Research Article Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel

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1 Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 29, Artcle ID 3958, 2 pages do:.55/29/3958 Research Artcle Multdmesoal Hlbert-Type Iequaltes wth a Homogeeous Kerel Predrag Vuovć Faculty of Teacher Educato, Uversty of Zagreb, Savsa cesta 77, Zagreb, Croata Correspodece should be addressed to Predrag Vuovć, predrag.vuovc@vus-c.hr Receved July 29; Revsed November 29; Accepted 8 November 29 Recommeded by Radu Precup We cosder the Hlbert-type equaltes wth ocojugate parameters. The obtag of the best possble costats the case of ocojugate parameters remas stll ope. Our geeralzato wll clude a geeral homogeeous erel. Also, we obta the best possble costats the case of cojugate parameters whe the parameters satsfy approprate codtos. We also compare our results wth some ow results. Copyrght q 29 Predrag Vuovć. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal wor s properly cted.. Itroducto Let /p /q p >, f,g, < f p xdx <, < g q xdx <.. The well-ow Hardy-Hlbert s tegral equalty see s gve by fxg y x y dx dy < π /p /q s π/p f p xdx g xdx q,.2 ad a equvalet form s gve by fx p [ ] p x y dx π dy < s π/p f p xdx,.3 where the costat factors π/sπ/p ad π/sπ/p p are the best possble.

2 2 Joural of Iequaltes ad Applcatos Durg the prevous decades, the Hlbert-type equaltes were dscussed by may authors, who ether reproved them usg varous techques or appled ad geeralzed them may dfferet ways. For example, we refer to a paper of Yag see 2.If N\{}, p >, /p, s >, f, satsfy < x p s f p xdx <, 2,...,,.4 the, f x s dx dx < s /p Γ x p s f p j x xdx,.5 Γs p j where the costat factor /Γs Γs/p s the best possble. Our geeralzato wll clude a geeral homogeeous erel Kx,...,x : R R, where 2, wth beg ocojugate parameters. The techques that wll be used the proofs are maly based o classcal real aalyss, especally o the well-ow Hölder s equalty ad o Fub s theorem. The obtag of the best possble costats the case of ocojugate parameters seems to be a very dffcult problem ad t remas stll ope. Let us recall the defto of ocojugate expoets see 3. Letp ad q be real parameters, such that p>, q >, p q,.6 ad let p ad q, respectvely, be ther cojugate expoets, that s, /p /p /q /q. Further, defe ad λ p q.7 ad ote that <λ for all p ad q values as.6. I partcular, λ holds f ad oly f q p, that s, oly whe p ad q are mutually cojugate. Otherwse, <λ<, ad such cases p ad q wll be referred to as ocojugate expoets. Cosderg p, q,adλ as.6 ad.7, Hardy et al., proved that there exsts a costat C p,q, depedet oly o the parameters p ad q, such that the followg Hlbert-type equalty holds for all oegatve fuctos f L p R ad g L q R : fxg y λ dx dy C p,q f L x y p R g L q R..8

3 Joural of Iequaltes ad Applcatos 3 Covetos Throughout ths paper we suppose that all the fuctos are oegatve ad measurable, so that all tegrals coverge. We also troduce the followg otatos: R {x x,x 2,...,x ; x,x 2,...,x > }, x x x 2 x /, >,.9 ad let S 2 Γ // Γ/ be a area of ut sphere R vew of orm. 2. Ma Results Before presetg our dea ad results, we repeat the oto of geeral ocojugate expoets from 3. Letp,, 2,...,,be the real parameters whch satsfy, p >,, 2,...,. 2. p Further, the parameters p,, 2,...,are defed by the equatos p p,, 2,...,. 2.2 Sce p >,, 2,...,, t s obvous that p >,, 2,...,. We defe λ :. 2.3 p It s easy to deduce that <λ. Also, we troduce the parameters q,, 2,...,, defed by the relatos λ q p,, 2,...,. 2.4 I order to obta our results we eed to requre q >,, 2,...,. 2.5 It s easy to see that the above codtos do ot automatcally apply 2.5. Further, t follows λ, q q λ p,, 2,...,. 2.6

4 4 Joural of Iequaltes ad Applcatos Of course, f λ, the /p ; so the codtos reduce to the case of cojugate parameters. Results ths secto wll be based o the followg geeral form of Hardy-Hlbert s equalty prove 4. All the measures are assumed to be σ-fte o some Ω measure space. Theorem 2.. Let, N, 2, ad λ, p,p,q,, 2,...,, be real umbers satsfyg LetK : Ω R ad φ j : Ω R,, j,...,, be oegatve measurable fuctos such that,j φ jx j. The, for ay oegatve measurable fuctos f,, 2,...,,the followg equaltes hold ad are equvalet: K λ x,...,x f x dμ x dμ x Ω Ω K λ x,...,x f x dμ x dμ x Ω φ F x Ω Ω /p p φ F f x dμ x, /p p φ F f x dμ x, 2.7 p dμ x /p 2.8 where F x Kx,...,x Ω φ q j,j / j x jdμ x dμ x dμ x dμ x /q,,...,. 2.9 I the same paper the authors dscussed the case of equalty equaltes 2.7 ad 2.8. They proved that the equalty holds 2.7 ad aalogously 2.8 f ad oly f f x C φ x q / λq F x λq, C,,...,. 2. I the followg theorem we gve the most mportat case where Ω R,the measures μ,,...,, are Lebesgue measures, K :, R s a oegatve homogeeous fucto of degree s, s >, ad the fuctos φ j represet the form φ j x j x j A j where A j R,, j,...,. I order to obta the geeralzatos of some ow results we defe β,...,β :, K,t,...,t t β tβ dt dt, 2. where we suppose that β,...,β < for β,...,β > adβ β <s.

5 Joural of Iequaltes ad Applcatos 5 Due to techcal reasos, we troduce real parameters A j,,j, 2,...,satsfyg A j, j, 2,...,. 2.2 We also defe A j,, 2,...,. 2.3 j Theorem 2.2. Let, N, 2, ad λ, p,p,q,, 2,...,, be real umbers satsfyg LetK :, R be oegatve measurable homogeeous fucto of degree s, s >, ad let A j,,j,...,, ad,,..., be real parameters satsfyg 2.2 ad 2.3. Iff : R R, f /,,...,are oegatve measurable fuctos, the the followg equaltes hold ad are equvalet: K x λ,..., x f x dx dx <L R R /p, x p /q sp f p x dx x p /q s p p K x λ,..., x f x dx dx dx R R <L p R p x p /q sp f p /p x dx, 2.4 where L S λ 2 λ q A 2,..., q A /q s q2 2 A 22, q 2 A 23,..., q 2 A 2 /q2 2.5 q A 2,..., q A,,s q A /q, q A j >, / j ad q A > s.

6 6 Joural of Iequaltes ad Applcatos Proof. Set Kx,...,x K x,..., x ad φ j x j x j A j Theorem 2., where A j for every j,...,. It s eough to calculate the fuctos F x,,...,. By usg the -dmesoal sphercal coordates we fd F q x K x,..., x R xj q A j dx 2 dx j2 S K 2 x,t 2,...,t t q A j, j dt 2 dt. j2 2.6 Usg homogeety of the fucto K ad the substtutos u t / x, 2,...,, we have F q x S 2, x s K,u 2,...,u q A x u j j x du 2 du S 2 x sq A q A 2,..., q A. 2.7 j2 Smlarly, by applyg the -dmesoal sphercal coordates ad homogeety of the fucto K we have F q 2 2 x 2 K x,..., x R j,j / 2 xj q 2 A 2j dx dx 3 dx t s 2, K, x 2, t 3,..., t t t t t q 2A 2j j,j / 2 j dt dt 3 dt. 2.8 Usg the chage of varables t x 2 u 2, t x 2 u 2 u, 3,...,, so t,t 3,...,t u 2,u 3,...,u x 2 u 2, 2.9

7 Joural of Iequaltes ad Applcatos 7 where t,t 3,...,t / u 2,u 3,...,u deotes the Jacoba of the trasformato, we have F q 2 S 2 x 2 2 x 2 sq 2 2 A 22 K,u 2,...,u u s q22 A22, 2 2 x 2 s q 2 2 A 22 j3 u q 2A 2j j du 2 du s q2 2 A 22, q 2 A 23,..., q 2 A I a smlar maer we obta S F q x 2 x sq A q A 2,..., q A,,s q A, q A,,..., q A 2.2 for 3,...,. Ths gves equaltes 2.4 wth equalty sg. Codto 2. mmedately gves that otrval case of equalty 2.4 leads to the dverget tegrals. Ths completes the proof. Remar 2.3. Note that the erel K x,..., x x β s fucto of degree βs. I ths case we have s a homogeeous β,...,β, tβ tβ β Γs Γ s s dt dt β β Γ β β, 2.22 where we used the well-ow formula for gamma fucto see, e.g., 5, Lemma 5..Now, by usg Theorem 2.2 ad 2.22 we obta the result of Krć etal.see The Best Possble Costats the Cojugate Case I ths secto we cosder the equaltes Theorem 2.2. I such a way we shall obta the best possble costats for some geeral cases. It follows easly that Theorem 2.2 the cojugate case λ, p q becomes as follows.

8 8 Joural of Iequaltes ad Applcatos Theorem 3.. Let, N, 2 ad let p,...,p be cojugate parameters such that p >,,...,. Let K :, R be oegatve measurable homogeeous fucto of degree s, s>, ad let A j,,j,...,,ad,,...,be real parameters satsfyg 2.2 ad 2.3. If f : R R, f /,,..., are oegatve measurable fuctos, the the followg equaltes hold ad are equvalet: /p, K x,..., x f x dx dx <M x sp f p R x dx R x p s p p K x,..., x f x dx dx dx R R <M p R p x sp f p /p x dx, 3. where M S 2 p A 2,..., p A /p s p2 2 A 22, p 2 A 23,..., p 2 A 2 /p2 3.2 p A 2,..., p A,,s p A /p, p A j >, / j ad p A > s. To obta a case of the best equalty t s atural to mpose the followg codtos o the parameters A j : p j A j s p A, j/,, j {, 2,...,}. 3.3 I that case the costat M from Theorem 3. s smplfed to the followg form: M 2 à 2,..., Ã, 3.4 where à p A for /, à p A. 3.5

9 Joural of Iequaltes ad Applcatos 9 Further, by usg 3.4 ad 3.5, the equaltes 3. wth the parameters A j, satsfyg the relato 3.3, become K x,..., x f x dx dx <M R R x p p p à K x,..., x f x dx dx R R <M R /p. x p à f p x dx /p, x p à f p x dx 3.6 /p dx 3.7 Theorem 3.2. Suppose that the real parameters A j,,j,...,satsfy codtos Theorem 3. ad codtos gve 3.3. If the erel K t,...,t s as Theorem 3. ad for every 2,..., K,t 2,...,t,...,t CK,t 2,...,,...,t, t, t j, j/ 3.8 for some C>, the the costat M s the best possble equaltes 3.6 ad 3.7. Proof. Let us suppose that the costat factor M gve by 3.4 s ot the best possble the equalty 3.6. The, there exsts a postve costat M <M, such that 3.6 s stll vald whe we replace M by M. We defe the real fuctos f,ε : R R by the formulas, x <, f,ε x à x ε/p, x,,...,, 3.9 where < ε < m {p p à }. Now, we shall put these fuctos equalty 3.6. By usg the -dmesoal sphercal coordates, the rght-had sde of the equalty 3.6 becomes M [ x ] /p x ε dx M S 2 t ε dt M S ε

10 Joural of Iequaltes ad Applcatos Further, let J deotes the left-had sde of the equalty 3.6, for the above choce of the fuctos f,ε. By applyg the -dmesoal sphercal coordates ad the substtutos u t /t, / 2, we fd J K x,..., x x x x à ε/p dx dx 2 K t,...,t t à ε/p dt dt 3. 2 t ε/β /t K,u 2,...,u /t u à ε/p 2 du 2 du dt. Now, t s easy to see that the followg equalty holds: J 2 2 t ε t ε I j t dt, j2 K,u 2,...,u 2 u à ε/p du 2 du dt 3.2 where for j 2,...,, I j t s defed by I j t K,u 2,...,u du 2 du, 3.3 D j u à ε/p 2 satsfyg D j {u 2,...,u ;<u j < /t, <u l <, l/ j}. Wthout losg geeralty, we oly estmate the tegral I 2 t. For 2 we have I 2 t /t K,u 2 u à 2 ε/p 2 2 du 2 C C à 2 ε t ε/p 2 à 2 p, 2 /t u à 2 ε/p 2 2 du 2 3.4

11 Joural of Iequaltes ad Applcatos ad for >2wefd I 2 t C [, 2 K,,u 3,...,u ] /t à u ε/p à du 3 du u 2 ε/p 2 2 du 2 3 C εp2 ε/p à 2 t 2 à 2 à 3 εp3,..., à εp, 3.5 where à 3 ε/p 3,..., à ε/p s well defed sce obvously à 3 à <s 2. Hece, we have I j t t ε/p j à j O j, for ε,j {2,...,}, ad cosequetly t ε I j t dt O. j2 3.6 We coclude, by usg 3., 3.2, ad 3.6, that M M whch s a obvous cotradcto. It follows that the costat M 3.6 s the best possble. Fally, the equvalece of the equaltes 3.6 ad 3.7 meas that the costat M s also the best possble the equalty 3.7. That completes the proof. Remar 3.3. If we put 2, K x, y l x / y / x s y s, à s/q ad à 2 s/p the equaltes 3.6 ad 3.7 applyg Theorem 3.2, we obta the result of Baoju Su see 7. Further, by puttg Theorems 3. ad 3.2 we obta approprate results from 8. More precsely, the equalty 3.6 becomes K x,...,x f x dx dx <M, x p à f p x dx /p. 3.7 If the erel K x,...,x ad the parameters A j satsfy the codtos from Theorem 3.2, the the costat M à 2,...,à s the best possble. For example, settg K x,...,x x x s,s>, à s p /p, 2,...,, the equalty 3.7, we obta Yag s result.5 from troducto. Acowledgmet Ths research s supported by the Croata Mstry of Scece, Educato ad Sports, Grat o

12 2 Joural of Iequaltes ad Applcatos Refereces G. H. Hardy, J. E. Lttlewood, ad G. Pólya, Iequaltes, Cambrdge Uversty Press, Cambrdge, UK, 2d edto, B. Yag, O a ew multple exteso of Hlbert s tegral equalty, Joural of Iequaltes Pure ad Appled Mathematcs, vol. 6, o. 2, artcle 39, pp. 8, F. F. Bosall, Iequaltes wth o-cojugate parameters, The Quarterly Joural of Mathematcs, vol. 2, pp. 35 5, I. Bretć, M. Krć, ad J. Pečarć, Multple Hlbert ad Hardy-Hlbert equaltes wth ocojugate parameters, Bullet of the Australa Mathematcal Socety, vol. 7, o. 3, pp , B. Yag ad Th. M. Rassas, O the way of weght coeffcet ad research for the Hlbert-type equaltes, Mathematcal Iequaltes & Applcatos, vol. 6, o. 4, pp , M. Krć, J. Pečarć, ad P. Vuovć, O some hgher-dmesoal Hlbert s ad Hardy-Hlbert s tegral equaltes wth parameters, Mathematcal Iequaltes & Applcatos, vol., o. 4, pp. 7 76, B. Su, A multple Hlbert-type tegral equalty wth the best costat factor, Joural of Iequaltes ad Applcatos, vol. 27, Artcle ID 749, 4 pages, I. Perć adp.vuovć, Hardy-Hlbert s equalty wth geeral homogeeous erel, Mathematcal Iequaltes & Applcatos, vol. 2, o. 3, pp , 29.