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292 XIV INTEGRATION IN LOCALLY COMPACT GROUPS
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t e l(x\ F(*, /) > a. (Use Problem 2, and observe that, by virtue of the relation
rn(\ — /) = -rn(t\ for a given integer h E [1, n] the set of / e [0, 1] such that the numbers rh(t)g(shx) and Y rk(t)g(SkX) have the same sign, has measure i> ^.) |
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4. Let G be a compact commutative group, /? the Haar measure on G for which /?(G) = 1 .
Let (Un) be a sequence of continuous endomorphisms of La(G, /3), each of which commutes with all translations /->(yW/)~ for all s e G. For each /e o£?|(G, /?) we denote by Un -/any function belonging to the class Un •/ and we put
£/*•/= sup (/„•/.
n
For each a > 0, let E«(/) be the set of points x e G such that (U* -f)(x) > a.
(a) Let /e J^(G, j8) be such that N2(/) g 1, and let w be an integer such that
n - £(Ea(/)) > 1 . Let Ji, . . . , j,, be points of G such that the union B of the sets sj^EJJ)
n
has measure j8(B) > J (Problem 2), and let F(x, 0 = S rk(t)f(skx). Show that for each
/c = i
* eB there exists a finite union of intervals I(x) c [0, 1] such that A(I(jt)) ^> J and
such that x e Ea(F( • , t)) for all /• e l(x). (Observe that if A: e B there exists an integer m and an integer j e [1, n] such that (Um -f)(sjx) = 0, and apply Problem 3 to g = £/,„•/)
(b) Let S c [0, 1] be a A-integrable set such that A(S) ^ t. Show that there exists
t e S such that j8(Ea(F( * , 0)) > i- (If H is the set of (x, /) e G x [0, 1] such that (fj* - F( • , t))(x) > a, remark that for each x e B we have A(H(x)) ^> J, and deduce that(j8<8>A)(H)>J.)
(c) For each M > 0, let SM be the set of / e [0, 1] such that N2(F( • , /)) ^ M. Show
that A(SM) ^ 1 — w/M2. Deduce from (b) that if M2 ^ 4n there exists t e [0, 1] such that both N2(F( - , 0) ^ M and ]8(E«(F( - 0)) > i
(d) Suppose that for each/ e J^R(G, /3) the function £7* • /is finite almost everywhere.
Show that there exists a constant C > 0 such that, for each/e ^JCG, jS) and each a > 0, we have (£(Ea(/)))1/2 < Ca-^af/) (E. Stein's theorem). (Deduce from Section 13.12, Problem 12 that there exists a constant c>0 such that for each function h e &i(G, ft) satisfying N2(/z) g M, we have £(EcM(/z)) < i Then make use of (c) above, taking M = oc/c, h = F(-,t) and n = [JM2].)
5. Let A be Lebesgue measure on R and let / be a compactly supported A-integrable
function. Put |
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p~ f
e 2H J _/i
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for each e > 0, and
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these are lower semicontinuous functions on R (Problem 1) with compact support.
6(f) is the Hardy-Littlewood maximal function relative to/.
(a) For each a> 0, let Ea,«(/) be the set of x eR such that 6&(f)(x) > a. Every
compact set K c Ee,«(/) is contained in the union of a finite number of compact
intervals Ik(l^k^n) such that a • A(Ifc)g | |/(f)I i/r.
Ji*
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