By Sergei Mihailovic Nikol’skii (auth.)

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4. Averaging of Functions According to Sobolev This process is due to Sobolev [4J. Denote by a8 = {lxl ~ s}, a1 = a, the ball in IR = IRn of radius s and with center at the origin. Suppose that 1p(t) is an infinitely differentiable even nonnegative function of one variable t(-oo < t <(0), equal to zero for It I ;;S 1 and such that (1 ) f 1p(lxj) dx = 1, 1"'1~1 where "n is the volume of the unit (n - i)-dimensional sphere in IRn. 28 1. Preparatory information I As "P we may choose the function °~ It I < 1, ~ el'~l ' An "P(t) = 0, 1 ~ Itl, where the constant An is chosen so that condition (1) is satisfied.

Fm (for k j = the corresponding sum is extended only to Vi = 0); or - is entered according as k j is positive or negative. Transform a function of period 2n of the type (see 1. 2. (7)) ° + (2) f(x) = ~ Ckeik3J = ~ Ok (f) E Lp* < P< (1 (0) k by means of the Marcinkiewicz multipliers Ak: 0) F(x) = L: }'kckeikaJ = L: ok(F) . k Then F E L: and there exists a constant cp depending only on p such that (4) . Proof. We restrict ourselve;;; to the case n = 2. Moreover, we will suppose that in (2) the series are extended only over k ;::;; 0, which does not affect the generality.

J K(S) I(). - = S) dg = K@ I()· - S) d; = K *1. Since IE 5, then the integral in '1'1 in the top line is a function of u, belonging to 5 c L. ' The product belongs to L, so that after multiplication by ei "''' and integration with respect to u we obtain a /'.. continuous function fcJ of x. The change of the order of integration relative to sand), is legitimate, since K, I E L. Here we use Fubini's theorem. 5. Generalized functions The integral in the next-to last term of these relations is called the convolution of K and I.