Related papers: A remark on sums of squares of complex vector fiel…

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevi\v{c} operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey…

We argue that a necessary condition for hypoellipticity is that the polar is not a spiral domain.

We consider an operator $ P $ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $ \sigma $ is not constant on $ \Char P $.…

This is the first in a series of three papers dealing with sums of squares and hypoellipticity in the infinite regime. We give a sharp sufficient condition on a smooth nonnegative function f on n-dimensional Euclidean space so that it can…

It is shown using Schur complement techniques that on finite dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two variables with degree $(d_1,d_2)$ can be written as a finite sum of hermitian squares of…

Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is…

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata…

A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in q-alg/9609011 In this paper we give an outline of the construction of a noncommutative analogy of the algebra of partial…

We present an abstract result that characterizes the coincidence of certain classes of linear operators with the class of Cohen strongly summing linear operators. Our argument is extended to multilinear operators and, as a consequence, we…

The purpose of this paper is to construct a new class of discrete generalized Korn's inequalities for piecewise H2 vector fields in three-dimensional space. The resulting Korn's inequalities are different from the standard Korn's…

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy H\"ormander's bracket generating…

In this paper, we investigate the spectrum of the self adjoint differential operator with operator coefficitent in a separable Hilbert space. We also derive asymptotic formulas for the sum of eigenvalues of this operator.

This study is an attempt at generalizing the class of partially hypoelliptic differential operators to a class of pseudodifferential operators, Symbol ideals are formed on the set of lineality and we discuss suitable topologies that allow…

In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer-manin obstruction (see \cite{CTX}). The…

We give an index formula for the class of all *-maximally hypoelliptic differential operators on any closed manifold with vector bundle coefficients, generalising previous index formulas by Atiyah-Singer and van Erp. Using this formula, we…

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.

We consider the problem of decomposing a regular non-negative function as a sum of squares of functions which preserve some form of regularity. In the same way as decomposing non-negative polynomials as sum of squares of polynomials allows…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum…

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible…