Spaces Of Pseudodifferential Operators Research Articles

Defect Distributions Related to Weakly Convergent Sequences in Bessel-Type Spaces $${H_{\Lambda }^{-s,p}}$$ H Λ - s , p

We consider microlocal defect distributions associated to a weakly convergent sequences $$u_n$$ in $$H^{-s,p}_{\Lambda }$$ and $$v_n$$ in $$H^{s+m,q}_{\Lambda }$$ through the space of pseudo-differential operators with the symbols in $$(s^{m,N+1}_\Lambda )_0$$ . Symbols correspond to a weight function $$\Lambda $$ determining a quasi-elliptic symbol. Results are applied to partial differential equations with symbols related to weights of the type $$\Lambda $$ .

Deforming 𝔥-trivial the Lie algebra Vect(S1) inside the Lie algebra of pseudodifferential operators Ψ𝒟𝒪

In this paper, we consider the action of Vect(S1) by Lie derivative on the spaces of pseudodifferential operators [Formula: see text]. We study the [Formula: see text]-trivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle [Formula: see text]. We classify the deformations of this action that become trivial once restricted to [Formula: see text], where [Formula: see text] or [Formula: see text]. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

Equivalence classes of subquotients of pseudodifferential operator modules

Consider the spaces of pseudodifferential operators between tensor density modules over the line as modules of the Lie algebra of vector fields on the line. We compute the equivalence classes of various subquotients of these modules. There is a 2-parameter family of subquotients with any given Jordan-Hölder composition series. In the critical case of subquotients of length 5, the equivalence classes within each non-resonant 2-parameter family are specified by the intersections of a pencil of conics with a pencil of cubics. In the case of resonant subquotients of length 4 with self-dual composition series, as well as in those of lacunary subquotients of lengths 3 and 4, equivalence is specified by a single pencil of conics. Non-resonant subquotients of length exceeding 7 admit no non-obvious equivalences. The cases of lengths 6 and 7 are unresolved.

Perturbation of sectorial projections of elliptic pseudo-differential operators

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderón projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley’s original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.

Lie Algebras of Formal Power Series

Pseudodifferential operators are formal Laurent series in the formal inverse -1 of the derivative operator whose coefficients are holomorphic functions. Given a pseudodifferential operator, the corresponding formal power series can be ob tained by using some constant multiples of its coefficients. The space of pseu dodifferential operators is a noncommutative algebra over C and therefore has a natural structure of a Lie algebra. We determine the corresponding Lie algebra structure on the space of formal power series and study some of its properties. We also discuss these results in connection with automorphic pseudodifferen tial operators, Jacobi-like forms, and modular forms for a discrete subgroup of SL(2, R).

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E → M . We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin–Kravchenko–Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M = S 1 , we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S 1 -pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111–127]. When dim ( M ) > 1 , we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.

Fractional analytic index

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators `acting on sections of the projective bundle' in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e. the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of K-theory twisted by w; hence the index is a rational number but in general it is not an integer.

Rankin–Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL ( 2 , R ) correspond to certain sequences of modular forms for Γ. Rankin–Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin–Cohen brackets. We also discuss generalized Rankin–Cohen brackets on modular forms and use these to construct certain types of modular forms.