Regular nonlinear generalized functions and applications

2008

In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz’ result is contained in our main theorem.

Proceedings of the Edinburgh Mathematical Society, 2005

We characterize microlocal regularity, in the G ∞ -sense, of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic regularity result for distributional solutions and consider propagation of G ∞ -singularities for homogeneous first-order hyperbolic equations.

Monatshefte für Mathematik

We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise criteria for detecting Besov regularity of distributions.

Linear and Non-Linear Theory of Generalized Functions and its Applications, 2010

In analogy to the classical isomorphism between L(D(R n), D (R m)) and D (R m+n) (resp. L(S(R n), S (R m)) and S (R m+n)), we show that a large class of moderate linear mappings acting between the space GC (R n) of compactly supported generalized functions and G(R n) of generalized functions (resp. the space GS (R n) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n) of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of G(R m+n) (resp. Gτ (R m+n)). The main novelty is to use accelerated δ-nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.

Mathematical Proceedings of The Cambridge Philosophical Society, 2007

In analogy to the classical isomorphism between L (S (R n ) , S ′ (R m )) and S ′ R n+m , we show that a large class of moderate linear mappings acting between the space GS (R n ) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n ) of temperate ones admits generalized integral representations, with kernels belonging to Gτ R n+m . Furthermore, this result contains the classical one in the sense of the generalized distribution equality. : 45P05, 46F05, 46F30, 47G10

Doklady Mathematics, 2008

2015

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties with respect to nonlinear operations. In this spirit we give several examples of propagation of singularities through nonlinear operators.

Journal of Integral Equations and Applications, 2003

In this article the notion of multiplicative regularizator, a smooth function that by multiplication allows the extension of operators in spaces of distributions, is introduced, and several of the properties are obtained. Applications to Hilbert transforms, Carleman operators, fractional integration operators and generalized Abel operators are given.

Journal of Pseudo-Differential Operators and Applications, 2016

We define and study classes of smooth functions which are less regular than Gevrey functions. To that end we introduce two-parameter dependent sequences which do not satisfy Komatsu's condition (M.2)', which implies stability under differential operators within the spaces of ultradifferentiable functions. Our classes therefore have particular behavior under the action of differentiable operators. On a more advanced level, we study microlocal properties and prove that WF 0,∞ (P (D)u) ⊆ WF 0,∞ (u) ⊆ WF 0,∞ (P (D)u) ∪ Char(P), where u is a Schwartz distribution, P (D) is a partial differential operator with constant coefficients and WF 0,∞ is the wave front set described in terms of new regularity conditions. For the analysis we introduce particular admissibility condition for sequences of cutoff functions, and a new technical tool called enumeration.

Acta Mathematica Academiae Scientiarum Hungaricae, 1982

New Developments in Pseudo-Differential Operators, 2008

The aim of this paper is to give a review of local and global properties of Fourier integral operators with real and complex phases, in local L p , global L 2 , and in Colombeau's spaces. P u(t, x) = 0, t = 0, ∂ j t u| t=0 = f j (x), 0 ≤ j ≤ m − 1. The loss of regularity for solutions u(t, ·) compared to the Cauchy data depends on the operator P and on the function spaces. For example, the usual energy conservation that holds in L 2 fails in L p for p = 2.

2007

2006

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau's theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau's theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau's theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau's community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau's theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics. MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05).

Analysis Mathematica, 1997

An integral transform of generalized functions. II ANIL KUMAR MAHATO D(I) will denote the standard union space (see [7, pp. 32, 33]) of countably multinormed spaces DK (I) of all complex-valued smooth functions defined on I = (0, c¢), which vanish on those points of I that are not in a compact subset K of I, with seminorms defined by ~k(¢) = suplDkC(t)l, ¢ ~ DK(I), tEI k c¢ and with the topology generated by the countable multinorms {7 }k=0 assigned to the corresponding linear space with usual pointwise operations of addition and multiplication of functions. E(I) denotes the space of smooth functions on I. Its dual E'(I) is the space of distributions with compact support on I.

Russian Mathematical Surveys, 2000

In the paper the regularity properties of Fourier integral operators are considered in different function spaces. The most interesting case are L p spaces, for which a survey of recent results is made. Thus, the sharp orders are known for operators, satisfying the so-called smooth factorization condition. Further in the paper this condition is analyzed in both real and complex settings. In the last case conditions for the continuity of Fourier integral operators are related to the singularities of affine fibrations in (subsets of) C n , defined by the kernels of Jacobi matrices of holomorphic mappings. Singularities of such fibrations are analyzed in this paper in the general case. In particular, it is shown that for small dimensions n or for small ranks of the Jacobi matrix, all singularities of an affine fibration are removable. Fourier integral operators lead to fibrations, given by the kernels of the Hessian of a phase function of the operator. Based on the analysis of singularities for operators, commuting with translations, in a number of case the factorization condition is shown to be satisfied, which leads to L p estimates for operators. In the other cases, the failure of the factorization condition is exhibited by a number of examples. Results are applied to derive L p estimates for solutions of the Cauchy problem for hyperbolic partial differential operators.