For a few hundred years theoretical physics has been developed on the basis of real and, later, complex numbers. This mathematical model of physical reality survived even in the process of the transition from classical to quantum physics – complex numbers became more important than real, but not essentially more so than in the Fourier analysis which was already being used, e.g., in classical electrodynamics and acoustics. However, in the last 20 years the field of p-adic numbers ℚp (as well as its algebraic extensions, including the field of complex p-adic numbers ℂp) has been intensively used in theoretical and mathematical physics (see [1]–[3], [10]–[15], [24], [35], [38], [43], [44], [55], [64], [65], [77]–[79], [89], [94], [115]–[122], [123], [124], [133], [157], [158], [174], [198], [241]–[246] and the references therein). Thus, notwithstanding the fact that p-adic numbers were only discovered by K. Hensel around the end of the nineteenth century, the theory of p-adic numbers has already penetrated intensively into several areas of mathematics and its applications.

The starting point of applications of p-adic numbers to theoretical physics was an attempt to solve one of the most exciting problems of modern physics, namely, to combine consistently quantum mechanics and gravity, thus to create a theory of quantum gravity. In spite of the considerable success of somemodels, there is still no satisfactory general theory.

Introduction

In standard mathematical physics (in the real setting) there are problems which require the definition of products of distributions (generalized functions) [66], [68], [86], [97], [194]. Such problems appear in quantum mechanics [53], [9], [27], quantum field theory, some problems of gas dynamics, elasticity theory, and also in the description of, e.g., shock waves, δ-shock waves, and typhoons. In the framework of the approaches connected to problems of multiplications of distributions, a theory of singular solutions of non-linear equations has been developed [8], [68]–[71], [149], [150], [195], [220]. Solving problems of this kind requires the development of special analytical methods, the construction of algebras containing the space of distributions, and the development of a technique for constructing singular asymptotics. As a result, the demand arises for a construction of a nonlinear theory of generalized functions. Besides, the development of nonlinear theories of distributions is of great interest in itself.

Since p-adic mathematical physics is a relatively young science, p-adic analogs of the above mentioned problems have not been studied so far (to the best of our knowledge). The problems of p-adic analysis related to the theory of p-adic distributions which have been solved up to now are of the linear type. To deal with nonlinear singular problems one needs some additional technique similar to that developed in the usual real mathematical physics mentioned above.

Introduction

Tauberian theorems is a generic name used to indicate results connecting the asymptotic behavior of a function (distribution) at zero with the asymptotic behavior of its Fourier, Laplace or other integral transforms at infinity; the inverse theorems are usually called abelian. In the real setting Tauberian theorems have numerous applications, in particular, in mathematical physics (for example, see Drozzinov and Zavyalov [80], [81], Korevaar [160], Nikolić-Despotović, Pilipović [191], Vladimirov, Drozzinov and Zavyalov [240], Yakymiv [248] and the references cited therein). Multidimensional Tauberian theorems for distributions are treated in the fundamental book [240]. Some of them are connected with the fractional operator. In [240], as a rule, theorems of this type are proved for distributions whose supports belong to a cone in ℝn (semi-axis for n = 1). This is related to the fact that such distributions form a convolution algebra. In this case the kernel of the fractional operator is a distribution whose support belongs to the cone in ℝn or a semi-axis for n = 1 [240, §2.8.]

p-adic analogs of Tauberian theorems do not seem to have been discussed so far except for [140], [141], [21]. In this chapter, we present a first study of them based on the above papers.

In the beginning, in Sections 12.2 and 12.3, we introduce the notion of the p-adic distributional asymptotics [140], [141]. In Section 12.2, the definition of distributional (stabilized) asymptotic estimate at infinity is introduced.

Introduction

In this appendix we construct and study associated homogeneous distributions (AHDs) and quasi associated homogeneous distributions (QAHDs) for the real case. These results are based on the paper [223]. The results of this appendix are used in Chapter 6 to develop the theory of p-adic associated and quasi associated homogeneous distributions.

The concept of AHD was first introduced and studied in the book [95, Ch.I, §4.1.] (see Definitions A.2 and A.3 by analogy with the notion of the associated eigenvector (A.2.2)). Later the concept of an AHD was introduced in the paper [232, Ch.X, 8.] by Definition A.4, and in the books [87, (2.6.19)], [88, (2.110)] by Definition A.5. In the book [95, Ch.I, §4] and in the paper [232, Ch.X, 8.] a theorem was given (without proof), in which all AHDs were described (see Proposition A.2.1). In Section A.2.2 we discuss and analyse Definitions A.3, A.4, A.5, (A.2.9) of an AHD and show that they are selfcontradictory for k ≥ 2. Moreover, these definitions come into conflict with Proposition A.2.1. According to Section A.2.2, there exist only AHDs of order k = 0, i.e., homogeneous distributions (HDs) (given by Definition A.1) and of order k = 1 (given by Definition (A.2.3) or Definition A.2). Thus one can see that the concept of an AHD requires a special study.