On u-weak stability of coarse isometries between Lp spaces

In this paper, we study u-stability and weak u-stability of coarse isometrics between Banach spaces for a free ultrafilterti on N. As a result, for a coarse isometry f : L-p (Omega(1), Sigma(1), mu(1)) -> L-p (Omega(2), Sigma(2),mu 2)( 1 < p < infinity), where (Omega(j), Sigma(j), mu(j)) (j = 1, 2) are two measure spaces, we prove that the following conditions are equivalent: ( )(i) The mapping Phi defined by Phi(x) = w - lim(u) f(nx)/n for any x is an element of L-p (Omega(1), Sigma(1), mu(1)) is a linear isometry ; (ii) For any x is an element of L-p (Omega(1), Sigma(1), mu(1)), lim(u) f(nx)/n exists and lim(u) f(nx)/n = Phi(x); (iii) f is U-weakly stable; (iv) There exists a T is an element of R (L-p (Omega(2), Sigma(2), mu(2)), L-p L-p (Omega(1), Sigma(1), mu(1) With parallel to T parallel to = 1 such that parallel to Tf (nx) - nx parallel to = o(n)u.