A Parameter-Dependent Symbolic Calculus for Pseudo-Differential Operators With Negative-Definite Symbols

Jacob, Niels; Tokarev, A. G.

doi:10.1112/s0024610704005769

Cited by 4 publications

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that there exist other calculi for pseudodifferential operators with nonsmooth symbols, which were elaborated by J. Marschall [35][36][37], W. Hoh [21,22], and by N. Jacob and A. G. Tokarev [25].…”

Section: Introductionmentioning

confidence: 99%

“…It is based on the Lévi-Khinchin representation for a continuous negative-definite function (see, for example, [21]), which gives a decomposition of symbols into a differentiable part and a remainder part considered as a low order perturbation. In [25] Hoh's calculus for pseudodifferential operators with negative-definite symbols extends to the case of parameter-dependent symbols. This parameter-dependent calculus is applied in [25] to studying subordinate sub-Markovian semigroups.…”

Section: Introductionmentioning

confidence: 99%

“…In [25] Hoh's calculus for pseudodifferential operators with negative-definite symbols extends to the case of parameter-dependent symbols. This parameter-dependent calculus is applied in [25] to studying subordinate sub-Markovian semigroups.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Algebra of Pseudodifferential Operators With Slowly Oscillating Symbols

Karlovich¹

2006

Proc. Lond. Math. Soc.

View full text Add to dashboard Cite

Let $V(\mathbb{R})$ denote the Banach algebra of absolutely continuous functions of bounded total variation on $\mathbb{R}$, and let $\mathcal{B}_p$ be the Banach algebra of bounded linear operators acting on the Lebesgue space $L^p(\mathbb{R})$ for $1 < p < \infty$. We study the Banach algebra $\mathfrak{A}\subset\mathcal{B}_p$ generated by the pseudodifferential operators of zero order with slowly oscillating $V(\mathbb{R})$-valued symbols on $\mathbb{R}$. Boundedness and compactness conditions for pseudodifferential operators with symbols in $L^\infty (\mathbb{R}, V(\mathbb{R}))$ are obtained. A symbol calculus for the non-closed algebra of pseudodifferential operators with slowly oscillating $V(\mathbb{R})$-valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra $\mathfrak{A}^\pi = {\mathfrak A} / \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators in $\mathcal{B}_p$, is commutative and involutive. An isomorphism between the quotient Banach algebra $\mathfrak{A}^\pi$ of pseudodifferential operators and the Banach algebra $\widehat{\mathfrak{A}}$ of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators $A \in \mathfrak{A}$ are obtained in terms of their Fredholm symbols.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Algebra of Pseudodifferential Operators With Slowly Oscillating Symbols

Karlovich¹

2006

Proc. Lond. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Pseudodifferential operators with non-regular symbols are intensively studied nowadays in view of extension of their applications (see, e.g., [10,12,13,[45][46][47] and the references therein). Recently several calculi for pseudodifferential operators with non-smooth symbols were elaborated in [22,23,26,[28][29][30][31][39][40][41]. Applications of pseudodifferential operators to the theory of singular integral operators with non-regular data are given, e.g., in [2][3][4][5]29,34,43].…”

Section: Introductionmentioning

confidence: 99%

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Karlovich

2012

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S * related to the Carleson-Hunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in L ∞ (R, V (R)), P C(R, V (R)) and Λγ(R, V d (R)) on the weighted Lebesgue spaces L p (R, w), with 1 < p < ∞ and w ∈ Ap(R). The Banach algebras L ∞ (R, V (R)) and P C(R, V (R)) consist, respectively, of all bounded measurable or piecewise continuous V (R)-valued functions on R where V (R) is the Banach algebra of all functions on R of bounded total variation, and the Banach algebra Λγ(R, V d (R)) consists of all Lipschitz V d (R)-valued functions of exponent γ ∈ (0, 1] on R where V d (R) is the Banach algebra of all functions on R of bounded variation on dyadic shells. Finally, for the Banach algebra Ap,w generated by all pseudodifferential operators a(x, D) with symbols a(x, λ) ∈ P C(R, V (R)) on the space L p (R, w), we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators A ∈ Ap,w. Mathematics Subject Classification (2010). Primary 47G30;Secondary 42B20, 42B25, 47A30, 47G10.

show abstract

Algebras of Pseudo-differential Operators with Discontinuous Symbols

Karlovich

Modern Trends in Pseudo-Differential Operators

View full text Add to dashboard Cite

A Parameter-Dependent Symbolic Calculus for Pseudo-Differential Operators With Negative-Definite Symbols

Abstract: Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.

Cited by 4 publications

References 12 publications

An Algebra of Pseudodifferential Operators With Slowly Oscillating Symbols

An Algebra of Pseudodifferential Operators With Slowly Oscillating Symbols

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Algebras of Pseudo-differential Operators with Discontinuous Symbols

Contact Info

Product

Resources

About