Alessandro Oliaro

Local solvability and hypoellipticity for semilinear anisotropic partial differential equations

We propose a unified approach, based on methods from microlocal analysis, for characterizing the local solvability and hypoellipticity in C°° and Gevrey G σ classes of 2-variable semilinear anisotropic partial differential operators with multiple characteristics. The conditions imposed on the lower-order terms of the linear part of the operator are optimal.

A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform

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.

Local solvability for semilinear anisotropic partial differential equations

In this paper we introduce a class of Gevrey-Sobolev anisotropic spaces and, by means of the application of the Banach fixed point Theorem, we obtain the solvability of a related semilinear equation.

Two-Window Spectrograms and Their Integrals

We analyze in this paper some basic properties of two-window spectrograms, introduced in a previous work. This is achieved by the analysis of their kernel, in view of their immersion in the Cohen class of time-frequency representations. Further we introduce weighted averages of two-window spectrograms depending on varying window functions. We show that these new integrated representations improve some features of both the classical Rihaczek representation and the two-window spectrogram which in turns can be viewed as limit cases of them.

Wigner Representations Associated with Linear Transformations of the Time-Frequency Plane

In this paper we define a variation of theWigner form depending on a linear transformation of the time-frequency plane and study the corresponding properties. This construction yields a natural geometric interpretation of the so-called “ghost frequencies” showed, among others, by the Wigner quadratic representation. In particular we prove that the representations in this new class which satisfy the support properties are just the so-called t-Wigner forms. On the other hand for these new forms an “essential” re-adjustement of the supports is showed to be possible, whereas their features of moving almost arbitrarily the ghost frequencies is used to define representations with no interferences at all for a certain class of signals.

Local solvability for powers of anisotropic operators

We consider a class of semilinear partial differential equations whose linear part is the power of an anisotropic operator in n variables and whose nonlinear term is allowed to be nonanalytic with respect both to the variables and the covariables; for such equations we prove local solvability in Gevrey classes. We shall mention, in the last section, a possible generalization of this result to mixed Gevrey-C ∞ classes.

Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order

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 G s for small s, and local nonsolvability in G s for large s.

Regularity of partial differential operators in ultradifferentiable spaces and Wigner type transforms

Abstract We study the behaviour of linear partial differential operators with polynomial coefficients via a Wigner type transform. In particular, we obtain some results of regularity in the Schwartz space S and in the space S ω as introduced by Bjorck for weight functions ω. Several examples are discussed in this new setting.

On a Gevrey-Nonsolvable Partial Differential Operator

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.

On the solvability of the characteristic Dirichlet problem for linear degenerate parabolic equations

We investigate the classical solvability for some classes of linear, degenerate equations in divergence form with prescribed Dirichlet data. Since the boundary value problem is characteristic according to Fichera on a part of the boundary, some typical nonlinear phenomena at these points are observed as boundary gradient blowups of the classical solutions in space directions. The regularity results explain the lack of hypoellipticity for special right-hand sides or boundary data for linear degenerate parabolic equations.