ARKIV FOR MATEMATIK Band 3 nr 19. Communicated 9 February 1955 by OT:ro FROSTMAN. On the uniform convexity ofl ~ and F

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1 ARKIV FOR MATEMATIK Band 3 nr 9 Communicated 9 February 955 by OT:ro FROSTMAN On the uniform convexity ofl ~ and F By OLOF HANNER CLARKSO~ defined in 936 the uniformly convex spaces [2]. The uniform convexit.~ asserts that there is a function ~ (~) of s > 0 such that x ] =, tl Y ] =, and [Ix - y II => e imply II ~ (x + y)ii =< - 6 (e), where x and y are elements of the space. CLARKSO~ proved that the well-known spaces L ~ and ~ are uniformly convex if p>. The purpose of this note is to give the best possible function (~(e) for these spaces, i.e. to find for each p>l and s>0 n'( reader the conditions ]] x I] =, II y II =, II ~- y II =>*. We need two inequalities, which are given in Theorem, formula (). I have been informed that the left-hand side inequality of this formula was proved by B~tmLI~O at a seminar in Uppsala in 945, but it does not seem to be in print. The right-hand side inequality is proved by CLARKSON ([2] p. 400) and BoAs ([] p. 305). We give here a reconstruction of BEVRLINO'S proof and also for completeness a simple proof of the other inequality. Let the functions in L ~ be defined over 0<t<l. The norm of x=x(t) is then given by I~~= flz(t)l~'dt. o In v the norm of x=@l, x~... ) is given by jlxr= t=l ix, t Theorem. For p > 2 the /ollowing inequalities hold (HxH-t-IlylD~+lllxll-ltylllp~llx+yllp+llx-yHV~2xllp+2y[I ~. () For < p < 2 these inequalities hold in the reverse sense. The equality sign holds [or L p [[or I p] in the le.{t-hand side of () q and only i/ x=o, or y=o, or there is a number a>o such that (x(t)-ay(t))(x(t)+ +ay(t))=o /or almost every t [such that (xi-ay~)(x~+ayi)=o /or every i], and in the right-hand side o/ () i/ and only i/x(t)y(t)=0/or almost every t[x~y~=o /or every i]

2 O. HAI'qNER, On the uniform convexity of L ~ and I T It is easy to show that for given Ilxll and II y ll each of these conditio~ for equality can be satisfied by suitable x and y. Hence the inequalities in Theorem give the maximum and the minimum of [[ x + y l[" + [I x- y I" for fixed II ~ll and Y [I- Bemark. For p=2 the three terms in () are equal for any x and y. This is the relationship II~+yll~ +ll~-yll~=2~ll~ + 2yll ~ well known in the theory of Hilbert spaces. Proof. A. The left-hand side of (). Let l<p<2 and consider LL We have to prove that flx(t)wy(t)[ v +]x(t)-y(t)[ ~ dt~(ll~ll+llyll) ~ I II~[I- I[ y Ill ~. 0 (2) Let us first show that it is sufficient to prove (2) for non-negative functions. Consider d = [~ + ~ I ~ + I ~ - ~ I ~, (3) where z and z 2 are complex numbers. Let I zj] and ]z 2[be fixed and let us calculate the minimum of d. If zt=o this minimum is 2[z2[ ~ and if z 2=0 this minimum is 2[ z [P. Take ]z [= a > 0 and z~= zla - b e ~, b > 0. Then p p d (~0) = [ a + b e' ~ ]~ + I a - b e' v ]v = (a* + b a b cos ~0) ~ + (a* + b 2-2 a b cos ~0) ~. The minimum of d (T) is (a + b)v+la-b [" and is reached for T=0, 7I. Thus IZI+Z2[P+IZI--Z2[P~([Z]+]Z2[)P+[[Z [--]Z2 [[P, (4) where equality holds if and only if z and z 2 have a real quotient or one of them is zero. Let x*(t)=ix(t)] and y* (t) ---- ] y (t) l. Put z l=x(t) and z~=y(t) in (4) and integrate. fix(t)+y(t)l'+tx(t)-y(t)lvdt~ f[x*(t)+y*(t)[v+lx*(t)-y*(t)lvdt. (5) o 0 Here equality holds if and only if for almost every t such that x (t)#0 and y (t) 4= 0, the quotient of x (t) and y (t) is real. Because of (5), since II x [[ = [[ x* ] and IIY II = [I Y* II, we only have to prove (2) for the non-negative functions x* (t) and y (t). Now introduce 240 ~(u, v)=(u~+v~f +lu~-v~[", u>o, v>o,

3 /krkiv FOR MATEMATIK. Bd 3 nr 9 and let /(t)=(x* (t)) ~ and g(t)=(y* (t)) ". Then (2) may be written f ~(f(t), g(t))dt>=~ (f/(t)dt, f g(t)dt). (6) We shall show below that ~ is convex. (6) is an immediate consequence of this fact. For the three integrals in (6) are the w-, u-, and v-coordinates of the center of gravity for the distribution of mass given by u = f (t), v = g (t), 0 ~_ t <= on the surface w=~ (u, v). Hence we only have to prove the convexity of $. We have (a) ~ (u, v) = $ (v, u), (b) $(0, 0)=0, (c) ~(tu, tv)=t~(u, v) for t~0. Thus w = ~ (u, v) is a cone with its center in the origin. The convexity of w = ~ (u, v) will therefore follow from the convexity of w-----~(u; ). But for the second derivative is (u) = ~ (~, )= (l + u~)" + - ~ I T,~ #" (u) = p- u~-. (I - u ~ [~-2 _ [ + u; ~-2), P which is strictly positive for every u > 0. For u= we have ~"= ~, but ~' is continuous. Thus ~ (u), and therefore also ~ (u, v), are convex. This proves (2). In order to get equality in (2) we must have equality in both (5) and (6). Since ~" is never 0, equality in (6) holds if and only if the point (/(t), g (t)) for almost every t lies on one single line through the origin in the u v-plane, i.e. /(t)=0 for almost every t, g (t)=0 for almost every t, or there is a positive number, say a T, such that for almost every t we have ] (t)=ap9 (t), i.e. x* (t)= =ay* (t). Combined with the condition for equality in (5) this gives the condition in Theorem. The case p> 2 is proved similarly. For these p-values d (T)reaches its maximum for ~=0, 7e and ~ is concave. The proof for l p is ana]ogous. B. The right-hand side of (). Let p>2 and consider L p. We have to prove that flx(t)+y(t)lp+]x(t)-y(t)l~dt>= 52x(t) lp+2y(t)l~dt. (7) 0 0 Introduce as before d by (3). Then the minimum of d(~) is 2 (a s +b2) ~ and is P reached for ~=+~. But, since t ~ is a convex function (a>0, b>0), P (a ~ + b2) ~ > a T + b Y. 24

4 O. HANNER, On the uniform convexity of L v and l v Hence I~, +z2 ~ + I z,- z. I~ ~ 2 zll ~ + 2~, ~, (8) where equality holds if and only if at least one of zj and z 2 is 0. Put in (8) Zl=X (t) and zp=y (t) and integrate. This proves (7). It also shows, that equality in (7) holds if and only if ] (t)g (t)=0 for almost every t. The remaining eases are proved similarly. Theorem where 0 < e < 2. Then 2. Let x and y be two elements o/ L ~ or o/ l p. Suppose that Ilxll=l, Ilyll=l, IIx-Yll~e, (9) where ~=d(e) is determined in the /ollowing way: when <p<2: l-s+.~ + e p l-d-} =2, when p=>2: 8=- For each e, we can chose x and y such that equality holds in (9). Proof. Put x* = ~ (x + y) and y* = (x - y). Thus A. l<p<2. Let x=x* +y* and y=x*-y*. (0) ~(u, v)=(u+v)'+lu-vl" for u_>_0, v>=0. Then ~ is symmetric in the variables u and v, and if one of these remains fixed, ~= is strictly increasing in the other one. The left-hand side inequality of Theorem may be written Hx+yli'+ilx-yll~>=~:(llxll, lull) Apply this formula on x* and y*. Then (u) 2>--~(ll~*ll, Ily*ll)~-~(llx*ll, ~)" (2) Since ~(,0)=2, we get ~(,2)> 2 and ~(0,2)<2. uniquely determined solution d of Thus there is a positive 242

5 ARKIV FOR M)~TEMATIK. Bd 3 nr 9 Hence, because of (2), The last two formulas prove formula (9) for < p< 2. To get equality in (9) we may take in L ~ x* (t)= -(~ for O<t<l, y* (t)= ~ e for 0<tg~, -- 2 for ~<t<l. Then IIx*=-~, Ily*ll= ~- Let and y be defined by (0). Hence Ilxll=llull. By Theorem (or by simple calculation) we have equality in () for these x* and y*, and we get In l ~ we may take x*=2-~ (-~, -~, 0, 0, 0,...), ( ) o,o... B. p > 2. We have by Theorem (and the remark following the theorem) Hence IIx* +y* II ~ + IIx* -y* II ~ >~ 2* II ~ + 2 II y* II ~- 2~-2~*"+2y*">=2x*"+2 ~, (3) Put Then I~*" < (-~)', which implies (9). To get equality in (9) we may take in L ~ x* (t) = 2 ~ ( - ~) for oztz~, =0 for ~<t<l, 243

6 o. HAr~EE, On the uniform convexity of L p and I p Then II~*ll=l-~, y* (t)=0 for 0<t<~, =2 ~ ~- for ½<t<l. 2 Ily*ll = ~. Let and y be defined by (0). Hence Ilxll=liYll- By Theorem (or by simple calculation) we have equahty in (3) for these x* and y*. Thus 8 p In l p we may take x*=(-~, 0, 0, 0... ), Remark. For fixed e lira ~ (E)= 0 and lim ~ (e)= 0. For small ~ > 0 we have p--}l p--} oo ~(e)= +-.- for l<p<2, (~)=~\~] +.~. for p=>2. REFERENCES. BOAS, R. P., Jr., Some uniformly convex spaces, Bull. Amer. Math. Soc. 46, (940). 2. CLA~KSON, JA~S A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40, (936). Tryck~ den 7 maj Uppsa]a 955. Almqvist & Wiksells Boktryckeri AB