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New presheaves of hyperfunction spaces with the growth estimates with respect to x → ∞ and y → 0 in a cone Γ are introduced. Then it is shown that the Laplace transform is a bijective mapping of the space of tempered ultradistributions on Rn of non-quasianalytic class onto the corresponding hyperfunction space of sections over Dn, the compactification of Rn. Microlocalization of tempered ultradistributions at (x0∞, ξ0) is introduced as well as a new microlocalization within some classes of hyperfunctions. [ABSTRACT FROM AUTHOR]

Let M(ω)p,q be the modulation space with parameters p, q and weight function ω. We prove that if p, q, p1, p2, q1, q2 ∈ [1,∞], ω1, ω2, ω and h1, h2 are appropriate, and a ∈ M(ω)p,q, then the Toeplitz operator $$ Tp_{h_1 ,h_2 } (a):M_{(\omega _1 )}^{p_1 ,q_1 } \to M_{(\omega _2 )}^{p_2 ,q_2 } $$ is continuous. If in addition p1 = p2 = q1 = q2 = 2, then we present sufficient conditions on p, q, h1 and h2 in order for $$ Tp_{h_1 ,h_2 } (a) $$ should be a Schatten-von Neumann operator of certain degree. [ABSTRACT FROM AUTHOR]

We present new results concerning pseudo-differential operators in the function spaces Sμμ(ℝn) of Gelfand and Shilov. In particular we discuss Sνμ(ℝn)-regularity of solutions to SG-elliptic pseudo-differential equations, allowing lower order semilinear perturbations. The results apply to SG-elliptic partial differential equations with polynomial coefficients. We also study the action of Weyl operators and localization operators on Sνμ(ℝn). [ABSTRACT FROM AUTHOR]

We provide examples of the product of two localization operators. As a special case, we study the composition of Gabor multipliers. The results highlight the instability of this product and underline the necessity of expressing it in terms of asymptotic expansions. [ABSTRACT FROM AUTHOR]

A large class of time-varying filters can be described via pseudo-differential operators belonging to the Hörmander class OPS00, 0. The questions whether and how an input signal can be reconstructed from a known output lead to the problems of invertibility of pseudo-differential operators in that class and of (at least, numerical) solution of pseudo-differential equations. We are going to derive effective conditions for the invertibility for pseudo-differential operators with globally slowly varying symbols as well as for causal pseudo-differential operators, and we study the stability of the finite sections method with respect to time and frequency for these operators. [ABSTRACT FROM AUTHOR]

Signals in real applications are typically finite in duration, dynamic and non-stationary processes with frequency characteristics varying over time. This often requires techniques capable of locally analyzing and processing signals. An integral transform known as the Stockwell transform is a combination of the classic Gabor transform and the current and versatile wavelet transform. It allows more accurate detection of subtle changes and easy interpretation in the time-frequency domain. In this paper, we study the mathematical underpinnings of the Stockwell transform. We look at the Stockwell transform as a stack of simple pseudo-differential operators parameterized by frequencies and give a complete description of the Stockwell spectra. [ABSTRACT FROM AUTHOR]

Motivated by problems in signal analysis, we define a class of time-frequency representations which is based on the short-time Fourier transform and depends on two fixed windows. We show that this class can be viewed as a link between the classical Rihaczek representation and the spectrogram. Correspondingly we formulate for this class a suitable general form of the uncertainty principle which have, as limit case, the uncertainty principles for the Rihaczek representation and for the spectrogram. We finally consider the questions of marginal distributions. We compute them in terms of convolutions with the windows and prove simple conditions for which average and standard deviation of the distributions in our class coincide with that of their marginals. [ABSTRACT FROM AUTHOR]

Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem we prove the boundedness and study the compactness of pseudo-differential operators a(x,D) with bounded measurable V (ℝR)-valued symbols a(x, ·) on the Lebesgue spaces Lp(ℝ) with 1 < p < δ, where V (ℝ) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation allows us to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Three different Banach algebras of pseudo-differential operators with discontinuous symbols acting on the spaces Lp(ℝ) are studied. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators in the first algebra we also obtain an index formula. An application to the Haseman boundary value problem is given. [ABSTRACT FROM AUTHOR]

Let M(ω)p,q be the modulation space with parameters p, q and weight function ω. We prove that if t ∈ R, p, pj, q, qj ∈ [1, ∞], ω1, ω2 and ω are appropriate, and a ∈ M(ω)p,q, then the pseudo-differential operator at(x,D) is continuous from M(ω)p1,q1 to M(ω)p2,q2. If in addition pj = qj = 2, then we establish necessary and sufficient conditions on p and q in order to at(x,D) should be a Schatten-von Neumann operator of certain order. [ABSTRACT FROM AUTHOR]

In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type. [ABSTRACT FROM AUTHOR]

For the differential operator P of order m and the inhomogeneous data f ∈ S′ on Rn, we say that the Lp-extension of the solution holds if u ∈ Lp, m ≤ n(1 − 1/p), and Pu = f on Rn / 0 imply Pu = f on Rn. In this article, we discuss which kind of inhomogeneous terms f ∈ S′ admit the Lp-extension of the solution. In previous works, this problem was studied by using classical Bochner's method ([1]) or a new method developed by the author and Uchida ([5], [4]). We consider inhomogeneous terms which are not covered by these results. [ABSTRACT FROM AUTHOR]

In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces Gs for small s, and local nonsolvability in Gs for large s. [ABSTRACT FROM AUTHOR]

We study the exponential decay of solutions to differential equations of hypoelliptic type (see Definition 2.1). In particular we find sufficient conditions on the differential operator A in order for estimates of the kind $$ e^{\varepsilon \left\langle x \right\rangle ^r } Au \in V \Rightarrow e^{\varepsilon \left\langle x \right\rangle ^r } u \in V $$ to hold, for several types of functions and distribution spaces V. For example, V may be . [ABSTRACT FROM AUTHOR]

We construct the wave kernels of the non-isotropic twisted Laplacian by means of its heat kernel. We then express the wave kernels at the origin in terms of complex Fourier integrals and we exploit the connections of the phase functions with the complex integrals. [ABSTRACT FROM AUTHOR]

Basic properties of Fourier integral operators on the torus $$ \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n $$ are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

The Plancherel formula and the inversion formula for Weyl transforms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels. [ABSTRACT FROM AUTHOR]

The method of construction of the fundamental solution for heat equations using pseudo-differential operators with parameter time variable is discussed, which is applicable to calculate traces of operators. This gives extensions of both the Gauss-Bonnet-Chern Theorem and the Riemann-Roch Theorem. [ABSTRACT FROM AUTHOR]

We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays. [ABSTRACT FROM AUTHOR]

We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantization of edge-degenerate symbols which lead to continuous operators in weighted edge spaces. [ABSTRACT FROM AUTHOR]