INTRODUCTION. NOTATION 3

Lp,\oc — the space of measurable functions on R^, belonging to LP(A) for all

compact A c

RiV.

V — CQ°, V', P

/ m

— the usual spaces of test functions, distributions, and

distributions of order at most m.

S, S' — the Schwartz spaces of test functions and temperate distributions.

T,

T~x

— the Fourier transformation, and its inverse.

Mr,df — the maximal function, defined for measurable functions / , and r 0,

d 0 by

Mr4f(x)=sup(a-N f \f(x + y)\r(l + \y\)-rddy) .

a0\ JB(0,a) /

Mr$f is denoted Mrf.

The action of a distribution / on a test function p is denoted (/, (f).

The convolution of a distribution / and a test function ip is denoted / * (p and

defined by / * ip(x) = {f,ip(x- •)).

^Pm5 TU — —1, 0, 1, 2, ..., is the set of polynomials of degree at most m. (We

define ^3_i = {0}). The notation / _ L ^3

m

for a function or distribution / means

that JnN f-K dx = 0, or (/, 7r) = 0 , for all polynomials ir G £pm.

Let A C R ^ be bounded, let m = 0, 1, 2, ..., and let X be a normed or

quasi-normed space of functions defined on R^ . The degree of local approximation

is

£m(f,A,X)= mf \\f-ir\\x{A),

where X(A) is the restriction of X to A.

We adopt the convention that the letter C (without any index) is used to denote

a constant, whose exact value is irrelevant and can change even within a chain of

inequalities.