 ( x , y ) x q − 1 D y , Z j = a j ( x , y ) x p j − 1 y k j D y , X=D_x, \quad Y= a_{0}(x, y) x^{q-1}{D_y}, \quad Z_j= a_{j}(x, y) x^{p_j-1}y^{k_j}\,D_y, \] with a j ( 0 , 0 ) ≠ 0 a_{j}(0, 0) \neq 0 , j = 0 , 1 , … , L j = 0, 1, \ldots , L and q > p j , { k j } q>p_j, \{k_j\} arbitrary. The theorem we prove is that P P is Gevrey-s hypoelliptic for s ≥ 1 1 − T , T = max j q − p j q k j . s\geq \frac {1}{1-T}, T = \max _j \frac {q-p_j}{q k_j}.

We study the hypoellipticity of (pseudo)differential operators in one variable when a positivity assumption is made on a suitable principal part. We prove that the found Gevrey regularity is optimal.