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In this work, the quadratic-phase Fourier transform (QPFT) on Schwartz space is defined and its necessary results are derived, including adjoint formula, Parseval identity, and continuity property. Furthermore, the continuity property on tempered distribution space is discussed, and also an example of the QPFT is provided. The present analysis is applied to develop a new class of pseudo-differential operators and prove an important theorem on Schwartz space. The boundary value problems of generalized partial differential equations (i.e. generalized telegraph, wave, and heat equations) are discussed using QPFT, and graphical plots are provided. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator involving the Kontorovich–Lebedev transform is defined. The boundedness of pseudo differential operator in a certain Sobolev-type space is shown. Composition of pseudo-differential operatorsis defined in terms of the two symbols, and also its boundedness in certain Sobolev-type space is given. Some special cases are also discussed. Finally, the Kontorovich–Lebedev transform is used in the solution of some partial differential equations. [ABSTRACT FROM PUBLISHER]

The goal of the present paper is to derive a simultaneous description of the decay and the regularity properties for elliptic equations in n with coefficients admitting irregular decay at infinity of the type O(|x|-σ), σ > 0, filling the gap between the case of Cordes globally elliptic operators and the case of regular/Fuchs behavior at infinity. Representative examples in n are the equations [image omitted] where 0 < σ <2, 〈x〉 = (1 + |x|2)1/2, ω(x) a bounded smooth function, f given and F[u] a polynomial in u, and similar Schrodinger equations at the endpoint of the spectrum. Other relevant examples are given by linear and nonlinear ordinary differential equations with irregular type of singularity for x → ∞, admitting solutions y(x) with holomorphic extension in a strip and sub-exponential decay of type |y(x)| ≤Ce-ε|x|r; 0 < r < 1. Sobolev estimates for the linear case are proved in the frame of a suitable pseudodifferential calculus; decay and uniform holomorphic extensions are then obtained in terms of Gelfand-Shilov spaces by an inductive technique. The same technique allows to extend the results to the semilinear case. [ABSTRACT FROM AUTHOR]

On the torus, pseudo-differential operators can be presented globally by Fourier series, without local coordinate charts. Periodization of partial differential equations leads to investigating the correspondence between toroidal and Euclidean symbols of operators. Fourier series operators—the analogues of Fourier integral operators—are introduced, and formulae for their compositions with pseudo-differential operators are presented. The L2 boundedness of pseudo-differential and Fourier series operators follows from certain conditions on phases and amplitudes. [ABSTRACT FROM AUTHOR]

The authors consider pseudodifferential operators whose symbols have decay at infinity of quasi-homogeneous type and study their behaviour on the wave front set of distributions in weighted Zygmund-Holder spaces and weighted Sobolev spaces in Lp-framework. Then microlocal properties for solutions to linear partial differential equations with coefficients in weighted Zygmund-Holder spaces are obtained. [ABSTRACT FROM AUTHOR]

In asymptotic expansions of resolvent traces Tr( A ( P - λ) -1 ) for classical pseudodifferential operators on closed manifolds, the coefficient C 0 ( A , P ) of ( - λ) -1 is of special interest, since it is the first coefficient containing nonlocal elements from A ; moreover, it enters in index formulas. C 0 ( A , P ) also equals the zeta function value at zero when P is invertible. C 0 ( A , P ) is a trace modulo local terms, since C 0 ( A , P ) - C 0 ( A , P ′) and C 0 ([ A , A ′], P ) are local. By use of complex powers P s (or similar holomorphic families of order s ), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from A , A ′, log P and log P ′. The present paper has two purposes. One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of P as in the original proofs. We also give here a simple direct proof of a recent residue formula of Scott for C 0 ( I , P ). The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems. [ABSTRACT FROM AUTHOR]

In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a 1 ( x , D x ) a 2 ( x , D x ) coincides with a Green operator a ( x , D x ) up to order m 1 + m 2 - Θ, where Θ &Element; (0, τ 2 ) is arbitrary, a j ( x , ξ) is in C τ j (&reals; n ) w.r.t. x , and m j is the order of a j ( x , D x ), j = 1, 2. Moreover, a ( x , D x ) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m 1 + m 2 - Θ. This result is used to construct a parametrix of a uniformly elliptic Green operator a ( x , D x ). [ABSTRACT FROM AUTHOR]

We give here a class of counterexamples to the Fefferman–Phong inequality for systems of pseudodifferential operators, which contains Brummelhuis’ one as a particular case. The main ingredient in the proof is the use of“localized operators” associated with the system, and Hörmander's example of a positive-semidefinite matrix whose Weyl quantization is not nonnegative. For the considered class, in the“isotropic” case, the Sharp Gårding inequality cannot be improved. [ABSTRACT FROM AUTHOR]

The aim of this article is to provide a new approach for the classical Fefferman–Phong inequality and some other related inequalities concerning pseudodifferential operators. The idea is to conjugate these estimates with respect to the FBI transform and obtain equivalent formulations which require only a few derivatives of the symbol. Modulo some integration by parts, these are reduced to certain decomposition results for C1, 1, respectively C3, 1 functions. As a byproduct of this method, we get to extend the inequalities to (possibly sharp) classes of symbols with limited smoothness. [ABSTRACT FROM AUTHOR]