By Mark Kac

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**Extra resources for Integration in function spaces and some of its applications (Lezioni fermiane)**

**Example text**

Furthermore. 17). 2 that llHyf-Qy fll~ ~ l[0fll~ ~ 321]fI]~ < +~ . and thus llHyf -Qy f]I2 § 0 as y ~0+ by dominated convergence. 14). P2yf dx . 15). 16). 4. 16). 13) proved . 39 Proof. 13). is of weak type is of type 1 . 15) in 2 , and hence also of weak type 2 . e. Hf dx[ = < cp llfllp llgllq we conclude fs and all and g~Lq(R) H . Hgdx[ = llfIlp<__l proving that (I

Is continuous. 20) is unaffected, if we from any continuous function fo f subtract of compact support. , proving the lena. 4. e. the operator Proof. P for all P Y is of strong type p Using H~ider's inequality we get for for all p ~ ]I,+ ~] p ~ ]I,+~[ IPyf(x)]P< ([ +~k (x-t). 5. 2. In fact, llPyfllp ~ II0 flip ~ Cpllfllp , p E ]I,+~] However~ we do not get that the constant may be chosen equal I . 6. (5. If fE for some ~) p e ]I, + ~[ , then lim llPyf - fIlp= 0 y~O+ and (5. ~S) Proof. 23) are obvious if tinuous function of compact support.

Due to the facts that f has compact support and gx(t) tends to 0 as Itl ->+~ vals mentioned we shall never need to consider the two infinite inter- above. We define g(x) = ~I J f(t)dt 9 x-c(J) I and gj (x) = ~I J . = j The function J f(t)dt 9 x-c(J) 36 g(x) - gj(x) ffi i f(t)dt 9 x-c(J) Jn J is clearly decreasing in the interval get for x=c(lj) I. 10) we J , g(c(lj)) - gj(c(lj)) > (I-6)X , for x in the left half of I.. I ~m(lj) < m({x I g(x) > 89 ~=i = + n I m({x ] gj(x) < j=l - ~(I-~)}) 2 . llm(lj)< __ 2 11 j =~(--(T~-6)J ~ 2 ~i fJ f(t)dt+ j=l~~ X(I-~) J I.

### Integration in function spaces and some of its applications (Lezioni fermiane) by Mark Kac

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