Gevrey Local Solvability for Semilinear Partial Differential Equations

2002 Mathematics Subject Classification: 35S05In this paper we deal with a class of semilinear anisotropic partial differential equations. The nonlinearity is allowed to be Gevrey of a certain order both in x and ∂au, with an additional condition when it is GScr in the (∂au)-variables for a critical index scr. For this class of equations we prove the local solvability in Gevrey classes.The author is supported by NATO grant PST.CLG.97934

Two Aspects of the Donoho-Stark Uncertainty Principle

We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $\varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical Donoho-Stark estimate in two different aspects: a better general lower bound and a lower bound in dependence on the signal itself.Comment: 20 page

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension $2$. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree $2$. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin

Greedy expansions with prescribed coefficients in Hilbert spaces for special classes of dictionaries

Greedy expansions with prescribed coefficients have been introduced by V. N. Temlyakov in the frame of Banach spaces. The idea is to choose a sequence of fixed (real) coefficients $\{c_n\}_{n=1}^\infty$ and a fixed set of elements (dictionary) of the Banach space; then, under suitable conditions on the coefficients and the dictionary, it is possible to expand all the elements of the Banach space in series that contain only the fixed coefficients and the elements of the dictionary. In Hilbert spaces the convergence of greedy algorithm with prescribed coefficients is characterized, in the sense that there are necessary and sufficient conditions on the coefficients in order that the algorithm is convergent for all the dictionaries. This paper is concerned with the question if such conditions can be weakened for particular dictionaries; we prove that this is the case for some classes of dictionaries related to orthonormal sequences