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