Maste Thesis Maste s Degee in Advanced Mathematics Weighted Ineualities fo the Hady Oeato Autho: Segi Aias Gacía Sueviso: D. F. Javie Soia de Diego Deatment: Matemàtiues i Infomàtica Bacelona, June 27
Contents Abstact Acknowledgments iii v Intoduction 2 Peliminay concets 3 2. Lebesgue Saces.............................. 3 2.2 Duality inciles and Minkowski s integal ineuality......... 5 2.3 Distibution functions, deceasing eaangements and Loentz saces 6 3 Classical Hady ineualities 3 3. The classical Hady s integal ineuality................ 3 3.2 Hady s ineualitiy with owe weights................. 5 4 Hady ineualities with weights 7 4. Fist esults involving weights...................... 7 4.2 Chaacteization of weighted Hady ineualities............ 2 4.3 Weighted Hady ineualities of (, ) tye............... 26 5 Hady ineualities fo monotone functions 43 5. Fist esults fo monotone functions................... 43 5.2 Maimal oeato and the Hady ineuality fo monotone functions. 46 5.3 Chaacteization of the weighted Hady ineuality fo monotone functions.................................... 5 5.4 Alications................................ 6 5.4. Nomability of Loentz saces.................. 6 5.4.2 Nomability of weak Loentz saces............... 69 5.4.3 Maimal oeato and nomability of Loentz saces..... 78 Bibliogahy 8 i
Abstact This oject evolves aound Hady s integal ineuality, oved by G. H. Hady in 925. This ineuality has been studied by a lage numbe of authos duing the twentieth centuy and has motivated some imotant lines of study which ae cuently active. We study the classical Hady s integal ineuality and its genealizations. We analyse some of the fist esults including weighted ineualities and ove the key theoem of B. Muckenhout, who chaacteized Hady s integal ineuality with weights fo the diagonal case in 972. Afte this fundamental esult, diffeent authos consideed the geneal contet and new chaacteizations aeaed until closing definitely the oblem in 2. Also we study Hady s integal ineuality in the cone of monotone functions. This oint of view is eally inteesting and has a lot of suising conseuences. Fo eamle, M. A. Aiño and B. Muckenhout ealized in 99 that Hady s ineuality in the cone of monotone functions is euivalent to the boundedness of the Hady- Littlewood maimal oeato between Loentz saces. Just afte E. Sawye oved that the classical Loentz sace Λ (w) is nomable if, and only if, Hady s integal ineuality in the cone of monotone functions is satisfied fo w. We study also the nomability of both saces Λ (w) and Λ, (w) in tems of the boundedness of the maimal oeato. iii
Acknowledgments To my maste thesis adviso, Pof. Javie Soia, fo his dedication and atience, and fo choosing such an inteesting toic. v
Chate Intoduction The oject that follows coesonds to the Maste Thesis in Mathematics of the Faculty of Mathematics of the Univesity of Bacelona by Segi Aias. This Maste Thesis is oganized in fou chates. In Chate 2 we give some eliminay concets and esults. Thoughout the oject we will need to wok with some secific saces such as weighted Lebesgue saces, weak-tye Lebesgue saces, classical Loentz saces o weak-tye Loentz saces, which ae defined in this chate. It is necessay to intoduce some concets in ode to define Loentz saces, and theefoe the distibution function and noninceasing eaangement function ae esented, as well as some imotant oeties. It is also esented some duality inciles needed in Chate 5. This oject evolves aound Hady s integal ineuality, discoveed in 925 by G. H. Hady. This ineuality has been studied by a lage numbe of authos duing the twentieth centuy and has motivated some imotant lines of study which ae cuently active. In Chate 3 we study the classical Hady s integal ineuality and its fist genealization, studied by Hady himself, including owe weights. We also study in both cases if the constants aeaing ae sha. In addition, Hady s ineuality can be genealized even moe. It can be studied with weights instead of owe weights, which at the same time can be consideed diffeent in both sides of the ineuality, as well as woking with diffeent indees. In Chate 4 we deal with this kind of oblem. We esent some of the fist esults consideing weights until eaching the key theoem of B. Muckenhout, who chaacteized Hady s integal ineuality with weights fo the diagonal case (when the indees ae the same) in 972. Afte this fundamental esult, the authos studied the geneal case and new esults aeaed until closing definitely the oblem in 2. Finally, in Chate 5 we study Hady s integal ineuality in the cone of monotone functions, that is, Hady s ineuality consideed just fo ositive deceasing functions. This oint of view is eally inteesting and has a lot of suising conseuences. M. A. Aiño and B. Muckenhout ealized in 99 that the boundedness of Hady s ineuality in the cone of monotone functions is euivalent to the boundedness of the Hady-Littlewood maimal oeato between Loentz saces. Thus, the chaacteization of the weighted Hady ineuality fo ositive deceasing functions
has been widely studied. M. A. Aiño and B. Muckenhout oved that the chaacteization of Hady s ineuality in the cone of monotone functions fo the diagonal case is not euivalent to the one given by B. Muckenhout in 972. Hee a new class of weights, called B, lay a cucial ole. But this aoach of Hady s ineuality has anothe suising conseuences. Fo eamle, E. Sawye ealized that the classical Loentz sace Λ (w) is nomable if, and only if, Hady s integal ineuality in the cone of monotone functions is satisfied fo the weight w and the inde. Actually, the nomability of the classical Loentz sace Λ (w) is euivalent to the weak boundedness of the Hady oeato in the cone of monotone functions (when ) and, similaly, the nomability of the weak-tye Loentz sace Λ, (w) is euivalent to Hady s integal ineuality fo ositive deceasing functions. Futhemoe, the weak boundedness of the Hady-Littlewood maimal oeato is euivalent to the weak boundedness of the Hady oeato in the cone of monotone functions. In the eecution of this oject, the chonological evolution of the study of Hady s ineuality has been etacted fom A. Kufne, L. Maliganda and L.-E. Peson s book [9] and A. Kufne and L.-E. Peson s book []. The main esults aeaing along this Maste Thesis have been consulted diectly fom the oiginal aticles. In some cases, the oofs have been etacted fom othe aticles, whose easoning wee easie to undestand. Also, some geneal concets have been studied in C. Bennett and R. Shaley s book [3]. 2
Chate 2 Peliminay concets In this chate we ae going to esent some definitions and esults that we will need in the subseuent chates. We stat defining the Lebesgue saces, weighted Lebesgue saces and weak-tye Lebesgue saces. The weighted Lebesgue saces will be vey imotant thoughout the oject and they will be used constantly. The weak-tye Lebesgue saces will aea in Chate 5. Net we esent some duality inciles that will be useful in Chate 5 and the Minkowski s integal ineuality. Finally, we define the concets of distibution function and noninceasing eaangement, we give some basic oeties and we define both the Loentz saces and the weak-tye Loentz saces. These mateials will be used as well in Chate 5. 2. Lebesgue Saces We define the classical Lebesgue saces. We esent the geneal definition fo an abitay measue sace although we will mostly wok with R n. Definition 2... Given a measue sace (X, µ) and <, we define the Lebesgue sace L as the set of measuable functions on X such that ( f : when < <, and such that X ) f() dµ() <, f : ess su f inf{a R : µ ({ X : f() > a}) } <. X The net definition states what we will conside by a weight function. Definition 2..2. Conside the inteval (a, b) R with a < b +. We say that w : (a, b) R is a weight function if it is measuable, w() a.e. (a, b) and it is locally integable on (a, b). 3
2.. Lebesgue Saces 4 Definition 2..3. Given a weight function w and a eal numbe < <, we define, fo measuable functions f : (a, b) R, ( b ) f,w : f() w()d. Poosition 2..4. f,w is a uasinom. a Poof. We obseve that f,w and λf,w λ f,w fo all λ R. In addition, as two eal numbes y satisfy the ineuality we have So using (2..), we conclude + y + y 2 + y (2 ) 2 ( + y ). (2..) ( b f + g,w a ) f() + g() w()d ( b 2 f() w()d + a 2 2 ( ( b a b ) f() w()d + 2 + ( f,w + g,w ). a ( b a g() w()d ( b a ) 2 ( f() + g() ) w()d ) ) ) g() w()d Finally, if f then f,w. Futhemoe, if f,w, we deduce f() a.e. (a, b). Definition 2..5. Fo a given weight function w and a eal numbe < <, we define the weighted Lebesgue sace L (a, b; w) as (classes of euivalent) functions f on (a, b) such that f,w <. Remak 2..6. We will usually wok with the inteval (, ) and then the weighted Lebesgue sace will be denoted by L (w). We finally define the weak-tye Lebesgue saces. Definition 2..7. Given < < and a weight w, we define the weak-tye Lebesgue sace as { } L, (w) f : f L, (w) su t t> w(s)ds <, {s: f(s) >t} whee f is a measuable function defined on R n o R +. Remak 2..8. We obseve that the. L, (v)-nom can be also witten as ( t ) f L, (v) su f(t) w(s)ds. t>
5 Chate 2. Peliminay concets 2.2 Duality inciles and Minkowski s integal ineuality We esent this well-known duality incile fo L saces. Poosition 2.2.. Let v and g be measuable functions in (, ) with v ositive. Then f()g()d ( ) su ( f f() v()d ) g() v() d, whee +. Poof. Fist of all we obseve that, by Hölde s ineuality, f()g()d f()v() g() v() d ( ) ( f() ) v()d g() v() d, fo any f. On the othe hand, if we conside f() sign(g())v() g(), then and Theefoe, ( f()g()d g() v() d ) f() v()d ( f()g()d ( ( f() v()d ) ) g() v() d. ) g() v() d. Anothe duality incile, which can be oved in a simila way, is the following. Poosition 2.2.2. Given a function f L, < <, we have that f()g() d R f n. g su g L,g As a conseuence of this last duality incile (Poosition 2.2.2), we can deduce the Minkowski s integal ineuality.
2.3. Distibution functions, deceasing eaangements and Loentz saces 6 Theoem 2.2.3. (Minkowski s Integal Ineuality) Fo, F (, y)dy F (, y) dy. R n R n Poof. The case coesonds to Fubini s Theoem. Fo the case we just notice that F (, y)dy su F (, y) dy. R R n n R n If < <, alying Fubini s Theoem and Poosition 2.2.2, we get F (, y)dy F (, y)dy g() R d n R n R n g su g L,g su g L,g R n R n su g L,g F (, y)g() ddy R n g F (, y)g() d R n dy g R n F (, y) dy. 2.3 Distibution functions, deceasing eaangements and Loentz saces We esent the notion of distibution function. wok on a measue sace (X, µ). Thoughout this section we will Definition 2.3.. Given a measuable function f, we define its distibution function as λ f (t) µ({ X : f() > t}), with t. Eamle 2.3.2. ([3, Eamle I..4]) Let us conside a ositive simle function f() n a j χ Ej (), j whee a >... > a n > and the sets E j ae aiwise disjoint with finite measue. Then, we have n λ f (t) m j χ [aj+,a j )(t), whee j m j j µ(e i ). i
7 Chate 2. Peliminay concets We give some oeties of the distibution function (cf. [3, Poosition II..3]). Poosition 2.3.3. The following oeties ae satisfied: (i) If g f µ a.e., then λ g λ f. (ii) If f lim inf n f n, then λ f lim inf n λ fn. Poof. Fo (i) we just notice that g() > t f() > t fo almost evey X. Hence λ g λ f. To ove (ii) we fi t > and we define the sets E : { X : f() > t} and E n : { X : f n () > t}, n N. Notice that µ(e) λ f (t) and µ(e n ) λ fn (t). Now by hyothesis we deduce that thee is an m N such that fo all n > m we have f() f n () and, theefoe, E m N n>m E n. We also notice that ( ) µ E n inf µ(e n) su inf µ(e n) : lim inf µ(e n) (2.3.) n>m n>m n n>m fo all m N. Finally, as n>m m N E n inceases with m, we conclude, by the Monotone Convegence Theoem and (2.3.), that ( ) µ(e) µ E n χ m n>m E ()dµ() n lim m m n>m X X χ n>m E n ()dµ() lim µ m ( n>m E n ) lim inf n µ(e n). A concet elated to the distibution function is the noninceasing eaangement function. Definition 2.3.4. Given a measuable function f, we define its noninceasing eaangement as f (t) inf{s > : λ f (s) t}, with t. We esent some oeties of noninceasing eaangement functions (cf. [3, Poosition I..7]). Poosition 2.3.5. Let f, f n and g be measuable functions. The following oeties ae satisfied: (i) f is deceasing.
2.3. Distibution functions, deceasing eaangements and Loentz saces 8 (ii) We have fo any t, t 2 >. (f + g) (t + t 2 ) f (t ) + g (t 2 ) (iii) If g f µ-a.e., then g f. (iv) λ f (f (t)) t wheneve f (t) <. (v) If f lim inf n f n, then f lim inf n f n. (vi) If f n f, then f n f. Poof. Poeties (i), (ii) and (iv) ae immediate conseuences of the definition. To see (iii) we notice that, by Poosition 2.3.3 (i) we have λ g λ f and hence {s > : λ f (s) t} {s > : λ g (s) t}, fom which the oety follows. Now, to ove (v) we obseve that λ f lim inf n λ fn by Poosition 2.3.3 (ii). So thee eists an m N such that fo all n > m we have λ f (t) λ fn (t) fo all t > and, theefoe, {s > : λ fn (s) t} {s > : λ f (s) t} fo all n > m, fom which the oety can be deduced. Finally we oof (vi). Fist of all, by (iii) we deduce that f n f and hence lim inf f n lim su fn f. (2.3.2) n n Futhemoe, as f lim inf n f n lim n f n f, by (v) we deduce that The esult follows by combining (2.3.2) and (2.3.3). f lim inf n f n. (2.3.3) Eamle 2.3.6. Fo the function defined in Eamle 2.3.2, we have f (t) n a j χ [mj,m j )(t), j whee we define m. The following oosition [3, Poosition II..8] is a well known oety of the noninceasing eaangement functions, which states that f and f have the same. -nom.
9 Chate 2. Peliminay concets Poosition 2.3.7. If < < and f is a measuable function, then X Futhemoe, when, f() dµ() t λ f (s)ds ess su f() f (). X f (t) dt. G. H. Hady and J. E. Littlewood ovided the following ineuality ([3, Theoem II.2.2]), which bounds the. -nom of the oduct f g of two functions by the. -nom of the oduct f g of thei noninceasing eaangement functions. Poosition 2.3.8. If f and g ae measuable functions, then X f()g() dµ() f (s)g (s)ds. We define the function f, the aveage of the noninceasing eaangement function f. Definition 2.3.9. Fo a measuable function f, we define f as with t >. f (t) t t f (s)ds, The oeato f f is subadditive ([3, Theoem II.3.4]). Poosition 2.3.. If f and g ae measuable functions, then fo all t >. (f + g) (t) f (t) + g (t) Now we esent a theoem [3, Theoem II.6.2] that ovides a useful eession fo f in tems of the Peete s K-functional (cf. [3, Definition V..]) fo L and L, K(t, f, (L, L )), which is a vey common oeato in Inteolation Theoy. Theoem 2.3.. If f is a measuable function, then t K(t, f, (L, L )) : inf { g L fg+h + t h L } f (s)ds tf (t), (2.3.4) fo all t >.
2.3. Distibution functions, deceasing eaangements and Loentz saces Poof. The last identity in (2.3.4) follows fom the definition of f. In ode to ove the second identity in (2.3.4), we fi a measuable function f and t >, denoting by a t the infimum on (2.3.4). We want to ove fist that t f (s)ds a t. (2.3.5) Let us assume that f L + L since, othewise, the infimum would be infinite and (2.3.5) would hold tivially. So eessing f as f g + h with g L and h L, and alying Poosition 2.3., we get t f (s)ds Now, by Poosition 2.3.7, we have and Theefoe, t t t g (s)ds g (s)ds + t h (s)ds. g (s)ds g L h (s)ds th () t ess su h() t h L. X t f (s)ds g L + t h L, and taking the infimum ove the eesentations of f, we get (2.3.5). In ode to ove the evese ineuality, a t t f (s)ds, we ae going to constuct functions g L and h L such that g L + t h L t f (s)ds. (2.3.6) Assuming that the ight hand side on (2.3.6) is finite (othewise thee is nothing to ove), Poosition 2.3.8 and Eamle 2.3.6 ovides that f is integable ove any subset of X of measue at most t. Now if we define E : { X : f() > f (t)} and we denote t µ(e), by Poosition 2.3.5 (iv) it must be t t, concluding that f is integable ove E. As a conseuence, the function is integable, as well as g() : ma{ f() f (t), } sign() h() : min{ f(), f (t)} sign()
Chate 2. Peliminay concets is a function in L bounded by f (t). We obseve that f g + h. Hence, by Poosition 2.3.8, t g L ( f() f (t))dµ() f() dµ µ(e)f (t) f (s)ds t f (t), and so E g L + t h L t E f (s)ds + (t t )f (t) t f (s)ds, whee in the last euality we have used that f (s) is constant when t s t with value f (t). Finally, we esent the classical Loentz saces and the weak-tye Loentz saces. Definition 2.3.2. Given a weight w in R + and < <, we define the weighted Loentz sace Λ (w) as the set of measuable functions satisfying ( f Λ(w) : f,w (f (t)) w(t)dµ(t) X ) <. Definition 2.3.3. Given a weight w in R + and < <, we define the weaktye Loentz sace as ( ) λf (t) Λ, (w) f : f Λ, (w) su t w(s)ds < t>, whee f is a measuable function defined on R n. Remak 2.3.4. We obseve that the. Λ, (w)-nom can be also witten as ( t ) f Λ, (w) su f (t) w(s)ds. t>
Chate 3 Classical Hady ineualities In this shot chate we esent the classical Hady ineuality, an histoical esult that G. H. Hady oved in 925, and its genealization with owe weights, also studied by G. H. Hady in 928. 3. The classical Hady s integal ineuality The following theoem is known as the Hady s integal ineuality. Theoem 3... If f(), > and f ()d is convegent, then ( ( ) f(t)dt) d f() d. Poof. Changing the vaiable (t s) we get ( ( ) f(t)dt) d ( ( ) f(s)ds) d. Using Minkowski s integal ineuality (Theoem 2.2.3) and changing again the vaiable u s, we conclude ( ( ) f(s)ds) d ( ) f(s) d ds ( f(u) du s ( ) ( ) ds ) f() d. Definition 3..2. We define the classical Hady oeato as Hf() : 3 f(t)dt.
3.. The classical Hady s integal ineuality 4 Remak 3..3. (See Eamle in [9]) The constant in Theoem 3.. is sha. Indeed, H is bounded on L (, ) and H L (, ) L (, ) but, actually, we ae going to ove that the nom of H is eactly. To see this, we can take the functions defined as f ɛ (t) t +ɛ χ (,a) (t) with < ɛ < ( a f ɛ and, changing vaiables ( s t ), ( ( Hf ɛ ( ( ) +ɛ d ([ ɛ ɛ ] a ) t +ɛ χ (,a) (t)dt) d ) (s) +ɛ χ a (, ) (s)ds d ( a ( (s) ds) +ɛ d + a and a >. Then ) a ɛ ) (ɛ) ( a (s) ds) +ɛ d ( ( ) a +ɛ ( ) a +ɛ + ɛ d + d) a + ɛ [ ] ɛ a (a +ɛ) [ ] ( + + ɛ) ɛ ( + ɛ) As a conseuence ( a ɛ + ɛ ɛ + ). a ) H L (, ) L (, ) Hf ɛ f ɛ and then H L (, ) L (, ). ( + ɛ + ɛ ) ɛ Remak 3..4. Theoem 3.. is not tue fo. If it would eist a constant C such that ( ) f(t)dt d C f()d, we could choose f() χ (,) () and we would get a contadiction, since { if < <, Hf() if,
5 Chate 3. Classical Hady ineualities and d + d (Hf)()d C whee the left hand side of the ineuality is not convegent. f()d C, 3.2 Hady s ineualitiy with owe weights The following theoem (cf. [9, Theoem 2]) genealizes the classical Hady s integal ineuality by intoducing owe weights α. Theoem 3.2.. If f is a ositive function, and α <, then ( ( ) f(t)dt) α d f () α d. α Poof. Changing the vaiable (t s) we get ( ( ) f(t)dt) α d ( ( ( ( ) f(s)ds) α d ) ) f(s) α ds d. Now, alying Minkowski s integal ineuality (cf. Theoem 2.2.3) and changing the vaiable (u s), we conclude ( ( ) ) f(s) α ds d ( ) f (s) α d ds ( ( y ) α f dy (y) s s ( ) ( s (α+) ds α ( f (y)y α dy ) ds ) f (y)y α dy ). Remak 3.2.2. The constant in Theoem 3.2. is sha. Indeed, oceeding α in the same way as in Remak 3..3, we can conside the functions with a >. Then ( a f ɛ, α f ɛ (t) t α +ɛ χ (,a) (t), ) α+ɛ α d ([ ɛ ɛ ] a ) a ɛ (ɛ)
3.2. Hady s ineualitiy with owe weights 6 and, changing vaiables ( s ) t, ( ( Hf ɛ, α t α +ɛ χ (,a) (t)dt) α d ( ( ) (s) α +ɛ χ a (, ) (s)ds α d Finally, ( a ( (s) α ds) +ɛ α d + a ) ) ( a (s) α ds) +ɛ α d ( ( ) a α +ɛ ( α + ɛ α a α +ɛ d + a α + ɛ ( [ ] ɛ a [ a α+ɛ α + ( α + + ɛ) ɛ ( α + ɛ) α + ( ) a ɛ α + ɛ ɛ +. α H L (, ; α ) L (, ; α ) Hf ɛ, α f ɛ, α ɛ α and hence the constant in Theoem 3.2. is sha. ( α + ɛ ) α d) ] a + ɛ α Remak 3.2.3. (cf. [9, Theoem 2]) Conditions and α < ae essential in Theoem 3.2.. Indeed, if we conside the functions f a () χ (a,a+) () with a >, then < a, a Hf() a a +, a +, and, if α, Hf a () α d a+ Hf a () α d whee this last integal is not convegent since α. On the othe hand, if < < and α, then Hf a () α d f a () α d a+ α d a+ α d a+ α d (a + ) α and Theoem 3.2. does not hold. a α a+ α d, (a + ) α + (a + ) α a, ) ) )
Chate 4 Hady ineualities with weights Hady s integal ineuality can be genealized by consideing diffeent weights (instead of just owe weights) in both sides of the ineuality. In Section 4. we esent some of the fist esults consideing the Hady ineuality with weights. Seveal authos consideed this kind of ineualities until B. Muckenhout gave the fist esult chaacteizing comletely the Hady s integal ineuality with weights in the diagonal case ( ). In Section 4.2 we study this esult. Finally, anothe lage amount of authos studied the geneal case ( ), which is esented in Section 4.3. 4. Fist esults involving weights Definition 4... A weighted Hady ineuality is an ineuality of the fom ( ( ) f(t)dt) u()d C ( whee u and v ae weights, and, ae ositive eal numbes. ) f() v()d, The fist esult chaacteizing comletely the weighted Hady ineuality fo the case v() and 2 aeaed in 958 and it is due to Kac-Kein (cf. [9, Theoem 3]). Theoem 4..2. The ineuality ( f(t)dt) 2 u()d C holds fo evey f L 2 (, ) if, and only if, the suemum is finite. A : su > 7 u() d 2 f() 2 d (4..)
4.. Fist esults involving weights 8 Remak 4..3. Euivalently, we can wite u() instead of u() and then (4..) 2 becomes ( 2 f(t)dt) u()d C f() 2 d (4..2) and A is now A : su > u()d. Poof. We ae going to wok with the changes done in Remak 4..3. Fist we assume ineuality (4..2) holds fo evey f L 2 (, ). We conside, fo, h >, the functions f h () : 2 χ(,] () 2 h χ [+h,+2h) (), which ae in L 2 (, ). Now, we notice that f h () 2 d d + +2h +h h 2 d + h and f h (t)dt 2 χ(,] (t)dt 2 χ(,] () + 2 χ(, ) () 2 h χ [+h,+2h) (t)dt 2 h ( h)χ (+h,+2h] () 2 χ(+2h, ) () 2 χ(,] () + 2 χ(,+h] () ( ) + 2 2 h ( h) χ (+h,+2h] (). Then + f h (t)dt +h +h We conclude 2 u()d 2 u()d u()d u()d. +h +2h +h u()d C ( ) 2 2 h ( h) χ (+h,+2h] () 2 u()d and hence, making h, we obtain f h () 2 d C u()d C, ( + ), h
9 Chate 4. Hady ineualities with weights which imlies A su u()d <. > Now we assume that A <. Given the integal ( 2 f(t)dt) u()d, we ae going to integate by ats with dv u(). Fist we notice that, by the Lebesgue Diffeentiation Theoem, if v u(t)dt then dv u() a.e.. So we have ( [ 2 ( ) 2 ( ) f(t)dt) ] u()d f(t)dt u(t)dt 2 + 2 ( ) ( ) f(t)dt f() u(t)dt d ) ( ) f(t)dt f() u(t)dt d. Now, alying Hölde s ineuality and Theoem 3.., we get ( ) ( ) 2 f(t)dt f() u(t)dt d ( ( ) 2 f(t)dt) ( 2 2 2 d f() ( u(t)dt) 2 d ( ) 2 ( 2A ( ( 4A f() 2 d f(t)dt) 2 d ) 2 ( ( ) f() 2 2 d ) f() 2 2 d 4A f() 2 d ) 2 Definition 4..4. Fo a given weight u, we define the modified Hady oeato as H u f() : u() f(t)u(t)dt. The following theoem was oved by N. Levinson in 964 (cf. [3, Theoem 4]). Theoem 4..5. Let > and f. Let () be an absolutely continuous function defined fo >. Assume + (4..3) λ fo almost evey > and fo some λ >. Then H f() d λ f() d.
4.. Fist esults involving weights 2 Poof. We conside < a < b < and let h,a f() : () a (t)f(t)dt. Then, defining H,a f() h,af() and integating by ats (with u (h,a f) and dv ), we obtain b a H,a f() d b + a b Now we notice that the tem [ (h,af) d a + (h,af()) [ ] b + (h,a f)() ( (h,a f()) f() (h,a f()) () () ] b a b (h,af(b)) a ) d. is negative, since >, h,a f and b >. Hence, b b ( ) (H,a f()) + d (h,a f()) f() (h,a f()) () d, () a o euivalently, b ( a a ) + () (H,a f()) d () Now, using (4..3) and Hölde s ineuality, we get that is, λ b If we take c > a then a a b ( b ) (H,a f()) d (H,a f()) d b c b a (H,a f()) d ( b (H,a f()) d a a ) (H,a f()) d λ f() d. b a (H,af ()) f()d. ( b a ( f() d (H,a f()) d λ f() d, ) f() d ), and, by the Dominated Convegence Theoem, making a we get b c (H f()) d λ f() d
2 Chate 4. Hady ineualities with weights fo all c, b >. Finally, letting b and c, (H f()) d λ f() d. Coollay 4..6. Let > and f. function defined fo >. Assume Let u() be an absolutely continuous u u λ (4..4) fo almost evey > and fo some λ >. Then ( Poof. We define () Hf() u()d λ f() u()d. u() ) and then (4..4) becomes + () () λ, since () () u () u(). Now, we eess f() ()g() fo the suitable g() and we aly Theoem 4..5 to g, obtaining ( ) f(t)dt u()d λ f() u()d. 4.2 Chaacteization of weighted Hady ineualities Some authos like G. Tomaselli, G. Talenti o M. Atola woked in the weighted case, giving some imotant esults. Howeve, B. Muckenhout was the fist one who gave the comlete chaacteization of this kind of ineualities (cf. [5, Theoem ]). Theoem 4.2.. Let. Fo a given two weights u and v, we can find a finite constant C > such that ( f(t)dt ) u()d C ( ) f() v()d (4.2.)
4.2. Chaacteization of weighted Hady ineualities 22 if, and only if, ( B su > ) u() ( d ) v() d <. In addition, if C is the sha constant fo (4.2.), then B C ( ) B, if < <, and C B if o. Remak 4.2.2. Euivalently, we can call U() u(), V () v() (4.2.) becomes ( U() f(t)dt and B is now ) d C ( ) V ()f() d ( ) ( B su U() ) d V () d <. > This is the notation used by Muckenhout in his ae [5]. and then (4.2.2) Poof. We ae going to use the notations given in Remak 4.2.2. We want to ove fist that ( ) ( U() ) f(t)dt d ( ) B V ()f() d fo < <, which would imly that C ( ) B. We define h() ( V (t) dt ) and, by Hölde s ineuality, we have U() f(t)dt d U() f(t) V (t)h(t) V (t)h(t) dt d ( ) ( ) U() f(t)v (t)h(t) dt V (s)h(s) ds d. Now, alying Fubini s theoem we get ) V (s)h(s) ds dt d ( U() f(t)v (t)h(t) ( ( ) f(t)v (t)h(t) U() V (s)h(s) ds) d dt. t
23 Chate 4. Hady ineualities with weights We want to bound ( ( ) f(t)v (t)h(t) U() V (s)h(s) ds) d dt. (4.2.3) Fist we notice that, by hyothesis, and ( t ( t ) ( V (s) ) ds B U(s) ds, ) U(s) ds B ( t Hence ( ( ( s V (s)h(s) ds) V (s) Then, (4.2.3) is bounded by and hence ([ ( s ) V (s) ds B h(t). ) ] V (t) dt ( ( ) V (s) ds ) ) V (t) dt ds s ) ( (B ) U() t ( (B ) [ U(s) ds ( (B ) U(s) ds B ( ) h(t). t ) ) ) U(s) ds f(t)v (t)h(t) B ( ) h(t) dt B ( ) f(t)v (t) dt, ) ] t U() f(t)dt d B ( ) f(t)v (t) dt. Now we want to see that C B when o. In the case we have, alying Tonelli s theoem, U() f(t)dt d U() f(t) dt d f(t) U() d dt, t d
4.2. Chaacteization of weighted Hady ineualities 24 and since V (t) U() d B, we conclude that t On the othe hand, when, we have ( U() f(t)dt d B f(t) V (t) dt. B su > ess su U() > ) ( ) V () d, and then U() f(t)dt U() V (t) f(t) V (t) dt ( ) ess su f(t)v (t) t> ( U() ) B ess su f(t)v (t). t> V (t) dt Finally, we want to see that B C. Fist we notice that fo non negative f and fo > we have ( U() f(t)dt ) d and taking V ()χ (,) () as V (), (4.2.) becomes ( ) U() d Now we ae going to ove that ( ( U() f(t)dt ( f(t)dt C ) d, ) V ()f() d. (4.2.4) ) ( U() ) d V () d C, (4.2.5) which would imly B C. In the case and < V () d <, taking f() V () we get, fom (4.2.4), that ( and (4.2.5) holds. ) ( U() ) ( d V () d C ( C ) V () d ) V () d,
25 Chate 4. Hady ineualities with weights When and < ess su << <, we can take f the chaacteistic V () function of the set and, by (4.2.4), we get Theefoe, ( { : ) U() d ( V () + ess su n << }, V () f(t)dt C V ()f() d C n + ess su << V () ) ( ) U() d + ess su C, n << V () f(t)dt. and letting n we get (4.2.5). Finally, we obseve that if V () d then (4.2.5) is obvious. If we can conside the functions V () d, f n () V () χ An (), whee A n : { > : n V () n }. Then, (4.2.4) becomes ( that is, ) ( U() ) ( ) d V () χ An ()d C V () χ An ()d, ( ) U() d C ( ) V () χ An ()d. Then, by the Monotone Convegence Theoem, if we let n, we get and so (4.2.5) holds. ( ) U() d, Remaks 4.2.3. The conditions given in Theoem 4..2 o Theoem 3.2. ae euivalent to the condition given in Theoem 4.2..
4.3. Weighted Hady ineualities of (, ) tye 26. If the take, in Theoem 4.2., 2 and v(), then ( B su > which is finite if, and only if, ) u() ( d 2 ) d 2 su > ( 2 su > ) u() d 2 ( ) u() d 2, 2 is finite. But this last condition is eactly the one aeaing in Theoem 4..2. 2. If we take, in Theoem 4.2., u() v() α, fo some α eal, we have But B is finite if, and only if, ( ) ( B su α ) d α d. > α < and α >, that is, α <, which is eactly the condition in Theoem 3.2.. 4.3 Weighted Hady ineualities of (, ) tye Now we focus in the geneal case with two diffeent indees and. The following esult (cf. [4, Theoem ]) chaacteizes the weighted Hady ineuality (, ) fo and it is due to J. S. Badley. It is an etension of Muckenhout s diagonal case (cf. Theoem 4.2.). Theoem 4.3.. Let. Given two weights u and v, we can find a finite constant C > such that ( ( ) f(t)dt) u()d C ( ) f() v()d (4.3.) fo all ositive function f if, and only if, ( B su > ) u() ( ) d v() d <. In addition, if C is the sha constant fo (4.3.), then B C ( ) B, if < < <, and C B if o.
27 Chate 4. Hady ineualities with weights Remak 4.3.2. Euivalently, we can call U() u(), V () v() (4.3.) becomes and then ( ( U() ) f(t)dt) d C ( ) (V ()f()) d (4.3.2) and B is now ( ) ( B su U() ) d V () d <. (4.3.3) > Poof. The oof of (4.3.2) (4.3.3), that is, B C, is analogous to the one in Theoem 4.2.. We ae going to ove, then, the imlication (4.3.3) (4.3.2), o moe ecisely, C ( ) B. Fist we assume < < < and we define ( t h(t) ) V (s) ds. Now, alying Hölde s ineuality and Minkowski s integal ineuality (cf. Theoem 2.2.3), we get I ( U() f(t) V (t)h(t) ) V (t)h(t) dt d ( U() (f(t)v (t)h(t)) χ {<t<} (t, )dt ) ( (V (s)h(s)) ds ( ( U() (f(t)v (t)h(t)) χ {<t<} (t, ) (V (s)h(s)) ds ( ( ) (f(t)v (t)h(t)) U() (V (s)h(s)) ds t ) d ) d ) ) dt dt. d To efom the innemost integal, we obseve that, since by hyothesis ( ) ( V (u) ) du B U(s) ds,
4.3. Weighted Hady ineualities of (, ) tye 28 we have that ( ) ( (V (s)h(s)) ds ([ ( V (s) ( s ( s ( ( B) ( ) ) V (u) du ds ) ] V (u) du ) ) V (u) du s ) U(s) ds. ) Theefoe, I ( B) ( ( (f(t)v (t)h(t)) U() t ) ) U(s) ds d dt. Again, to efom the inne integal we obseve that, since by hyothesis we have that ( ( ( U() t t ) U(s) ds B ( t ) ) U(s) ds d ) V () d Bh(t), ([ ( ) ] U(s) ds ( U(s) ds Bh(t). t ) t ) Theefoe, and hence ( ) I ( ) B (f(t)v (t)) dt, I ( ) () B. Now, fo, we fist obseve that, by hyothesis, ( t U() d ) V (s) B
29 Chate 4. Hady ineualities with weights fo all < s t, and hence, by Minkowski s integal ineuality (cf. Theoem 2.2.3), we get ( ( U() ) f(t)dt) d ( ( B ( f(t) t f(t)v (t)dt. ) U()f(t)χ {<t<} (t, )dt) d ) U() d dt Finally, if, we aly Hölde s ineuality to get U()f(t)dt U() ( B ess su U(s) s ( f(t)v (t)dt V (t) ) ( dt V (t) (f(t)v (t)) dt ). ) ( (f(t)v (t)) dt ) Fo the case < <, Q. Lai oved (cf. [2, Theoem ]) that the weighted Hady ineuality cannot be tue, ecet fo tivial situations. His oof is based on a esult due to M. M. Day (cf. [8]), which states that the dual sace of a Lebesgue sace is the zeo sace when < <, unde some hyothesis on the measue. Theoem 4.3.3. (Day) Let µ be a nonatomic measue and let < <. Then evey linea and continuous functional, T : L (µ) R, must be zeo. Theoem 4.3.4. (Lai) Given < <, if < < then thee is no constant C > such that ( ( ) f(y)dy) w()d C ( ) f() v()d (4.3.4) fo all f L (v), ecet fo the tivial case w() a.e. (, ). Remak 4.3.5. If we conside w() as w() in Theoem 4.3.4, then (4.3.4) becomes ( ( ) f(y)dy) w()d C ( ) f() v()d. (4.3.5) We ove now Theoem 4.3.4 with the notations of Remak 4.3.5.
4.3. Weighted Hady ineualities of (, ) tye 3 Poof. We define c inf {τ > : w() a.e. (τ, )}. If c, then w() a.e. (, ) and (4.3.5) tivially holds. We assume < c < and hence w() a.e. (c, ), and c b w()d > (4.3.6) fo all < b < c. Now we obseve that it has to be v() > a.e. (, c). Indeed, if we assume that the set E { < < c : v() } has ositive measue, then we get a contadiction. It would eist an (, c) such that (, ) E > and, if f() χ E (), f(y)dy > fo all. But we would also have that so if (4.3.5) holds, we would get f(y) v(y)dy, ( f(y)dy) w()d, and necessaily w() a.e. (, ), in contadiction with the definition of c. Given < <, we define N,v(g) su f()g()v()d, f,v and we claim that, fo any inteval (a, b) with a < b < c, N,v ( ) χ(a,b) ( ) su v( ) f,v b a f()d <. (4.3.7) Othewise, thee would eist a seuence (f n ) n of ositive functions such that f n,v and b a f n ()d > n,
3 Chate 4. Hady ineualities with weights fo all n N. But then, if (4.3.5) holds, and since f n,v ( f n () v()d ) fo all n N, we have ( C C ( c ( b ( c > n w()d b ) f n () v()d ) f n (y)dy) w()d ) ( ( fo all n N. Howeve, this is only ossible if c b ) f n (y)dy) w()d ( c ( b f n (y)dy) w()d b w()d, in contadiction with (4.3.6). We obseve now that the condition (4.3.7) imlies that the oeato defined by T : L (v) R f b a f()d is linea and bounded. Theefoe, by Theoem 4.3.3, it must be T, that is, b a f()d fo all f L (v). Obviously, this is false since, fo eamle, we can conside the function f() χ (a,b) (), which is in L (v) but b a f()d >. So (4.3.5) cannot be satisfied. Finally, if c, we have that fo all b > thee eists a eal numbe d > b such that d w()d > and the agument of the case < c < can be alied. b When < the situation is slightly moe comlicated and diffeent aguments ae needed. W. Mazya and L. Rozin chaacteized the case < < in the eighties, G. Sinnamon chaacteized (987) the case < < < < and the case < < is due to G. Sinnamon and V. D. Steanov (996). G. Sinnamon and V. D. Steanov ublished a ae [8] whee they gave the oof of the < < case and a moe elementay oof of the case < <, < < (cf. [8, Theoem 2.4]). We give hee the oofs esented in this ae. Fist, in ode to deal with the case < <, < < we need some evious esults. a )
4.3. Weighted Hady ineualities of (, ) tye 32 Lemma 4.3.6. Assume α, β and γ ae ositive functions and γ is inceasing and absolutely continuous. If α(t)dt β(t)dt (4.3.8) fo all >, then γ(t)α(t)dt γ(t)β(t)dt. Poof. It is known that fo any ositive, inceasing and absolutely continuous function γ, we can wite γ(t) γ() + Hence, alying (4.3.9), Tonelli s Theoem and (4.3.8), we get t γ (s)ds. (4.3.9) γ(t)α(t)dt γ() γ() γ() α(t)dt + α(t)dt + β(t)dt + t s s γ(t)β(t)dt. γ (s)ds α(t)dt α(t)dt γ (s)ds β(t)dt γ (s)ds Poosition 4.3.7. Let u, b and F be ositive functions such that F is inceasing and absolutely continuous. We assume also that b satisfies b(t)dt < > but b(t)dt. (4.3.) If < < < and then ( ) F () u()d Poof. We define U() : ( ) ( ( ) ( ) ) u(t)dt b(t)dt b()d ( ) F () b()d. u(t)dt and B() : b(t)dt. We eess u as
33 Chate 4. Hady ineualities with weights u(t) u(t) + ( ( ( t and alying Hölde s ineuality with and, we get ) F () u()d ( ( t ) ( U(s) t B(s) b(s)ds F (t) ) ) U(s) B(s) b(s)ds u(t)dt ( ( t F (t) ) ) U(s) B(s) b(s)ds u(t)dt Now, alying Tonelli s Theoem we obtain ( t ) I U(s) B(s) b(s) u(t) ds dt ( U(s) B(s) b(s) s ) u(t)dtds ) ) U(s) B(s) b(s)ds u(t)dt ( I II. U(s) B(s) b(s)ds ). (4.3.) In ode to bound II, we want to aly Lemma 4.3.6 with ( t ) ( ) α(t) U(s) B(s) b(s)ds u(t), β(t) b(t) and γ(t) F (t). By hyothesis, γ(t) has to be inceasing and it emains to check that α β ( ) t fo all >. To see this, we obseve that as U(s) B(s) b(s)ds and U ae deceasing, we have ( t ) α(t)dt U(s) B(s) b(s)ds u(t)dt ( ( ( ) U(s) B(s) b(s)ds ) B(s) b(s)ds U() ) B(s) b(s)ds, and integation togethe with (4.3.) yields ( ) B(s) b(s)ds ( ( ) ) b(t)dt u(t)dt u(t)dt β(t)dt.
4.3. Weighted Hady ineualities of (, ) tye 34 Finally, alying (4.3.) and Lemma 4.3.6 we conclude ( ) ( F () ) u()d U(s) B() b()d ( ( ) ) F () b()d. Poosition 4.3.8. Suose that < < and let w be a ositive function such that Then ( ( with C. w(t)dt < > but ) ( ) f(t)dt w(t)dt) w()d w()d. (4.3.2) ( C ) f() w() d, (4.3.3) Poof. Accoding to Theoem 4.2., the weighted Hady ineuality (4.3.3) holds if, and only if, ( ( ) ( ) B su w(t)dt) w()d w()d < > and, moeove, C ( ) B. But alying (4.3.2) and taking into account that >, we have ( ( ) w(t)dt) w()d and hence In addition, ( ( ( ( w(t)dt) ( ) ( ) ( w(t)dt B ( ) <. ) )) w(t)dt ) ( ( ) C ( ). ) w(t)dt,
35 Chate 4. Hady ineualities with weights Theoem 4.3.9. Assume < <, < < and. Let u and v be two weights. Then, the weighted Hady ineuality ( ( ) f(t)dt) u()d C ( ) f() v()d (4.3.4) holds fo all ositive function f if, and only if, ( ( D : ) ( v(t) dt ) u(t) dt t Moeove, if C is the sha constant fo (4.3.4), then ( ) D C ) u() d ( ) D. <. (4.3.5) Remak 4.3.. As usual, if we conside u() as u() then (4.3.4) becomes ( ( ) f(t)dt) u()d C ( ) f() v()d (4.3.6) and (4.3.5) becomes ( ( D : ) ( v(t) ) ) dt u(t)dt u()d <. (4.3.7) Poof. We use notation intoduced in Remak 4.3.. Fist we assume that (4.3.6) holds fo all f. We define w() v() and we conside u and w integable functions such that u u and w w. We conside the function and we obseve that f(t)dt ( f() : ( ( u (s)ds ( t ) ( ) u (t)dt w (t)dt w (), ) ( t u (s)ds ) ) u (s)ds ( ( t ) w (s)ds w (t)dt ) w (s)ds w (t)dt [ ( ) ( t + ) ( u (s)ds w (s)ds ). ) ] + w (s)ds t
4.3. Weighted Hady ineualities of (, ) tye 36 Now, alying this last estimate, (4.3.6) and integating by ats, we get ( C ( ( ) ( ( ( f(t)dt) u ()d ( C f() v()d ( ( C C ) ( ( ) ) ( ) ) u (t)dt w (t)dt u ()d ( ) ( ) ( ) ) u (t)dt w (t)dt w ()d ( ) ( ) u (t)dt w (t)dt u ()d ( ) ( ) ) u (t)dt w (t)dt u ()d. ) f(t)dt) u()d But since u and w ae integable functions, the ight hand side of the ineuality is finite and hence ( ) ( ( ) ( ) ) u (t)dt w (t)dt u ()d ) C. (4.3.8) Finally, aoimating u and w fom below by an inceasing seuence of integable functions, and alying the Monotone Convegence Theoem, (4.3.8) becomes ( ) ( D ) D C. Now assume (4.3.7) holds. Fo the moment, we will also assume that w satisfies (4.3.2) and we define W () : w(t)dt. Given a ositive function f, we want to aly Poosition 4.3.7 with b() W () w() and F () f(t)dt. Notice that b is unde the hyothesis of Poosition 4.3.7 since and b(t)dt b()d W (t) w(t)dt W () < W (t) w(t)dt. So alying Poosition 4.3.7, integating, using Poosition 4.3.8 and integating
37 Chate 4. Hady ineualities with weights by ats, we conclude ( ( ( ) ) f(t)dt) u()d ( ( ) ( ) ) u(t)dt W (t) w(t)dt W () w()d ( ( f(t)dt) W () w()d ( ) ( ( ( ( ) ( ( ( ) ) ( ) ) u(t)dt W () w()d f(t)dt) W () w()d ) ( u(t)dt ( f() w() d ) ( ) ( ( ( ) ( ) f() w() d ( ) ( ) D f() v()d. ) ) ) W () w()d ) ) u(t)dt W () u()d Now we wok with a geneal w (not necessaily satisfying (4.3.2)). We fi u and w and fo each n N we define u n () u()χ (,n) () and w n () min(w(), n) + χ (n, ) (). Now these functions w n satisfy (4.3.2) since w n (t)dt (n + ) < fo all n N, > and w n ()d min(w(), n)d + n d fo all n N. So we can aly the evious agument fo each ai u n and w n to
4.3. Weighted Hady ineualities of (, ) tye 38 conclude that ( n ( f(t)dt) u()d ) ( ) ( n ( ) ( n ) ) min(w(t), n)dt u(t)dt u()d ( ) f() w n () d. (4.3.9) Now, fo a given ositive function g, we wite f() g() min(w(), n) χ (,n) () and (4.3.9) becomes ( n ( ( ) ( n ( ) g() d ) g(t) min(w(t), n) dt) u()d ( ) ( n ) ) min(w(t), n)dt u(t)dt u()d and letting n we have, by the Monotone Convegence Theoem, that ( ( ( ) ( ( ) g() d. ) g(t)w(t) dt) u()d ( ) ( ) ) w(t)dt u(t)dt u()d In aticula, if we take g of the fom g() f()w() fo a ositive function f, ( ) we get (4.3.6) with C. Now we chaacteize the weighted Hady ineuality fo the case < <. Fist we need some evious esults. Definition 4.3.. Given a ositive function v, we define v() ess inf <t< v(t) su{λ R : { < t < : v(t) < λ} }. Remak 4.3.2. v() is a deceasing function. Indeed, if < y, then { < t < : v(t) < λ} { < t < y : v(t) < λ}
39 Chate 4. Hady ineualities with weights and hence Theefoe, { < t < y : v(t) < λ} { < t < : v(t) < λ}. {λ R : { < t < y : v(t) < λ} } {λ R : { < t < : v(t) < λ} }, and taking suemum in λ we get v(y) v(). The following theoem is a technical esult due to G. Sinnamon and V. D. Steanov (cf. [8, Theoem 3.2]), and states that the weighted Hady ineuality has the same sha constant when we conside v instead of v. Theoem 4.3.3. Suose that < <. Then the best constant in the ineuality ( ( ) f(t)dt) u()d C f()v()d, f, is unchanged when v is elaced by v. The net oosition is due to V. D. Steanov (cf. [2, Poosition (b)]). It is a eal analysis esult which states that the inclusion L (v) L (u) holds when < < unde some conditions on the weights u and v. Poosition 4.3.4. Assume < <. Let us conside C as the best constant in ineuality ( ) f() u()d C f()v()d, f, and conside ( ( E Then E C. ) ( ) ) u(t)dt v(t)dt u()d. Finally, we give the chaacteization of Hady s ineuality when < < (cf. [8, Thoeem 3.3]). Theoem 4.3.5. Suose that < <. Let us conside C as the best constant in ineuality ( and define Then E C. ( E ) f(t)dt) u()d ( ( v() C f()v()d, f, (4.3.2) ) u(t) dt t ) u() d.
4.3. Weighted Hady ineualities of (, ) tye 4 Remak 4.3.6. Again, if we conside u() as u() then (4.3.2) becomes ( ( ) f(t)dt) u()d C f()v()d, f, and E is now E ( ( v() ) ) u(t)dt u()d. Poof. We use the notation intoduced in Remak 4.3.6. By Theoem 4.3.3 C is also the best constant in ineuality ( ( ) f(t)dt) u()d C f()v()d, (4.3.2) with f. We conside fist that v is of the fom v() b(t)dt whee b satisfies b(t)dt < > but By Tonelli s Theoem, (4.3.2) becomes ( ( f(t)dt) u()d ) b(t)dt. (4.3.22) C f() b(t)dt d ( t ) C f()d b(t)dt, and, by Poosition 4.3.4, E E C. Now we conside the case of a geneal v. Fo each n N, we define the function v n () v()χ (,n) (), which is finite, deceasing (cf. Remak 4.3.2) and tends to when. Fied n N, we can aoimate v n fom above by functions of the fom b(t)dt with b satisfying (4.3.22). Let us conside a deceasing seuence of such functions (v m ) m conveging ointwise to v n at almost evey >. We define also the function u n () u()χ (,n) () fo each n N. Then, by the fist at of the oof, the ineuality ( ( ) f(t)dt) u n ()d ( ( v m () f()v m ()d ) ) u n (t)dt un ()d holds fo all f. Hence, eessing f as f() g() v m(), whee g, we have ( ( ) g() v m () dt u n ()d) ( ( v m () g()d ) ) u n (t)dt un ()d
4 Chate 4. Hady ineualities with weights and, as both (v m ) and (v m ) we have, letting m, ( ( ) g() v n () dt u n ()d) ae inceasing, by Monotone Convegence Theoem ( ( v n () g()d. ) ) u n (t)dt un ()d Using Monotone Convegence Theoem again, the last eession becomes, letting n, ( ( ) g() u()d) v() dt ( ( v() g()d. ) ) u(t)dt u()d Eessing g as g() f()v() with f in the last eession, we get C E.
Chate 5 Hady ineualities fo monotone functions In this chate we ae going to study the weighted Hady ineualities esented in evious sections but, instead of woking just with ositive functions, we will conside monotone functions. Some authos as G. H. Hady aleady consideed the study of Hady ineualities when we imose the estiction of monotony. In Section 5. we will give some of the fist esults concening monotone functions. Howeve, the study of the Hady ineuality in the cone of the monotone functions stated to geneate a eal inteest when M. A. Aiño and B. Muckenhout oved, while they wee studying the boundedness of the Hady Littlewood maimal oeato, that the maimal oeato is bounded between Loentz saces if, and only if, the weighted Hady ineuality esticted to ositive and deceasing functions holds. We will see this fact in Section 5.2. Then it is inteesting to study the classical Hady oeato in the cone of monotone functions, which will be the main goal in Section 5.3. Finally, in Section 5.4 we give some alications of the study of the Hady ineualities in the cone of monotone functions. We study when the Loentz saces Λ (w) and the weaktye Loenz saces Λ, (w) ae Banach, and we elate the weak boundedness of the maimal oeato with the weak boundedness of the Hady oeato. 5. Fist esults fo monotone functions As has been said in the intoduction of the chate, the Hady ineualities in the cone of monotone functions wee consideed befoe M. A. Aiño and B. Muckenhout s aoach. In aticula, we have a simila esult to Theoem 3.2. with an estimate fom bellow fo monotone functions (cf. [9, Theoem 2]). Theoem 5... If, α < and f is a ositive deceasing function, then ( f(t)dt) α d 43 α f() α d.
5.. Fist esults fo monotone functions 44 Poof. We define F () f(t)dt and, by Lebesgue Diffeentiation Theoem, we have d dt (F (t)) f(t)(f (t)) f(t)(f(t) t ) t f(t) fo almost evey t. Integating fom to, we get F () t f(t) dt. Finally, alying Tonelli s Theoem we conclude ( f(t)dt) α d F () α d ( ) t f(t) dt α d ( ) α d t f(t) dt α t t α+ α t f(t) dt f(t) t α dt Remak 5..2. If we aly both Theoem 3.2. and Theoem 5.. when > and α, we get that, fo any ositive deceasing function f, and f() d ( ) f(t)dt d ( ) f() d. Unde some additional conditions, we can get that the integals ( ae comaable (cf. [9, Theoem 2]). f(t)dt) α d f () α d Theoem 5..3. If < <, α < and f is a ositive monotone function, then ( f(t)dt) α d f() α d.
45 Chate 5. Hady ineualities fo monotone functions Poof. Fist we assume f to be deceasing. Then one of the ineualities is Theoem 3.2. and the othe one is conseuence of f(t)dt f(). Now we assume that f is inceasing. Fist we notice that, fo any eal numbe α, we have f() α d k 2 α+ 2 2α+ 2 2α+ 2 k f() α d f(2 k )2 k 2 α(k ) 2 k k f(2 k )2 k(α+) f(2 k )2 α(k+) 2 k k k 2 2α+ k 2 k+ f(2 k ) α d 2 2 k 2α+ f() α. k 2 k+ 2 k f() α d Now we obseve that if f() is inceasing then so is and α <, we have f(t)dt. Then, if < < ( f(t)dt) α d k k ( ) 2 k f(t)dt 2 k(α +) ( k ) 2 m f(t)dt 2 k(α +) m 2 m k ( 2 m ) f(t)dt 2 k(α +) 2 m k m k k m m f(2 m ) 2 (m ) 2 k(α +) ( ) 2 k(α +) f(2 m ) 2 (m ) km 2 α + 2 2 α + m f() α d. m 2 m(α +) f(2 m ) 2 (m ) f(2 m ) 2 m(α+)
5.2. Maimal oeato and the Hady ineuality fo monotone functions 46 5.2 Maimal oeato and the Hady ineuality fo monotone functions We esent in this section M. A. Aiño and B. Muckenhout s aoach to Hady ineualities in the cone of monotone functions. We fist ecall the definition of the maimal oeato. Definition 5.2.. Given f a locally integable function in R n, we define the Hady- Littlewood maimal oeato as Mf() : su f(y) dy, Q Q Q whee R n and the suemum is taken ove all cubes in R n containing. We have the following bound (cf. [3, Theoem III.3.3]) fo the noninceasing eaangement of the Hady-Littlewood maimal oeato. Poosition 5.2.2. If f is an integable function in R n, then with t >. t(mf) (t) 4 n f L, The following technical lemma (cf. [3, Lemma III.3.7]), known as the Caldeón- Zygmund coveing Lemma, is needed late on. Lemma 5.2.3. Let Ω be an oen subset of R n with finite measue. Then thee is a seuence of dyadic cubes Q, Q 2,..., with aiwise disjoint inteios, that coves Ω and satisfies (i) Q k Ω fo all k, 2,... (ii) Ω k Q k 2 n Ω. Now we esent a theoem (cf. [3, Theoem III.3.8]) stating that the noninceasing eaangement of the Hady Littlewood maimal oeato (Mf) is euivalent to the Hady oeato alied to the eaangement of f, that is, f. Theoem 5.2.4. Thee eist constants c and c such that c(mf) (t) f (t) c (Mf) (t) (5.2.) fo any t > and any locally integable function f in R n. The constants c and c only deend on n.
47 Chate 5. Hady ineualities fo monotone functions Poof. We ove fist the left-hand ineuality in (5.2.). Let us fi t > and assume that f (t) < (othewise thee is nothing to ove). By Theoem 2.3., fo any ɛ > thee ae functions g t L and h t L such that f g t + h t and g t L + t h t L tf (t) + ɛ. Now, using Poosition 2.3.5 (ii) and (iii), Poosition 5.2.2 and taking into account that Mf L f L, we get ( s ) ( s ) (Mf) (s) (Mg t ) + (Mh t ) c 2 2 s g L + h t L c s ( g L + s h t L ) fo all s >. Hence, utting s t, we get (Mf) (s) c t ( g L + t h t L ) cf (t) + c t ɛ. Letting ɛ, we get the left-hand ineuality in (5.2.). Let us show now the ight-hand ineuality in (5.2.). Again, we assume (Mf) is finite since, othewise, thee is nothing to ove. We define the set Ω { R n : (Mf)() > (Mf) (t)}. This set is oen because if Ω, then f(y)dy > (Mf) (t) Q su Q and we can find a cube Q such that f(y)dy > (Mf) (t), Q Q Q that is, Q Ω. Then Ω is a measuable set and, alying Poosition 2.3.5 (iv) we deduce Ω λ Mf ((Mf) (t)) t. Now, alying Lemma 5.2.3 we know that thee is a seuence of cubes Q, Q 2,..., with aiwise disjoint inteios, that coves Ω, satisfying Q k Ω, (5.2.2) fo all k, 2..., and Q k 2 n Ω 2 n t. (5.2.3) k We define the set F ( k Q k) c and the functions g fχ F c fχ Qk and h fχ F, k
5.2. Maimal oeato and the Hady ineuality fo monotone functions 48 so that f g + h. Now, by Poosition 2.3. and Poosition 2.3.7 we get f (t) g (t) + h (t) g(s) ds + h () t R t g L + h L. (5.2.4) n We notice that by (5.2.2) each cube Q k contains a oint in Ω c and, at these oints, the maimal function is bounded by (Mf) (t) (because of the definition of Ω). Hence, f(y) dy (Mf) (t) Q k Q k fo all k, 2,... Theefoe, using this last estimate and (5.2.3), we get g L k Q k f(y) dy Q k (Mf) (t) 2 n t(mf) (t). (5.2.5) k On the othe hand, F is contained in Ω c and hence the maimal function is bounded by (Mf) (t) in F. Now, as f() (Mf)() fo almost evey R n, we deduce h L fχ F L (Mf)χ F L (Mf) (t). (5.2.6) Finally, using (5.2.5) and (5.2.6) in (5.2.4), we conclude f (t) (2 n t + )(Mf) (t), and the ight-hand ineuality in (5.2.) holds with c 2 n t +. We define the cone of deceasing functions on L (w). Definition 5.2.5. Given a weight w in R + and < <, we define the cone of deceasing functions as ( ) } L dec { (w) : f : f() w()d <. We aleady know that (Mf) f Hf and the following ste is to see that fom this fact we can deduce that the boundedness of the Hady-Littlewood maimal oeato M : Λ (v) Λ (u) between weighted Loentz saces (cf. Definition 2.3.2) is euivalent to the boundedness of the classical Hady oeato H : L dec (v) L (u) in the cone of monotone functions. Fist we need the following lemma, stating that evey ositive deceasing function on R + is the noninceasing eaangement function of a function on R n. Lemma 5.2.6. Let f be a ositive deceasing function and conside g() : f(a n ), R n, whee A B (). Then g (t) f(t) fo almost evey t >.
49 Chate 5. Hady ineualities fo monotone functions Poof. Fist we conside that f is of the fom m f() α j χ [aj,a j+ )(t), (5.2.7) with α >... > α m and a < a 2 <... < a m+. Then, if t α, a, if α 2 t < α, λ g (t) { R n : f(a n ) > t} a 2, if α 3 t < α 2,. a m+, if t < α m, that is, λ g (t) j m+ j a j χ [αj+,α j )(), whee we define α m+. Then if s a m+, α g m if a m s < a m+, (s). α if < s < a. f(s). Now we conside a ositive deceasing function f. Then we can find a seuence of functions as the one in (5.2.7), say (s m ) m, such that s m () f() fo almost evey >. Then if we define g m () s m (A n ) we have that g m () f(a n ) g() fo almost evey R n. By Poosition 2.3.5 (vi), we have g m(t) g (t) fo all t >. But, as s m (t) f(t) fo almost evey t >, we conclude that g (t) f(t) fo almost evey t >. Finally, we see that the boundedness of M : Λ (v) Λ (u) is euivalent to the boundedness of H : L dec (v) L (u). Theoem 5.2.7. The boundedness of the Hady-Littlewood maimal oeato M : Λ (v) Λ (u) is euivalent to the boundedness of the classical Hady oeato H : L dec (v) L (u) esticted to ositive functions, that is, the weights u and v fo which the ineuality ( ( ) ( ) f(t)dt) u()d C f() v()d holds fo all ositive and deceasing functions f.