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By Katz N.M.

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U(x), LP(G')\\ -> 0, h-*Q. 4) PROOF. 2) and a - (1, . . 3). , in any interior subdomain, the weak differentiation operators commute with the averaging operators provided that h is sufficiently small. 6) J If 0 < h < p(dG, dG'), then for any x 6 G1 the function K((x - y ) / h ) (regarded as a function of y) belongs to the class C^°(G). 5). 4). D Based on the above theorems, it is easy to establish properties 5-7 of weak derivatives formulated at the beginning of the section. PROOF OF PROPERTY 5.

M. Stein [1, 2]). § 5. Laplace Transform In this section, we study the Laplace transform of functions of a single real variable. 1. 1) is called the Laplace transform and the operator associating with u(t) the function v ( r ) , r =. , v(r) — £[u](r). 1) is defined for a > 7 for any locally summable function u(t] such that e~~1tu(t] G L p (lR^). Indeed, in this case, e-atu(t] € ^i(M|) for a > 7. We will study the action of the integral Laplace operator on functions u(t) G Li 0 c(K^") such that e~ 7 t u(£) G L 2 ( M ^ ) .

The study of conditions under which a function /j(£) is a multiplier on L p (IR n ) is very complicated and is not completely investigated yet. At present, there are a number of theorems about sufficient conditions for a function to be a multiplier. These theorems are used in calculus and the theory of partial differential equations. We formulate some of such theorems and give examples of multipliers below. The proof of these theorems can be found, for example, in S. M. Nikol'skh [1], E. M. Stein [1, 2], H.

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Algebraic solutions of ODE using p-adic numbers by Katz N.M.

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