Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions

Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions

• 1.
  IOSR Journal ofMathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. 1 (Jan - Feb. 2015), PP 83-86 www.iosrjournals.org DOI: 10.9790/5728-11118386 www.iosrjournals.org 83 | Page Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions Arun Pal Singh Department of Mathematics, Dyal Singh College (University of Delhi) Lodhi Road, New Delhi - 110 003, INDIA Email: [email protected] Abstract: In this paper, we obtain the characterization on pair of weights v and w so that the Hardy-Steklov operator dttf xb xa )( )( )( is bounded from )(0,, qp vL to )(0,, sr wL for <,,,<0 srqp . 2010 AMS Mathematics Subject Classification: 26D10, 26D15. Keywords: Hardy-Steklov operator, Lorentz spaces, non-decreasing. I. Introduction By a weight function u defined on )(0, we mean a non-negative locally integrable measurable function. We take ))(),((0,00 dxxu  MM to be the set of functions which are measurable, non-negative and finite a.e. on )(0, with respect to the measure dxxu )( . Then the distribution function u f of   0Mf is given by .0,)(=:)( }>)(:)(0,{   tdxxut txfx u f The non-increasing rearrangement * uf of f with respect to )(xdu is defined as 0.},)(:{inf=:)(*  yyttyf u fu  For  qp <0,<<0 , the two exponent Lorentz spaces )(0,, qp vL consist of   0Mf for which            =,)(*1/sup 0> <<0, 1/ )](*1/[ 0, =: qtvfpt t q q t dtqtvfpt p q qp vL f (1) is finite. In this paper, we characterize the weights v and w for which a constant 0>C exists such that 0,,,  ffCTf qp vLsr wL (2) where T is the Hardy-Steklov operator defined as .=))(( )( )( )( txTf dtf xb xa (3)
• 2.
  Hardy-Steklov operator ontwo exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 84 | Page The functions )(= xaa and )(= xbb in (3) are strictly increasing and differentiable on )(0, . Also, they satisfy .<<0for)(<)(and=)(=)(0;=(0)=(0)  xxbxababa Clearly, 1 a and 1 b exist, and are strictly increasing and differentiable. The constant C attains different bounds for different appearances. II. Lemmas Lemma 1. We have              .=,)]([sup <<0,)]([ = 1/ 0> 1/ /1 0 , stt sdttst f rv f t s rsv f s sr vL   (4) Proof. Applying the change of variable )(= ty v f to the R.H.S. of (1) and integrating by parts we get the lemma.  Lemma 2. If f is nonnegative and non-decreasing, then .)()()(= 1 0 , dxxvdttvxf r s f r s x ss sr vL        (5) Proof. We obtain the above equality by evaluating the two iterated integrals of )()( 1 1 xvxh r s st r s s         over the set }<0),(<<0;),{( xxfttx , so that we have dxdtxvxh r s st r s sxf )()( 1 1)( 00          ,)()(= 1 1 )(0 dtdxxvxh r s st r s s tx          (6) where })(:{sup=)( txfxtx  for a fixed t , and .)(=)( dttvxh x  Integrating with respect to ‘t ’ first, the L.H.S. of (6) gives us the R.H.S. of (5). Further rs tx r s r s tx dssvtxhdxxvxh r s / )( 1 )( )(=))((=)()(        .)]([=}]>)(:{[= r s v f r s ttxfxv  Hence the lemma now follows in view of Lemma 1.  III. Main Results Theorem 1. Let <,,,<0 srqp be such that .<<1  sq Let T be the Hardy-Steklov operator given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that )(<)( xbxa  for ).(0,x Then the inequality q pqq v s rss w x dx xxf p q C x dx xxTf r s 1/ /* 0 1/ /* 0 )]([)]([               (7)
• 3.
  Hardy-Steklov operator ontwo exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 85 | Page holds for all nonnegative non-decreasing functions f if and only if s r s y x t tbxa xt dyywdzzw r s 1/ 1 )(<)( <<<0 )()(sup                 .<)()( 1/1 1)( )(                            qq p q y tb xa dyyvdzzv (8) Proof. Using differentiation under the integral sign, the condition )(<)( xbxa  for )(0,x ensures that Tf is nonnegative and non-decreasing. Consequently, by Lemma 2, the inequality (7) is equivalent to q q ssxb xa dxxVxfCdxxWdttf 1/ 0 1/ )( )(0 )()()()(                  (9) where )()(=)( 1 xwdzzw r s xW r s x        and ).()(=)( 1 xvdzzv p q xV p q x        Thus it suffices to show that (9) holds if and only if (8) holds. The result now follows in view of Theorem 3.11 [2].  Similarly, in view of Theorem 2.5 [1], by making simple calculations, we may obtain the following: Theorem 2. Let <,,,<0 srqp be such that .<<,1<<0 qqs Let T be the Hardy-Steklov operator given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that )(<)( xbxa  for ).(0,x Then the inequality (7) holds for all nonnegative non-decreasing functions f if and only if         plpqpqplrsrst tab txxbta ////// )((10 ][)]()([            <)()()( 1/ 1 l p q x dttdxxvdyyv p q  and        plpqpqplrsrstba t xttbxa //////))((1 0 ][)]()([ ,<)()()( 1/ 1            l p q x dttdxxvdyyv p q  where , 11 = 1 , 11 = 1 qslpqr  and  is the normalizing function as defined in [3]. Remark. The condition )(<)( xbxa  for )(0,x cannot be relaxed since otherwise the monotonicity of Tf would be on stake. For example, consider the functions                  20 ,1)29( 10 20<10 ,910 10<0, 10 =)( xx x x x x xa and
• 4.
  Hardy-Steklov operator ontwo exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 86 | Page                 20. ,1)299(1010 20<10 ,99 10 10<0,1010 =)( x x xx xx xb Note that a and b satisfy all the aforementioned conditions, except that, we have )(>)( xbxa  for 20<10 x . Acknowledgement This work was supported in part by the Department of Science and Technology (DST), INDIA. References [1]. H.P.Heinig and G.J. Sinnamon, Mapping properties of integral averaging operators, Studia Math., 129(1998), 157-177. [2]. A. Kufner and L.E. Persson, Weighted Inequalites of Hardy Type, World Scientific Publishing Co. Pte. Ltd, Singapore, 2003. [3]. E. Sawyer, Weighted Lebesgue and Lorentz norm inequalites for the Hardy operator, Trans. of Amer. Math. Soc., Vol. 281(1984), 329-337 .