By Ross Honsberger
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Additional resources for Mathematical Gems: The Dolciani Mathematical Expositions
M. let W ~ denote the set of all y E LP(J) such that y(n-1) exists and is locally absolutely continuous and y(k~) E LqJ(J), j = 1 , . . , m, y(~) E L r ( J ) . AJ) = w~(J) Then and the two norms Ilylll = Ilyilp + Ely(~)ii~ iiytl2 = Ilylip + ~ rly(kJ)ilqj + Iry(~)li~ j=l are equivalent. Proof. 58). 4 Let n be a positive integer, let J -= R or R +, 1 ~_ p ~_ oc. Let W ~ ( J ) denote the set of all functions y such that y C LP(J), y(n-1) is locally absolutely continuous and y(k) e LP(J), for k = 1 , .
Consider y(t) = sin(;rt/2), t C [0,2]. Then y C M1 when the interval [0,1] is replaced by [0,2]. 3. 20) where r denotes the gamma function. Proof. Consider y(t) = t ( 1 - t), tE[0,2]. Then y is in M2 on the interval [0,2]. A calculation yields f02 yP = 22V+lF2(p + 1)/F(2p + 2) = v/~F(p + 1)/F(p + 1 + 1/2) ly'l" = 2p+1/(; + 1) f3 lu"l" -- 2 '+1. 3. o We will show in Section 9 that k(p, R) is a continuous function of p. 20) is a "good" lower bound when p is large and 1 is a good lower bound when p is near 2.
Such inequalities are basic to the s t u d y of Sobolev i m b e d d i n g theorems and are i m p o r t a n t in the t h e o r y of p a r t i a l differential equations. See A d a m s  and F r i e d m a n . 3 is also known [Liable 1964], but our proof is different t h a n the one given by Ljubic. It is more elementary. We mention some interesting questions and problems. 1. 21) extend to the case when the three norms are different? In other words, for w h a t values of p, q, and r do we have Ily(k)llq < elly('~)ll~ + t((e)[[yHp?