Fundamental solutions of pseudodifferential equations connected with $ p$-adic quadratic forms

p-Adic Fractional Pseudodifferential Equations and Sobolev Type Spaces overp-Adic Fields

“…Then, = − . By (11), Theorem 4 is proved. In what follows we will introduce some relevant spaces over -adic fields (see [20]).…”

“…In recent years, -adic analysis has received a lot of attention due to its applications in mathematical physics; see, for example, [1][2][3][4][5][6][7][8][9][10][11] and references therein. As a consequence, new mathematical problems have emerged, among them, the study of -adic pseudodifferential equations; see, for example, [10][11][12][13][14][15][16] and references therein. In this paper, we study the solutions of -adic fractional pseudodifferential equations on the Sobolev type spaces.…”

Wavelets on ultrametric spaces

“…Ultrametric pseudodifferential operators were considered in [8][9][10][11][12][13][14]. The simplest example among these operator is the Vladimirov p-adic fractional derivation operator, which can be diagonalized by the p-adic Fourier transform.…”

Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields

“…Vladimirov showed the existence of a fundamental solutions for symbols of the form |ξ| α K , α > 0 [10], [11]. In [7], [6] Kochubei showed explicitly the existence of fundamental solutions for operators with symbols of the [8] Khrennikov considered spaces of functions and distributions defined outside the singularities of a symbol, in this situation he showed the existence of a fundamental solution for a p−adic pseudodifferential equation with symbol |f | K = 0. The main result of this paper shows the existence of fundamental solutions for operators with polynomial symbols.…”