Multipliers spaces, Muckenhoupt weights and pseudo-differential operators

Let sigma (x,xi) be a sufficiently regular function defined on R-d x R-d. The pseudo-differential operator with symbol a is defined on the Schwartz class by the formula [GRAPHIC] where (f) over cap(xi) = integral(Rd) f(x)e(-2 pi ix xi) dx is the Fourier transform of f. In this paper, we shall consider the regularity of the following type: [GRAPHIC] (alpha is an element of N-d), where omega is suitable positive function and we prove boundedness results on multipliers spaces X-r = M (H-r -> L-2) for pseudo-differential operators whose symbol sigma (x,xi) satisfies the regularity condition on x. (c) 2005 Elsevier Inc. All rights reserved.