Asymptotically homogeneous generalized functions and boundary properties of functions holomorphic in tubular cones

The structure of quasiasymptotics of Schwartz distributions

“…Most known results are only valid under restrictions over the support of the distribution; some samples of these results can be found in [16,24]. The recent work of Drozhzhinov and Zavialov [5] is of relevance for this open question, their results suggest that spherical representations may be a path to follow in order to find an answer to such a important question. Open problem: To find the complete structure of quasiasymptotic behaviors of distributions in the multidimensional case.…”

“…Since its introduction, the study of the structure of the quasiasymptotics has deserved a special place [5,8,10,11,12,13,14,15,16,19,22,24]. S. Lojasiewicz introduced the value of a distribution at a point, and he provided the corresponding structural theorem for it.…”

“…The quasiasymptotics of distributions have shown to be of importance in several areas such as mathematical physics [24,29], abelian and tauberian theory of integral transforms [16,24,25], asymptotic behavior of solutions of partial differential equations [5,8,21,24,28] and summability of trigonometric series and integrals [6,8,21,22,26,27].…”

Regularizations at the origin of distributions having prescribed asymptotic properties

“…Scaling asymptotic properties of distributions have shown to have a valuable role in this respect, they bring new insights into the problem [1,23,24]. On the other hand, the relationship between regularizations and asymptotic properties of distributions is also of importance from the point of view of pure mathematics, for instance, in areas such as singular integral equations [8], the study of boundary properties of holomorphic functions [5], or in Tauberian theory for integral transforms [4,12,13,22,23]. In fact, as shown in recent studies [4,11,22], the asymptotic analysis of various integral transforms may be completely reduced to the study of asymptotic properties of regularizations of distributions; this is the case for the Laplace and wavelet transforms.…”

“…We emphasize that such a problem is essentially a Tauberian one and may be restated in terms of Mellin convolution type integral transforms: quasiasymptotic behavior is nothing but knowledge of asymptotic information over (Mellin) convolution transforms for all kernels in a Schwartz space of test functions. Recently, this problem has been investigated in [5,18,19,20,22]; we shall give extensions of those results, and in particular we provide more detailed asymptotic information for critical degrees than that from [5]. We shall consider distributions with values in a Banach space.…”