Pseudo-differential operator in the framework of linear canonical transform domain

The main goal of this paper is to study properties of the linear canonical transform (LCT) on Schwartz-type space [Formula: see text]. The symbol class [Formula: see text] is introduced. The pseudo-differential operator (p.d.o.) involving LCT is defined and also some of its properties including boundedness are investigated in Sobolev-type space. Kernel and integral representation of p.d.o. are obtained. Some applications of LCT to generalized partial differential equations and canonical convolution integral equation have been solved.

Two versions of pseudo-differential operators involving the Kontorovich–Lebedev transform in L 2(ℝ+;dx/x)

The Pseudo-differential operators (p.d.o.) {L(x,A_{x})} and {\mathcal{L}(x,A_{x})} involving the Kontorovich–Lebedev transform are defined. An estimate for these operators in the Hilbert space {L^{2}(\mathbb{R}_{+};\frac{dx}{x})} is obtained. A symbol class Λ is defined and it is shown that the product of any two symbols from this class is again in Λ. At the end, commutators for the p.d.o. and their boundedness results are discussed.

Operator calculus on the class of Sato’s hyperfunctions

We construct a functional calculus for generators of analytic semigroups of operators on a Banach space. The symbol class of the calculus consists of hyperfunctions with a compact support in $[0,\infty)$. Domain of constructed calculus is dense in the Banach space.

Compactly supported analytic indices for Lie groupoids

For any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

The pseudodifferential operatorA(x,D)

The pseudodifferential operator (p.d.o.)A(x,D), associated with the Bessel operatord2/dx2+(1−4μ2)/4x2, is defined. Symbol classHρ,δmis introduced. It is shown that the p.d.o. associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian spaceHμinto itself. An integral representation of p.d.o. is obtained. Using Hankel convolutionLσ,αp-norm continuity of the p.d.o. is proved.

On a symbol class of elliptic pseudodifferential operators

We consider a class of symbols with prescribed smoothness and growth conditions and give examples of such symbols. The introduced class contains certain polynomial symbols and symbols with more, than polynomial growth in phase space. The corresponding pseudodifferential operators defined as the Weyl transforms of the symbols are elliptic. As an application, we give a result on isomorphisms between modulation spaces. In particular, we show that the Bessel potentials establish such isomorphisms.