The paper deals with function spaces F>,?(R", a ) and P;X(Rn, a ) defined on the EucLIDean n-space R". These spaces will be defined on the basis of function spaces of BESOV-HARDY-SOBOLEV type F;,(Rn) and B:,JRn) -see-[25], and by appropriate pseudo-differential operators A(x, 0,). We get scales of spaces such as the standard BESOV-HARDY-SOBOLEV spaces indexed by the degree of smoothness and mapped to each other by pseudo-differential operators.
There are several papers which relate function spaces and classes of pseudodifferential operators. But in general the basic function spaces are L2, see for example UNTERBERG and BOKOBZA [ZS], UNTERBERG [26], HORMANDER [9], VOLEVIE and KAGAN [29], KUMANO-GO and TSUTSUMI [15], BEALS [l], [2], or L, (1 < p < -) BEAUZAMY [4], [5], BEALS [3] respectively the classical BESOV spaces ( l i p < -, I ~q ~m ) [4], [5] and HOLDER spaces [3]. At the beginning there were considered subclasses of hypoelliptic pseudo-differential operators with respect to the HORMANDER class f ! $ $ of pseudo-differential operators. But the development goes in the direction of more general classes of pseudo-differential operators-see for example [2], [lo] and [3].
In this paper we discuss the problem in a general class of function spaces. We have hypoelliptic pseudo-differential operators of the classbal H~EMANDER class Sy6, but we relate they to function spaces Fi,q and Bisp which are more complicated than L2 or Lp. These two scales of spaces contain a lot of well-known classical spaces as special cases, the BEssEL-potential spaces, SOBOLEV spaces, classical BEsov-spaces, local HARDY-spaces and others -see also [25, section 2.3.51. In section 1 we recall some facts about classical pseudo-differential operators and prove a crucial theorem on theboundedness of pseudo-differentialoperators in Fi,g and Bh,q.
Section 2 deals with the definition and some properties of the class 8(l, 6; m ' ; m ) of hypoelliptic symbols.
In section 3 we define the function spaces F;:l(R", a), give some examples and describe properties of these spaces.
Finally in section 4 we characterize for special symbols a(x, E ) the spaces F;x( Rn, a ) as weighted function spaces in the usual sense.