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We consider an operator whose principal part is the mth power of a Mizohata hypoelliptic operator. We assume that the lower order term vanishes at a small rate with respect to the principal part, and we prove the local nonsolvability in Gevrey classes, for large Gevrey index.

We present a brief introduction to Hochschild and cyclic homology designed for researchers interested in pseudodifferential operators and their applications to index theory, spectral invariants, and asymptotics.

The paper deals with a local study of the Banach algebra A [SO, PC] generated by the convolution type operators W a b = a F −1bF with data a E ∈ [SO,PC] n× n and b∈ [SO p ,PC] p n× n which act on the Lebesgue space L p n (ℝ) (1 < p < ∞, n ≥ 1). Here [SO,PC] n× n means the C*-algebra generated by slowly oscillating (SO)and piecewise continuous(PC) n × n matrix functions, and SOBA11020200787 is a Fourier multiplier analogue onL p (ℝ)of [SO,PC] n× n The work is based on the study of Fourier multiplier analogue SO p ofSOon the characterization of the multiplicative linear functionals of slowly oscillating functions, and on the compactness of the commutators AW a, b — W a, b A, where A ∈ A PC] a ∈ SOandb ∈ SO p. Making use of the Allan-Douglas local principle we construct homomorphisms of A [SO PC]onto local Banach algebras and establish a Fredholm criterion for operators A ∈ A [SO PC] in terms of the invertibility of their images in the local algebras.

In this chapter we consider symmetric differential Riccati equations with real, continuous and bounded matrix-valued coefficients. We investigate the existence of (global) stabilizing real solutions and extend the theory to the case of quadratic indefinite sign terms. This extension is formulated in the framework of the generalized Popov—Yakubovich theory where a major role is played by the so-called Popov operator, associated with a controlled exponentially dichotomic evolution and a quadratic cost. It is proved that the J—factorization of the Popov operator provides both necessary and sufficient conditions for the existence of a (global) stabilizing solution to the symmetric differential Riccati equation. The technical machinery is instrumented via the Kalman-Yakubovich system in J-form, which is a more handy way of writing Riccati equations, as it inherits from the Popov operator useful information about its signature. The newly born theory has many interesting applications in modern systems theory: optimal control, approximation theory, and robust control.

The paper is aimed at describing a local algebra of boundary value problems near a corner point of the boundary. We show a symbol which controls the local solvability of the problem at the corner.

A continuity result is stated, also from the microlocal point of view, for pseudodifferential operators with symbol in a suitable class of weighted function spaces, defined by means of quasi-subadditive weight functions. Applications to semilinear partial differential equations are also given.

One direct way to generalize fractional Brownian motion to multifractional Brownian motion is to replace the constant Holder exponent by a certain deterministic function of time. In this paper local asymptotic properties of two types of multifractional Brownian motion will be considered.

We solve the index problem for general elliptic pseudodifferential operators on toric manifolds with conical points.

Asymptotic types for cone conormal asymptotics are constructed refining Schulze’s notion of discrete asymptotic type. This extends previous joint work with Liu Xiaochun (Wuhan University) for the case of Fuchsian differential operators to general cone pseudodifferential operators.

In this chapter we study the asymptotics of solutions to the Dirichlet problem for a strongly elliptic equation of order 2m in a subdomain of ℝ n from which a small neighborhood of a smooth closed manifold M of codimension d > 0 is cut out. We generalize the algorithms described in 12.2 for the Poisson equation in the case n = 3, d = 2.

We show that the problem of studying the Bergman-Toeplitz operators with functional symbols on a plane domain is, in a certain sense, the problem of studying the pseudodifferential operators on a half-line.

We consider an algebra A p (Γ,ω) of singular integral operators with slowly oscillating bounded coefficients acting in L p (Γ,ω), 1 < p < ∞, where Γ is a composed Carleson curve with logarithmic whirl points and ω is a power weight. The local analysis of operators A ∈ A p (Γ,ω) at singular points of the contours is based on the Mellin pseudodifferential operators method. This method gives effective formulas for the local symbols. These formulas describe the influence on the local symbol of both the curve and the weight in an explicit form.

Functional-differential equations are equations with respect to an unknown function that connect values of the unknown function and of its derivatives for different values of the argument. Such equations arise in the mathematical description of many systems, when one takes into account that parts of the system interact with a certain delay rather than instantaneously. A large number of publications are devoted to various aspects of theory of functional-differential equations and their applications (here we indicate only the monographs and surveys [2, 43, 72, 121, 122, 155, 163, 164, 185, 213], referring the reader to the bibliography given therein). Problems in the mathematical investigation of which it is essential to take into account deviations of the argument arise in nuclear reactor theory, automatic control theory, immunology, epidemiology, mathematical economics and other fields.

Multi-quasi-elliptic polynomials p(ζ), (ζ) ∈ ℝd,are important examples in the class of the hypoelliptic polynomials of Hormander [12]; they contain as particular cases the elliptic and quasi-elliptic polynomials, as well as any product of polynomials of this type. Their detailed study was carried out in the years 1960–70 by Cattabriga [7], Pini [17], Volevic-Gindikin [20], Friberg [8] and others.

We consider one-dimensional integral operators with kernel of the form \( [K(x,y) = {\mkern 1mu} \frac{1}{{x - y}}\sum\limits_{{\alpha ,\beta = 1}}^{n} {{{c}_{{\alpha ,\beta {\kern 1pt} \varphi \beta }}}(x)} \varphi \alpha (y) \) acting on functions on the union of intervals \( J{\mkern 1mu} = {\mkern 1mu} \bigcup\limits_{{k = 1}}^{m} {[{{a}_{{2k - 1,}}}{{a}_{{2k}}}].} \).

We discuss pseudo differential operators a(x,D) with a symbol a(x,ξ) which is with respect to ξ a continuous negative definite function. Such operators do occur in probability theory and mathematical physics.

In this paper we derive an analogue for integral operators of the well-known heat expansion for the Laplacian or other operators of positive order. Previous work established a partial expansion of n + 1 terms for a single integral operator in n dimensions. This is here extended to systems of operators.