ABOUT THE UNIMPROVABILITY OF THE LIMITING EMBEDDING THEOREM FOR DIFFERENT METRICS IN THE LORENTZ SPACES WITH HERMITE'S WEIGHT

In this article we obtained inequality of different metrics in the Lorentz spaces with Hermit's weight for multiple algebraic polynomials. On this basis we established a sufficient condition of embedding of different metrics in the Lorenz spaces with Hermite's weight. Its unimprobality is shown in terms of the "extreme function". Let f is an element of L-p,L-theta (R-n; rho(n)), 1 <= p < +infinity, 1 <= theta <= +infinity. The sequense {l(k)}(k=0)(+infinity) subset of N is such that l(0) = 1 and l(k+1) . l(k)(-1) > a(0) > 1, for all k is an element of Z(+). f((x) over bar) = Sigma(+infinity)(k=0) Delta l(k),...l(k) (f; (x) over bar) is some presentation of the functions in the metric L-p,L-theta (R-n; rho(n)), where Delta l(0),...,i(0) (f; (x) over bar) = T-1,...,1, Delta l(k),...,l(k) (f; (x) over bar) = Tl-k,...,l(k) ((x) over bar) - Tlk-1,...l(k-1) ((x) over bar), for all k is an element of N. Here Tl-k,....l(k) ((x) over bar) = Sigma(lk-1)(m1=0) ... Sigma(mn=1) (lk-1) a(m1,...,mn) Pi(n)(i-1) x(i)(mi)- are algebraic polynomials for all k is an element of Z(+). 1(0). If the series A(f)(p0) = Sigma(+infinity)(k=0) l(k)(tau(n/2p-n/2q)) parallel to Delta l(k),...l(k)(f)parallel to(tau)(Lp,theta(Rn;rho n)) converge under some q and tau: p < q < +infinity, 0 < tau < +infinity, then f is an element of L-q,L-tau(R-n; rho(n)) and we have the inequality parallel to f parallel to L-q,L-tau(R-n; rho(n)) <= C-pq theta tau n x (A(f)(p theta))1/tau. 2(0). The condition 1(0) is unimprovable in the sense that there exists a function f(0) is an element of L-p,L-theta(R-n; rho(n)) and A(f(0))(p theta) diverges for it and f(0) is not an element of L-q,L-tau (R-n; rho(n)). At the same time, the function f(0) is an element of L-q-epsilon,L-tau(R-n; rho(n)) for all epsilon > 0 : < (q - epsilon) < q.