Math. Proc. Cambridge Philos. Soc. Vol. 5, No., 994, (75-90) Prerit Ser. No. 33, 992, Math. Ist. Aarhus Maximal Iequalities of Kahae-Khitchie s Tye i Orlicz Saces GORAN PESKIR Several imal iequalities of Kahae-Khitchie s tye i certai Orlicz saces are roved. The method relies uo Lévy s iequality ad the techique established i [4] which is obtaied by Haageru-Youg-Stechki s best ossible costats i the classical Khitchie iequalities. Moreover by usig Dosker s ivariace ricile it is show that the umerical costat i the iequality deduced by the reseted method is ear to be as otimal as ossible: If f " i j i g is a Beroulli sequece, ad k k deotes the Orlicz orm iduced by the fuctio (x) = e x2 0 for x 2 R ; the the followig iequality is satisfied: a i " i r 8 5 X 2 =2 j a i j for all a;... ; a 2 R ad all, ad the best ossible umerical costat which ca take the lace of 8=5 belogs to the iterval ] 8=3 ; 8=5 ]. Shar estimates of that tye are also deduced for some other imal iequalities i Orlicz saces which are discovered i this aer.. Itroductio Let f " i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), let (x) = e x2 0 for x 2 R, ad let k k deote the gauge orm o (; F ; P ), that is: k X k = if f a > 0 j E [ (X=a) ] g wheever X is a real valued radom variable o (; F ; P ), with if ; =. The it is well-kow that the followig iequality is satisfied: X a i " i X C j a i j 2 =2 for all a;... ; a 2 R ad all, where C is a umerical costat. Moreover, it is recetly show i [4] that the best ossible umerical costat which ca take the lace of C i is equal to 8=3. Let us P i additio cosider real valued radom variables ;... ; defied o j (; F ; P ), ad let S j = i for j = ;... ;. The Lévy s iequality may be formulated as follows, see [8]: If for every j < the radom vector (;... ; ) has the same distributio AMS 980 subject classificatios. Primary 4A44, 4A50, 46E30, 60E5, 60G50. Secodary 44A0. Key words ad hrases: The gauge orm, Orlicz orm, Beroulli sequece, Khitchie iequality, Haageru-Youg-Stechki s costats, Lévy s iequality, the Wieer rocess, Dosker s ivariace ricile, subormal, symmetrizatio. gora@imf.au.dk
as the radom vector ( ;... ; j ; 0 j+ ;... ; 0 ), the we have: (2) P f j S j j > t g 2 P f j S j > t g for all t 0. I articular, if ;... ; are ideedet ad symmetric, the (2) is valid. I other words, the imum of a fiite umber of artial sums is stochastically cotrolled by the last artial sum. This ricile is ideed well-kow ad is established i may differet forms, see [8] for a uifiable geeral aroach ad [4] for some close related facts i the oerator theory. Cosequetly, havig (2) i mid oe might very aturally guess that the followig imal iequality corresodig to should be satisfied: (3) a i " i X D j a i j 2 =2 for all a ;... ; a 2 R ad all, where D is a umerical costat. Ideed, it is easily verified by usig the itegratio by arts formula ad the fact that 2 0 (x= 2) 0 (x) for x 0, that this iequality follows immediately from ad (2) with D = 2 C. However, after a quick look o (3) it is ot quite clear what is the best ossible value for D. Ad this aer is devoted to the study of these questios. I additio, we shall also establish the related imal iequalities ivolvig some other Orlicz orms, which corresod to those with a sigle artial sum give i [4]. Our mai aim is to fid the shar estimates for the best ossible costats aearig i these iequalities ad i that way to show that may of the deduced estimates themselves are ear to be as otimal as ossible. For istace, we shall rove (3) by establishig the estimate which will rovide to deduce that the best ossible umerical costat which ca take the lace of D i (3) belogs to the iterval ] 8=3 ; 8=5 ]. The method used i the roofs relies uo Lévy s iequality ad the techique established i [4] which is obtaied by Haageru-Youg-Stechki s best ossible costats i the classical Khitchie iequalities. The fial coclusios o the best ossible costats are rovided by usig Dosker s ivariace ricile. Moreover by usig the classical symmetrizatio techique the give iequalities will be exteded i a aroriate way from the Beroulli case to the case of more geeral real valued radom variables. 2. Prelimiary facts I this aer we work withi the followig Orlicz orms ad saces: kx k = if f a > 0 j E [ (X=a) ] g L (P ) = f X 2M(P ) j lim "#0 k"x k = 0 g kx k T = if f a > 0 j E [ (X=a) ] a g L T (P ) = f X 2M(P ) j lim "#0 kx k 7 = E [ (X) ] L 7 (P ) = f X 2M(P ) j lim "#0 k"x k T = 0 g k"x k 7 = 0 g 2
where M (P ) deotes the set of all real valued radom variables defied o a robability sace (; F; P ), ad (x) = e x2 0 for x 2 R. Recall that the Orlicz sace (L (P ) ; k k ) is called the gauge sace, ad the Orlicz orm k k is called the gauge orm. We remark that the quatity k X k T has bee emerged i the study [6]. Its iterest relies uo the fact that for more geeral fuctios, the ma k k eed ot to be a Frechet orm, but k k T is so. For more details see [6] (.7,8). For more iformatios i this directio i geeral we shall refer the reader to [6], [4] ad [6]. Let us i additio remid that a real valued radom variable X defied o a robability sace (; F; P ) is said to be subormal, if its Lalace trasform L X is domiated o the real lie by the Lalace trasform of some ormally distributed radom variable. I other words X is subormal, if there exist 2 R ad 2 > 0 such that: L X (t) ex ( t + 2 2 t 2 ) for all t 2 R. If is fulfilled with = 0 ad 2 =, the X is said to be a stadard subormal radom variable. A sequece of ( stadard ) subormal radom variables will be called a ( stadard ) subormal sequece. If X is a subormal radom variable satisfyig, the usig Markov s iequality oe ca easily obtai, see [4]: (2) P f X t g ex 0 (t0)2 2 2 for all t 0. I articular, if X (3) P f j t g 2 ex 0 (t0)2 2 2 is subormal ad symmetric satisfyig, the we get: for all t 0. Iequalities (2) ad (3) form a art of so-called classical Kahae-Khitchie iequalities for subormal radom variables, see [0] (.62). A fiite or ifiite sequece of ideedet ad idetically distributed radom variables " ; " 2 ;... takig values 6 with the same robability =2 is called a Beroulli sequece. Let f " i j i g be a Beroulli sequece, P P ad let f a i j i g be a sequece of real umbers. Put S = a i" i ad A = j a i j 2 for, the we have, see [4]: (4) f S j g is a stadard subormal sequece. A Let f i j i g be a sequece of ideedet ad idetically distributed radom variables defied P o a robability sace (; F; P ) with mea 0 ad variace 2 > 0. Let us ut j S j = i for j, ad let us defie a radom fuctio X :! ( C[0; ] ; k k) by: X (t;!) = S [t] (!) + (t 0 [t]) [t]+ (!) for t 2 [0; ] ;! 2 ad, where [t] deotes the iteger art of t. The Dosker s ivariace ricile states, see [] (.68): (5) X 0! W 3
as!, where W = f W t j t 2 [0; ] g is the Wieer rocess. Sice x 7! su t2[0;] j x(t)j is cotiuous o C[0; ], the we have: (6) su t2[0;] as!. Hece we easily get: j X (t) j 0! su t2[0;] j W t j (7) j S j j 0! su t2[0;] j W t j as!. Moreover we have: (8) P f su t2[0;] W t x g = 2 2 x 0 e 0t2 =2 dt for all x 0, or i other words: (9) P f su t2[0;] W t x g = 2 P f N x g = P f j N j x g for all x 0, where N N (0; ) is a stadard ormal radom variable. By the symmetry of W uder reflectio through zero hece we easily fid: (0) P f su j W t j x g < 2 P f j N j x g t2[0;] for all x 0. These facts are well-kow, see [] (.70-72), ad their use will be essetial i the fial coclusios o the best ossible costats i the sequel. 3. Maximal iequalities i the gauge sace L (P ) I this sectio we rove a imal iequality of Kahae-Khitchie s tye i the gauge sace (L (P ); k k ), see i theorem 3.. The method relies uo Lévy s iequality ad Haageru- Youg-Stechki s best ossible costats i the classical Khitchie iequalities, see [4]. By usig Dosker s ivariace ricile we show that the costat aearig i the deduced iequality is ear to be as otimal as ossible, see corollary 3.4. Usig the classical symmetrizatio techique we exted the give results from the Beroulli case to more geeral cases, see theorem 3.6. Theorem 3.. ( A imal iequality i the gauge sace ) Let f " i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), ad let k k deote the gauge orm o (; F ; P ). The the followig imal iequality is satisfied: a i " i for all a ;... ; a 2 R ad all. r 8 5 X j a i j 2 =2 4
Proof. Give a ;... ; a 2 R for some, we deote A P = ja ij 2 ad M = j S j j with S P j j = a i" i for j. The by the defiitio of the gauge orm k k it is eough to establish the followig iequality: (2) ex C 2 (M ) 2 dp 2 A with C = 8=5. I order to obtai a aroriate estimate for the left side i (2) we shall exad the itegrad ito Taylor s series, ad the we shall aly the classical Khitchie iequalities (2.) with (2.7) i [4]. First ote that by Lévy s iequality (.2) we have: (3) E(M ) 2k = P f (M ) 2k > t g dt = P f M > t =2k g dt 2 0 0 P f js j > t =2k g dt = 2 0 0 P f (S ) 2k > t g dt = 2 E(S ) 2k for all k. By the classical Khitchie iequalities (2.) with (2.7) i [4] we have: (4) E(S ) 2k K(2k; 2) (A ) k with K(2k; 2) = 2 k 0(k + =2)= for k. Sice 0(k + =2) = (2k 0 )!! =2 k where (2k 0 )!! = (2k 0 ) (2k 0 3)... 3 for k ad j 2=C 2 j <, the by (3) ad (4) we may coclude: (5) ex (M C 2 ) 2 dp = E A = X k=0 + 2 h ex C 2 A (M ) 2 i k! (C 2 A ) k E (M ) 2k + 2 X k= = + 2 = + 2 X k= X k= k! (C 2 A ) k 2 k 0(k + =2) (A ) k X k= k! 2 C 2 k 0(k + =2) (2k 0 )!! 2 k k! Thus (2) is satisfied ad the roof is comlete. 2 C 2 k = 2 = k! (C 2 A ) k E (S ) 2k 0 2 C 2 0=2 0 = 2 I order to show that the uer boud aearig i i theorem 3. is shar, we shall first tur out two relimiary results which are also of iterest i themselves. Lemma 3.2. Let f X i j i g be a sequece of ideedet idetically distributed radom variables 5
P defied o a robability sace (; F; P ) with fiite variace 2 > 0, let S = X i ad = (= ) (S 0ES ), ad let M = (= ) js j 0 ES j j for. Suose that f j g is symmetric stadard subormal sequece, that is, is symmetric ad we have L (t) ex (t 2 =2) for all t 2 R ad all. The for every C > 2 the sequece of radom variables f ex (M =C) 2 j g is uiformly itegrable. Proof. It might be roved i the same way as lemma 3.2 i [4] by usig Lévy s iequality (.2) ad Kahae-Khitchie s iequality for subormal radom variables (2.3). We shall leave the details to the reader. Proositio 3.3. Let f X i j i g be a sequece of ideedet idetically distributed radom variables defied o a robability sace (; F; P ) with fiite variace 2 > 0, let S P = X i ad = (= ) (S 0ES ) for, let W = f W t j t 2 [0; ] g be the Wieer rocess, ad let k k deote the gauge orm o (; F; P ). If f j g is a symmetric stadard subormal sequece, the we have: j S j 0ES j j 0! su j W t j t2[0;] as!, where 8=3 < k su t2[0;] j W t j k < 8=5. Proof. Statemet might be roved i exactly the same way as statemet i roositio 3.3 i [4] by usig (2.7) ad lemma 3.2. We shall leave the details to the reader. Let us i additio deote = su j W t2[0;] t j, ad ut C = k k. We must show that 8=3 < C < 8=5, see [] (.79-80). The first iequality follows easily by (2.9) ad the fact that for a stadard ormal radom variable N N (0; ) we have k N k = 8=3. For the secod iequality ut D = 8=5, the by (2.0) we have: 2dP (2) ex = P f ex (=D) 2 > t g dt D 0 = + < + 2 = + 2 = 2 ex P f > D P f jn j > D log t g dt log t g dt P f ex (N=D) 2 > t g dt 2dP 0 = 2 0 2 0=2 0 = 2. D D 2 Hece C < D follows easily by the defiitio of the gauge orm k k, ad the roof is comlete. For results ad roblems related to those reseted i lemma 3.2 ad roositio 3.3 we shall refer the reader to [4], see roblem 3.5. 6
Corollary 3.4. The best ossible umerical costat which ca take the lace of 8=5 i iequality i theorem 3. belogs to the iterval ] 8=3; 8=5 ]. Moreover the give costat is ot less tha k su t2[0;] j W t j k, where W = f W t j t 2 [0; ] g is the Wieer rocess. ( Accordig to the referee s remark, comuter calculatios, cosiderig a simle radom walk with stes, show that we have k M 60 k > :807 with M as i the roof of theorem 3. for. This gives a better lower boud tha 8=3 = :633 ). Proof. Let C be such a costat, the obviously C 8=5. Takig a =... = a = = i iequality i theorem 3. with 8=5 relaced by C we get: j " i j C beig valid for all. Accordig to (2.4) the sequece f (= ) P " i j g is a symmetric stadard subormal sequece. Thus lettig! i we may easily comlete the roof by usig the result of roositio 3.3. Cojecture 3.5. The best ossible umerical costat which ca take the lace of 8=5 i iequality i theorem 3. is equal to k su t2[0;] jw t j k, where W = f W t j t 2 [0; ] g is the Wieer rocess. Usig the classical symmetrizatio techique we shall exted the result of theorem 3. from the Beroulli case to the case of more geeral real valued radom variables. This rocedure has several stes ad the fial result may be stated as follows. Theorem 3.6. Let f X i j i g be a sequece of ideedet a.s. bouded real valued radom variables defied o a robability sace (; F; P ), let k k deote the gauge orm, ad let k k deote the usual su-orm o (; F; P ). The for every > 0 ad every we have: j (X i 0EX i ) j X C () where C () is give by: C () = 8 < : k X i 0EX i k 72=5 ; if 0 < 2 72=5 20 ; if 2 < < : Moreover, if X; X2;... are symmetric, the for every > 0 ad every we have: (2) j where D () is give by: X i j D () X k X i k = = 7
D () = 8 < : 8=5 ; if 0 < 2 8=5 2 0 ; if 2 < < : Fially, if f Y i j i g is a sequece of ideedet ad symmetric real valued radom variables defied o (; F ; P ), the we have: (3) for all. X j Y i j 2 =2 j Y i j r 8 5 Proof. Iequality (2) for = 2 might be roved i exactly the same way as iequality i theorem 3.6 i [4] by usig theorem 3. ad workig with the fuctio f from R 2 R ito R defied by: f (x;... ; x ; ;... ; ) = ex P C0 jx i j 2 j x i i j 2 for (x;... ; x ; ;... ; ) 2 R 2 R with C0 = 8=5. Moreover i the course of this roof iequality (3) could be also established i the same maer as i the roof of theorem 3.6 i [4]. We shall leave the details to the reader. Iequality (2) for 6= 2 follows easily from iequality (2) with = 2 by usig iequalities (2.20) ad (2.2) i [4]. Similarly, iequality for 6= 2 follows easily from iequality with = 2 by usig exactly the same argumet. Therefore to comlete the roof it is eough to deduce iequality with = 2. Let be give ad fixed, ut X = (X;... ; X ), ad let Y = (Y;... ; Y ) be a radom vector such that X ad Y are ideedet ad idetically distributed. There is o restrictio to assume that both X ad P Y are defied o P (; F; P ), ad that we have EX i = 0 for i = ;... ;. Put j S j = X j i ad T j = Y i for j = ;... ;, ad defie M = j S j j ad ^M = j S j 0T j j. We shall begi the roof by verifyig the followig iequality: (4) k M k k ^M k. Put C() = k ^M k (5) E, the (4) will be satisfied as far as we have the followig iequality: ex M C() 2 2. ito R by: 2 g(s;... ; s ; t;... ; t ) = ex C() js j 0t j j I order to establish (5) we shall defie a fuctio g from R 2 R for (s;... ; s ; t;... ; t ) 2 R 2 R. The for ay fixed (s;... ; s ) 2 R the fuctio (t;... ; t ) 7! g(s;... ; s ; t;... ; t ) is obviously covex from R ito R. Furthermore by our assumtios we have T j 2 L (P ) with ET j = 0 for j = ;... ;. Therefore by Fubii s theorem ad Jese s iequality we may obtai: 8
Eg(S;... ; S ; ET;... ; ET ) Eg(S;... ; S ; T;... ; T ). Sice ET j = 0 for j = ;... ;, the by the defiitio of the Orlicz orm k k we get: E ex M C() 2 = Eg(S;... ; S ; ET;... ; ET ) Eg(S;... ; S ; T;... ; T ) = E ex ^M C() 2 2. This fact roves (5), ad thus (4) follows. Sice X ad Y are ideedet ad idetically distributed, ad X;... ; X are by assumtio ideedet, the X 0Y = (X 0Y;... ; X 0Y ) is sig-symmetric. Therefore by (4) ad iequality (2) with = 2 we may coclude: j k M k k ^M k = (X i 0Y i ) j X 8=5 k X i 0Y i k 2 =2 These facts comlete the roof. 8=5 72=5 X 2 ( k X i k 2 + k Y i k 2 =2 ) X k X i k 2 =2 4. Maximal iequalities i the Orlicz sace L T (P ) I this sectio we rove several imal iequalities of Kahae-Khitchie s tye corresodig to those from the revious sectio but this time ivolvig the Orlicz orm k k T as defied i sectio 2, see i theorem 4., i theorem 4.6, i theorem 4.7, ad +(2) i theorem 4.9. The method relies uo the facts obtaied i the revious sectio ad the rocedure that is established i [4] for similar questios cocered with sigle artial sums. The estimates aearig through the whole sectio are shar ad ear to be as otimal as ossible. Desite the fact that the Orlicz orm k k T is ot homogeeous, see [4], we may begi by establishig the followig aalogue of iequality i theorem 3. for this orm. Theorem 4.. Let f "i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), ad let k k T deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The the followig imal iequality is satisfied: X j ai j 2 =2 j for all a;... ; a 2 R ad all, where C a i " i j T C is the uique root of the algebraic equatio 9
x 4 + 4x 3 0 2x 2 0 8x 0 8 = 0 for x > 2. Proof. Give a;... ; a 2 R for some, we deote P A = ja ij 2 ad M = j S j j with P j S j = a i" i for j. The by the defiitio of the Orlicz orm k k T it is eough to establish the followig iequality: (2) ex C 2 A (M ) 2 dp + C. I order to deduce (2) we shall use the estimate for its left side that is established i the roof of theorem 3.. Namely, by (5) i the roof of theorem 3. we have: (3) ex x 2 A (M ) 2 dp 2 0 2 x 2 0=2 0 for all x > 2. Put (x) = 2 (02=x 2 ) 0=2 0 ad (x) = + x for x > 2, the oe ca easily verify that there exists a uique umber C > 2 such that > o ] 2; C [, < o ] C; [, ad (C) = (C). The give C satisfies the followig algebraic equatio C 4 + 4C 3 0 2C 2 0 8C 0 8 = 0. Hece (2) follows directly by (3). These facts comlete the roof. Remark 4.2. We have see i the last roof that the umerical costat C aearig i i theorem 4. is the uique solutio of the algebraic equatio x 4 + 4x 3 0 2x 2 0 8x 0 8 = 0 for x > 2. By the well-kow criterio for ratioal solutios for algebraic equatios with ratioal coefficiets, see [2], each ratioal solutio of the above equatio belogs to the set f 6; 62; 64; 68 g. Hece oe ca easily verify that the above equatio has o ratioal solutios at all. Therefore the umerical costat C aearig i i theorem 4. is ot a ratioal umber. However oe ca easily verify that we have C = :68398945... 357=22 with 357=220C = 0:00096809.... Thus iequality i theorem 4. is satisfied with C = 357=22. Proositio 4.3. Let f X i j i g be a sequece of ideedet idetically distributed radom P variables defied o a robability sace (; F; P ) with fiite variace 2 > 0, let S = X i ad = (= ) (S 0ES ) for, let W = f W t j t 2 [0; ] g be the Wieer rocess, ad let k k T deote the Orlicz orm o (; F; P ) as defied i sectio 2. If f j g is a symmetric stadard subormal sequece, the we have: j S j 0ES j j 0! T su j W t j T t2[0;] as!. Moreover, let C s deote the umerical costat give by (2) i theorem 4. i [4], ad let C m deote the umerical costat aearig i i theorem 4. above. The we have: (2) C s < k su jw t j k T < C m t2[0;] where C s = :53865763..., ad Cm = :68398945.... 0
Proof. Statemet might be roved i exactly the same way as statemet i roositio 4.3 i [4] by usig (2.7) ad lemma 3.2. We shall leave the details to the reader. Let us i additio deote = su t2[0;] j W t j, ad ut C = k k T. We must show that C s < C < C m, see [] (.79-80). The first iequality follows easily by (2.9) ad the fact that C s = k N k T, where N N (0; ) is a stadard ormal radom variable, see roositio 4.3 i [4]. For the secod iequality ote that by (2) i the roof of roositio 3.3 ad the defiitio of C m ex C m 2 dp < 2 0 2 C 2 m 0=2 0 = + C m. Hece C < C m follows easily by the defiitio of the Orlicz orm k k T is comlete. Corollary 4.4. we have:, ad the roof The best ossible umerical costat which ca take the lace of C i iequality i theorem 4. belogs to the iterval ] C s ; C m ] with C s ad C m as i roositio 4.3. Moreover the give costat is ot less tha k su t2[0;] j W t j k T, where W = f W t j t 2 [0; ] g is the Wieer rocess. ( Agai, accordig to the referee s remark, comuter calculatios show that the lower boud ca be relaced by :625 ). Proof. It might be roved i exactly the same way as statemet i corollary 3.4 by usig (2.4) ad the result of roositio 4.3. We shall leave the details to the reader. Cojecture 4.5. The best ossible umerical costat which ca take the lace of C i iequality i theorem 4. is equal to k su t2[0;] j W t j k T, where W = f W t j t 2 [0; ] g is the Wieer rocess. Usig the classical symmetrizatio techique we shall exted the result of theorem 4. from the Beroulli case to the case of geeral symmetric real valued radom variables. Theorem 4.6. Let f X i j i g be a sequece of ideedet symmetric real valued radom variables defied o a robability sace (; F ; P ), ad let k k T deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The the followig imal iequality is satisfied: X 2 =2 j j X i j X i j T C where C is the umerical costat aearig i i theorem 4.. Proof. It might be roved by usig theorem 4. i exactly the same way as it has bee suggested for the roof of iequality (3) i the course of the roof of iequality (2) for = 2 i the begiig of the roof of theorem 3.6 with C0 = C. We shall leave the details to the reader. We shall cotiue our cosideratios by tryig to move the exressio ( P ja ij 2 ) =2 i
iequality i theorem 4. from the left side of that iequality to the right oe, see i theorem 3.. Let f "i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), ad let k k T deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The by (4.2) i [4] ad i theorem 4. we have: ai"i X T C jaij 2 =2 beig valid for all a;... ; a 2 R ad all for which P ja ij 2, where C is the umerical costat aearig i i theorem 4.. Moreover uttig a =... = a = = for ad usig P (2.7) oe ca easily verify that does ot hold i geeral. Note that i this case we have ja ij 2 = =! 0 for!. However by (2.4) i [4] ad i theorem 3. we easily fid: T 8=5 h X jaij 2 =2 X _ jaij 2 =4 i (2) ai"i beig valid for all a ;... ; a 2 R ad all. I articular, we have: (3) a i " i T X 8=5 ja i j 2 =4 for all a ;... ; a 2 R ad all P P for which ja ij 2. Moreover, give a P ;... ; a 2 R for some, ut A = ja ij 2 j ad M = j S j j with S j = a i" i for j. The by (5) i the roof of theorem 3. we have: (4) ex C 2 (M ) 2 dp = A 2 ex 0 2 A C 2 0=2 0 A C 2 A (M ) 2 dp wheever 2 A =C 2 <. Sice C > 2, the the last iequality is valid i the case where A. Moreover oe ca easily verify that we have: for all 2 x 3 + 8 C x2 + 8 C 2 0 C 2 x 0 4C 0 0 x, ad thus the followig iequality is satisfied: (5) 2 for all 0 2 C 2 x 0=2 0 + C x 0 x. By (4) ad (5) we get: ex C 2 A (M ) 2 dp + C 4 A. Hece by the defiitio of the Orlicz orm k k T (6) ai"i T C X 2 we may deduce: 2 =4 jaij
beig valid for all a;... ; a 2 R P ad all for which ja ij 2. Now by ad (6) we may coclude: (7) h X ai"i C T jaij 2 =2 X _ jaij 2 =4 i beig valid for all a;... ; a 2 R ad all. Our ext aim is to show that the exoet =4 i iequality (7) may be relaced by the exoet =3 i a otimal way, see sectio 4 i [4]. We roceed these cosideratios uder the same hyotheses ad otatio as above. Suose that A, let < < be give, ad let q be the cojugate exoet of, that is = + =q =. The 2 (A) =q =C 2 < ad therefore by (5) i the roof of theorem 3. we have: (8) ex C 2 (A ) = (M (A ) 2 ) =q dp = ex C 2 (M ) 2 dp A Let us defie: 2 q 3 = su8 q 2 j 0 2 (A ) =q C 2 0=2 0. 0 2 C 2 x=q 0=2 + C 2 x=20=2q ; 8x 2 [0; ] 9 ad let 3 be the cojugate exoet of q 3, that is = 3 + =q 3 =. The by (8), the defiitio of q 3, ad the defiitio of the Orlicz orm k k T we may easily coclude: (9) k M k T C (A ) =23. Furthermore it is easily verified that the iequality i the defiitio of q 3 for q = 4 is equivalet to the followig iequality: x 6 0 C2 2 x4 + 4 C x3 0 2C x + 4 C 2 0 beig valid for all 0 x, which is obviously ot satisfied. Thus q 3 < 4. Moreover it is easily verified that the iequality i the defiitio of q 3 for q = 3 is equivalet to the followig easily checkig iequality: 4 x 2 + C 0 C2 x + 4 2 C 2 0 2C 0 beig valid for all 0 x. Therefore 3 q 3 < 4. Fially it is easily verified that the iequality i the defiitio of q 3 for q = 3 + " with 0 < " < is equivalet to the followig iequality: x 2+" 0 C2 2 x+" + 4 C x+"=2 0 2C x "=2 + 4 C 2 0 for all 0 < x. However the left side of this iequality takes the value 4=C 2 > 0 at x = 0 for every 0 < " <. Therefore q 3 = 3, ad by (9) we get: (0) k M k T C (A) =3. I this way we have roved the followig theorem. 3
Theorem 4.7. Let f "i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), ad let k k T deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The the followig imal iequality is satisfied: ai"i h X T C jaij 2 =2 X _ jaij 2 =3 i for all a;... ; a 2 R ad all, where C is the umerical costat aearig i i theorem 4.. Proof. Straight forward by ad (0) above. Problem 4.8. What is the best ossible exoet that ca take the lace of =3 i iequality i theorem 4.7? Note that accordig to results deduced above we may coclude that this umber belogs to the iterval [=3; =2[. For more details i this directio see roblem 4.8 i [4]. Usig the classical symmetrizatio techique we shall exted the result of theorem 4.7 from the Beroulli case to the case of more geeral real valued radom variables. Agai this rocedure has several stes ad the fial result may be stated as follows. Theorem 4.9. Let f X i j i g be a sequece of ideedet a:s: bouded real valued radom variables defied o a robability sace (; F ; P ), let k k T deote the Orlicz orm o (; F ; P ) as defied i sectio 2, let k k deote the usual su-orm o (; F ; P ), ad let C be the umerical costat aearig i i theorem 4.. The for every > 0 ad all we have: h j (X i 0EX i ) j X = C T () k X i 0EX i k _ where C () is give by: C () = 8 < : _ X k X i 0EX i k 2C ; if 0 < 2 2C 20 ; if 2 < < : 2=3 i Moreover, if X; X2;... are symmetric, the for every > 0 ad all we have: (2) j h X i j X D T () k X i k = X _ k X i k 2=3 i 4
where D () is give by: D () = 8 < : C ; if 0 < 2 C 20 ; if 2 < < : Proof. It might be roved i exactly the same way as iequalities ad (2) i theorem 3.6 by usig theorem 4.7 ad iequalities (2.20) ad (2.2) i [4]. For this urose the followig iequality is tured out to be valid: k M k T k ^M k T with M ad ^M as i the roof of theorem 3.6. We shall leave the details to the reader. 5. Maximal iequalities i the Orlicz sace L 7 (P ) This sectio cosists of imal iequalities ivolvig the Orlicz orm k k 7 as defied i sectio 2. The method relies uo the facts obtaied i the revious two sectios. The deduced estimates are shar ad ear to be as otimal as ossible. Theorem 5.. Let f " i j i g be a Beroulli sequece defied o a robability sace (; F ; P ), ad let k k 7 deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The for every C > 2 the followig imal iequality is satisfied: C X ja i j 2 =2 j for all a ;... ; a 2 R ad all. a i " i j 7 2 C C 2 0 2 0 2 Proof. Give a ;... ; a 2 R for some, we deote A = P P ja ij 2 ad j M = j S j j with S j = a i" i for j. The by (5) i the roof of theorem 3. we have: X a i " i j = ex (M ) 2 dp 0 for all C > Theorem 5.2. C X ja i j 2 =2 2 j 7 0 2 C 2 0=2 0 2 = 2. This fact comletes the roof. C 2 A 2 C 0 2 C 2 0 2 Let f X i j i g be a sequece of ideedet symmetric real valued radom variables 5
defied o a robability sace (; F ; P ), ad let k k 7 deote the Orlicz orm o (; F ; P ) as defied i sectio 2. The for every C > 2 ad all the followig imal iequality is satisfied: C X i j 2 =2 j X i j 7 2 C 0 2 C 2 0 2 Proof. It might be roved i exactly the same way as iequality i theorem 5.2 i [4] by usig theorem 5. ad workig with the fuctio g defied by: 8 h g(x; C) = E ex C 2 X jx i j 2 j i9 x i " i j 2 for x = (x ;... ; x ) 2 R ad C > 2, where " ; " 2... is a Beroulli sequece. We shall leave the details to the reader. Remark 5.3. By (4.8) we may easily deduce the followig "dual" estimate which exteds the result of theorem 5.: C X ja i j 2 =2 j a i " i j 7 2 0 2 X =q 0=2 C 2 ja i j 2 0 2 beig valid for all a ;... a 2 R, all, ad all C > 0 for which 0 P ja ij 2 =2 < 0 C= 2 =0 with > ad = + =q =. Ad as i the roof of theorem 5.2 oe might be able to coclude that the followig iequality exteds iequality i theorem 5.2: If X ; X 2 ;... are ideedet symmetric a:s: bouded real valued radom variables, the we have: (2) C X i j 2 =2 j 2 0 2 X i j 7 C 2 X k X i k 2 =q 0=2 0 2 for all C > 0 ad all for which 0 P k X i k 2 =2 < 0 C= 2 =0 with > ad = + =q =. The give estimates are shar ad ear to be as otimal as ossible. Ackowledgmet. The author would like to thak his suervisor, Professor J. Hoffma- Jørgese, ad Professor S. E. Graverse for istructive discussios ad valuable commets, as well as the referee for useful remarks. 6
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