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By Man Kam Kwong

Norm inequalities bearing on (i) a functionality and of its derivatives and (ii) a chain and of its adjustments are studied. distinctive straightforward proofs of easy inequalities are given. those are available to somebody with a history of complex calculus and a rudimentary wisdom of the Lp and lp areas. The classical inequalities linked to the names of Landau, Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and Caravetta, etc., are mentioned, in addition to their discrete analogues and weighted models. top constants and the lifestyles and nature of extremals are studied and plenty of open questions raised. an intensive checklist of references is equipped, together with a few of the giant Soviet literature in this subject.

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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 [1975] and F r i e d m a n [1969]. 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?

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