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**Extra info for A First Course in Infinitesimal Calculus**

**Example text**

Sbordone 546 Using iv), we have Using once more iv), we have so, there exists c > 0, such that Choosing A such that c/).. e. the result. e. (Mf)(x) = sup i f(y)dy Q~% Q where Q is a cube with sides parallel to the axes and lbl JQ f(y)dy. )(x) =sup{_[ f(y)dy: JQ/2 X E Q/2, Q c n} where Q /2 denotes the cube with the same center of Q and half side. 2. Qo f dx: l{xEQo:Mf(x)>t}i~~~ 2 t {f>t} fdx. g. [2]. 3. IfQo is a cube ofRn; f E L 1 (Q 0 ), f then, for any t > 0 wnj l{x E Q 0 : Mf(x) > t}l :S:t U>H f 2 0, dx.

Then we set A of A. A(s) = 1& a(u)du. The following Young inequality st ~ A(t) + A(s), Vs,t ~ 0 is well known. 1) A ( A;t)) ~ A(t) Vt > 0. Let n be a bounded open set in Rn. LA(n) iu(x)i { . A-)dx::; 1}. 4. 1, 8 =_{A ( In j A( In u(x) lluiiLA - E ) _j A( u(x) lluiiLA )dx -In iiuiiLA iiuiiLA - E < 8( lluiiLA )k-1 iiuiiLA- E - and the last term can be less than 1 for some E > 0, which is absurd. 1. If A verifies i), ii), iv), and 1 . mf A(t)A(-) ~ m O

A

. > 1 to be chosen. l} . 1, = 1, so c+ 1 A(A(a