Distributions Supported on Conical Surfaces and Generated Convolutions
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
DISTRIBUTIONS SUPPORTED ON CONICAL SURFACES AND GENERATED CONVOLUTIONS V. B. Vasil’ev Belgorod National Research University 14/1 Studencheskaya St., Belgorod 308007, Russia [email protected]
UDC 517.983
We describe the structure of distributions supported on conical surfaces and calculate the Fourier transform for some cones. The results are represented as convolutions with particular kernels. We use transmutation operators owing to which it is possible to clarify connections between the change of variables for a distribution and its Fourier transform. Bibliography: 17 titles.
A lot of examples of distributions supported on various type surfaces in m-dimensional spaces can be found in [1, 2]. However, in the literature, there are no results concerning distributions in the general form (counterparts of the Schwartz theorem on the general form of a distribution supported at a point in the one-dimensional case [3]). This paper is motivated by the recent results of [4]–[8], where pseudodifferential equations are studied in domains with conical points on the boundary in the multidimensional (m 3) case.
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Distributions and Change of Variables
1.1. Choice of test functions. Let C be an acute convex cone in Rm containing no entire line. Assume that the conical surface is given by the equation xm = ϕ(x ), x = (x1 , . . . , xm−1 ), where ϕ : Rm−1 → R is a smooth function on Rm−1 \ {0} such that ϕ(0) = 0. We introduce the change of variables t1 = x1 , t2 = x2 , . . . , tm−1 = xm−1 , tm = xm − ϕ(x ) and denote by Tϕ : Rm → Rm the change operator. It is obvious that this transformation is smooth except for the origin. We introduce the change of variables for the following class of distributions. For the space of test functions we take the Lizorkin space Φ(Rm ) [9] which is a subspace of the Schwartz space S(Rm ) of infinitely differentiable functions that are rapidly decreasing at infinity and vanish at the origin, together with all its derivatives. If Φ (Rm ) and S (Rm ) denote the corresponding spaces of distributions, then Φ (Rm ) ⊃ S (Rm ) and all operations with distributions in Φ (Rm ) are legitimate for distributions in S (Rm ).
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 63-70. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0885
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Definition 1.1. The variable change operator on functions in Φ(Rm ) is defined by (Tϕ ψ)(x) = ψ(Tϕ x). It is easy to see that the Jacobian of Tϕ is everywhere (except for 0) equal to 1. By the choice of the class of test functions, we can ignore the point 0. It is clear that the operator Tϕ −1 is invertible and its inverse is defined by (T−1 ϕ ψ)(x) = ψ(Tϕ x). Let f be a locally integrable function generating a distribution by (f, ψ) = f (x)ψ(x)dx. Rm
The functional Tϕ f is defined by (Tϕ f, ψ) = (f, T−1 ϕ ψ) since
(Tϕ f )(x)ψ(x)dx ≡
(Tϕ f, ψ) = Rm
f (Tϕ x)ψ(x)dx =
Rm
f (x)ψ(Tϕ−1 x)dx ≡ (f, Tϕ−1 ψ).
Rm
We use the following result based on the Schwartz theorem
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