A HARDY TYPE GENERAL INEQUALITY IN L p( ) (0, 1) WITH DECREASING EXPONENT

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1 Transacions of NAS of Azerbaijan, 23, vol. XXXIII, No, pp Farman I. MAMEDOV, Firana M. MAMEDOVA A HARDY TYPE GENERAL INEQUALITY IN L p ), ) WITH DECREASING EXPONENT Absrac We derive a Hardy ype inequaliy W.) σ.) f)d C ω.) L f.), f. L,),) for he exponen p :, ), ) is a decreasing funcion on some inerval, ɛ), ɛ > and σ = ω.) L, ), W x) = σ)d. We sudy a Hardy ype inequaliy W.) σ.) f)d C ω.) L,) f.) L,), f. ) in he norms of variable exponen Lebesgue space L, ), whenever he exponen p is a decreasing funcion on some inerval, ɛ), < ɛ < and he funcions σ = ω.) L, ), W x) = σ)d. As o he basic properies of spaces L, we refer o he works [2], [5], [7]. In his paper, we assume ha p x) is a measurable funcion on, ) and is values are in he inerval [, ). Also p + = sup {p x) : x, )} < and p = inf {p x) : x, )} >. The space L, ) is a class of measurable funcions fx) on, ) such ha he modular I f) = f dx is finie. ) } A norm in L, ) is defined as f L,) {λ = inf > : I f λ. For < p, p + < he space L, ) is a reflexive Banach space. For he funcion < p x) denoes he conjugae funcion of, + p x) = and p x) = if =. We denoe by C, C, C 2,... various posiive consans whose values may vary a each appearance. For a las ime he variable exponen Hardy ype inequaliies was sudied by several auhors see, f.e. [], [3], [4], [6], [7], [8], [], [], [2], [3], [4], [5], [6]). There are several sufficien condiions on he funcion p :, ), ) for he inequaliy x x f)d C f.) L,), f 2) L,) o hold. They are expressed in erms of regulariy condiions for p a he origin. The inequaliy 2) follows from ) for he case ω. I follows from he resuls of works [3], [8], [4] see, also [], [], [3]) ha he inequaliy 2) holds if p = inf p >, p + = sup < and he condiion is saisfied. A := lim sup p) log <. 3) x x

2 46 [F.I.Mamedov,F.M.Mamedova] Transacions of NAS of Azerbaijan In [] see, also [6]), Mamedov had proved ha he condiion [ p B := lim sup x x 2 )] log x < 4) is necessary for he inequaliy 2) if he exponen funcion p is increasing. The condiion 4) is sricly weaker han 3). This condiion is saisfied e.g. by = p) + C α and α >, C >. For he exponen, ha is increasing near he ln x) origin, he condiion 4) is also sufficien if he number B be B < p) p) ) see, []). Unforunaely, such good condiion 4) is no longer sufficien for he inequaliy ) o hold if he condiion on B be ignored. In his case, a necessary and sufficien condiion is sill an open problem. Seing ω = x β.) in ) we aain he inequaliy 9) if he condiion 8) be saisfied for he funcions p, β see, Corollary ). The inequaliy 9) also was much sudied for a las ime. Noe a necessary and sufficien condiion for 7) o hold is he condiion β) < 5) p) if he β, p saisfy 3)see, e.g. [], [3], [7], [8], [4]). I is ineresing ha he condiion σ L, ) replaces 5) in he Corollaries,. Noice he inequaliy ) conains no only power ype weighs x β.), x β.) from he lef and righ hand sides respecively he ype of inequaliy 9)). For example, we can ake a funcion σ for he inequaliies 7) and 6) no necessarily power ype. In Theorem below, we prove ha he regulariy condiion is no needed if he exponen p is decreasing a small neighborhood of he origin. Theorem. Le ω :, ), ) be a measurable funcion σ = ω L, ) and W x) = σ)d. Suppose p :, ) [, ) be decreasing on some inerval, ɛ), ɛ > ; hen i holds he inequaliy ) for a posiive measurable funcion f. Corollary. Le ω :, ), ) be a measurable funcion σ = ω L, ). Suppose p :, ) [, ) be decreasing on some inerval, ɛ), ɛ > and he condiion σ σ)d Cxσx), < x < ɛ 6) is saisfied; hen i holds he inequaliy p.).) x f)d C σ p.).)f.) L,) L,), f. 7) for a posiive measurable funcion f. Corollary 2. Le β :, ) R be a measurable funcion. Suppose p :, ) [, ) be decreasing on some inerval, ɛ), ɛ > such ha βx)p x) for < x < and he condiion β)p ) d Cx βx)p x), < x < ɛ 8) is saisfied. Then i holds he inequaliy xβ.) f)d C x β.) f.) L,) L,), f. 9)

3 Transacions of NAS of Azerbaijan [A Hardy ype general inequaliy in...] for a posiive measurable funcion f. Since here is a relaion beween modular and variable exponen norm see, f.e. [7]): f p+ L,l) I p f) f p L,l), f L,l) ) f p L,l) I p f) f p+ L,l), f L,l). ) we can perform our esimaes in erms of he modular. Proof of Theorem. To prove he inequaliy ), i suffices o consider he case when f is a posiive measurable funcion such ha ω f see, [2]). ) I follows from ) ha I ω f. In order o prove Theorem we have o show W σ Hf C. 2) L,) To prove 2), we shall esablish he esimae ) I W σ Hf C 2. Using he riangle inequaliy of -norms for δ, ), we have W.)σ.) Hf.) W.)σ.) Hf.) + ;,δ) + W.)σ.) Hf.) := i + i 2.q 3) ;δ,) Define by ww ) an inverse funcion for W w) = z = W x) in he inerior inegral below gives Hfx) = W x) w 47 σu)du. A change of variable fwz)) fww x)) dz = W x) σwz)) σww x)) d. Using his and Minkowskii s inequaliy for L norms see, f.e. [7], [2], [5], [9]), we see ha i = W.)σ.) Hf.) = ;,δ) σ.) fww.)) σww.)) d ;,δ) σ.) fww.)) σww.)) d. 4) ;,δ) Now esimae he norm W.)σ.) Hf.) for < <. Since p is )) ;,δ) decreasing on, δ), we have p w pv) for v, δ). Therefore, σx) p,δ) fww x)) σww x)) W v) dx = fww x)) σww x)) p,δ) ) σx)dx

4 48 [F.I.Mamedov,F.M.Mamedova] Transacions of NAS of Azerbaijan ) fww x)) [ ww x)) = v, W x) = W v) σx)dx = σww x)) σx)dx = dw v) = dv ww δ)) ) W v) fv) pw )) χ fv) { }dv + fv) pv) χ fv) ) { }dv + where p,δ) is minimum of p over, δ). This implies + W δ)) σx) p p,ɛ),δ) fww x)) σww x)) ww δ)) ] χ { fv) <}dv χ fv) { <}dv + W δ), Therefore and using he definiion of -norms, we ge W.) σ.).)f. ) + W δ)) ;,δ) p,ɛ) dx, < <. Using 5) and 4) for he firs summand in 3) we ge he esimae Le us esimae W.)σ.) Hf.). For x δ, ) using Young s inequal- ;δ,) iy, we ge since and Therefore, i + W δ)) p p,δ), < <. 5) ) p,δ) d p p,δ) + W δ)),δ) C. 6) W x)σx) Hfx)) = σx)w x) σx) + W )) p + p W p+ δ) Cσx) C σx), f σ ) ) f σs)ds σ fs) σs) σs)ds ) σs)ds + Cσx) 2 σ p.) f ) fs) σs) σs)ds p x) σs)ds σ 2 + W )) W x) W δ) for x δ, ). ) I ;δ,) W.)σ.) Hf.) C 2 W ). Hence W.)σ.) Hf.) C 3 ;δ,) p

5 Transacions of NAS of Azerbaijan [A Hardy ype general inequaliy in...] Insering his esimae and 6) in 3) we complee he proof of Theorem. Proof of Corollary 2. This Corollary follows from he Corollary 2 by using of he inequaliy 6) in he he esimae ). Proof of Corollary 3. This Corollary follows from he Theorem 2 by using of he inequaliy 8) in he he esimae 7) for σx) = x βx)p x). Acknowledgemen. We express our graiude o he anonym referee for useful commens and deail examinaion of he paper. 49 References []. Cruz-Uribe D., SFO, Mamedov F.I., On a general weighed Hardy ype inequaliy in he variable exponen Lebesgue spaces, Revisa Maemaica Compluense, doi:.7/s , in press 2. [2]. Diening L., Harjuleho P., Haso P. and Ruzicka M., Lebesgue and Sobolev Spaces wih Variable Exponens, Lecure Noes in Mahemaics, Springer, Heidelberg, Germany. 2, vol 27. [3]. Diening L., Samko S. Hardy inequaliy in variable exponen Lebesgue spaces, Fracional Calculus and Applied Analysis, 27, vol., No, pp. -7. [4]. Edmunds D.E., Kokilashvili V., Meskhi A. On he boundedness and compacness of he weighed Hardy operaors in spaces, Georgian Mahemaical Journal. 25, vol. 2, No, pp [5]. Fan X.L., Zhao D. On he spaces L Ω) and W m, Ω), Journal of Mahemaical Analysis and Applicaions, 2, vol. 263, No 2, pp [6]. Harman A. On necessary condiion for he variable exponen Hardy inequaliy, Journal of Funcion Spaces and Applicaions, vol. 22, Aricle ID , 6 pages, doi:.55/22/ [7]. Harjuleho P., Haso P., Koskinoja M. Hardy s inequaliy in variable exponen Sobolev spaces, Georgian Mahemaical Journal. 25 vol. 2, No 3, pp [8]. Harman A., Mamedov F.I. On boundedness of weighed Hardy operaor in L and regulariy condiion, Journal of Inequaliies and Applicaions. 2, vol. 2, Aricle ID 83795, 4 p. [9]. Kovacik O., Rakosnik J. On spaces L and W. Czechoslovak Mahemaical Journal, 99, vol. 46), pp []. Mamedov F.I. On Hardy ype inequaliy in variable exponen Lebesgue space L, ), Azerbaijan Journal of Mahemaics, 22, vol. 2, No., pp []. Mamedov F.I., Harman A. On a weighed inequaliy of Hardy ype in spaces L, Journal of Mahemaical Analysis and Applicaions, 29, vol. 353, No 2, pp [2]. Mamedov F.I., Harman A. On a Hardy ype general weighed inequaliy in spaces L, Inegral Equaions and Operaor Theory. 2, vol. 66, No 4, pp [3]. Mamedov F.I., Zeren Y. On equivalen condiions for he general weighed Hardy ype inequaliy in space L. Zeischrif fur Analysis und ihre Anwendungen. 22, vol. 3, No, pp

6 5 [F.I.Mamedov,F.M.Mamedova] Transacions of NAS of Azerbaijan [4]. Mashiyev R., Cekic B., Mamedov F.I., Ogrash S. Hardy s inequaliy in power-ype weighed L spaces, Journal of Mahemaical Analysis and Applicaions. 27, vol. 334, No, pp [5]. Rafeiro H., Samko S.G. Hardy inequaliy in variable Lebesgue spaces, Annales Academiae Scieniarium Fennicae 29, vol. 34, No, pp [6]. Samko S.G. Hardy inequaliy in he generalized Lebesgue spaces, Fracional Calculus and Applied Analysis. 23, vol. 6, No 4, pp [7]. Samko S.G. Convoluion ype operaors in L, Inegral Transforms and Special Funcions. 998, vol. 7, pp Farman I. Mamedov, Firana M. Mamedova Insiue of Mahemaics and Mechanics of NAS of Azerbaijan 9, B.Vahabzade sr., AZ4, Baku, Azerbaijan Tel.: 9942) off.). Received: Sepember, 22; Revised: December 5, 22.