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We give a criterion when function m : E -> C is a multiplicator in Holder spaces with nonhomogeneous metrics. This criterion is applied to nonstationary system of hydrodynamical type. I know two criterions when function m : E —> C is a multiplicator in the space I/(E). The first of them goes back to Marcinkievicz (see [1] — [3]). It has found numerous applications to the study of elliptic equations and systems. But it is not applicable to parabolic equations. Even for m(£, A) = ^^2 arising for heat equation this criterion does not fulfilled. For this m the criterion given by Lizorkin and Stein ([4], [5]) can be used. As it was shown in [6] the later can be also applied to functions m which we have got solving the Cauchy problem for the system (1) dtUj DjktimdlhXmui + dXjq = fc, j = 1,2,3, div u = 0 with coefficients Djkjm fixed at a point (x,t). System (1) is a linearization (up to lower nonsignificant terms) of the Modified Navier-Stokes system (see [7], [8], etc) (2) dtv + v-Vv-div ^ + Vp = g, divv = 0, OS £=e(v) where D : MfyJ^ -> E]_ = [0, oo) is a smooth convex function of the symmetric tensors e = (eij) and e(v) = |Vv + |(Vv) (here Vv is the gradient of v with respect to the space variables x = (xi,^,^) G E). In (2) velocity field v and pressure p are unknown functions and in (1) their variations u and q are the functions under consideration. At first, we reduce the nonhomogeneous initial data v = v to the data t=o (3) = 0 t=o and study the Cauchy problem ■(!), (3), supposing that f(x,0) = 0. After that we put u, q and / equal to zero for (x, t) € E* x E\. — (-oo, 0) and pass to their Fourier images u, q and /. For them we get the system (4) i\uj + Aji^)ui+i£>jq = fj, j = 1,2,3, &&*=(), where Aji(g) = DjkjmtkZmThe variables ^ = (fij&jfs) are dual to variables (xi,X2,xs) and A is dual to t. The conditions which are usually put on D (see [7]—[9], etc) generate for Djk,im the restrictions (5) VQ\X\ < DjkjmXjkXim < //QM, * Received December 25, 1999. t St.-Petersburg Branch of Steklov Mathematical Institute, Russian Academy of Science, Fontanka 27, St.-Petersburg, 191011, Russia (ladyzhen@pdmi.ras.ru). The work is partly supported by RFFI no. 99-01-00108 and by INTAS no. 96-0835.

On multiplicators in Hölder spaces with nonhomogeneous metric