Hardy type inequalities in L p with sharp remainders

Journal of Inequalities and Applications, Jan 2017

Sharp remainder terms are explicitly given on the standard Hardy inequalities in L p ( R n ) with 1 < p < n . Those remainder terms provide a direct and exact understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals. MSC: 26D10, 26D15, 46E35.

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Hardy type inequalities in L p with sharp remainders

Ioku et al. Journal of Inequalities and Applications Hardy type inequalities in Lp with Norisuke Ioku 0 1 Michinori Ishiwata 0 1 Tohru Ozawa 0 1 0 Engineering Science , Osaka 1 Innovation, Graduate School of Lp(Rn) with 1 < p < n. Those remainder terms provide a direct and exact Sharp remainder terms are explicitly given on the standard Hardy inequalities in understanding of Hardy type inequalities in the framework of equalities as well as of the nonexistence of nontrivial extremals. Hardy's inequalities; remainders - · ∇f We revisit this famous inequality. Particularly, we present equalities which fill the gaps between the right- and left-hand sides of (.)-(.) with explicit remainder terms for p > . Those equalities yield (.)-(.) by dropping remainder terms. Moreover, we give a characterization of functions which leads to vanishing remainders. The study of the Hardy inequalities which is based on the viewpoint of the equality leads to a direct and explicit understanding of the Hardy type inequalities as well as of the nonexistence of nontrivial To state our main theorems, we introduce some necessary notation. In this paper, we deal with real-valued functions and we argue with sufficiently smooth functions with compact support in Rn \ {} so that the standard density argument goes through. Let us intro© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. for p >  and ξ , η ∈ R, where /p =  – /p and Rp(ξ , ξ ) makes sense only if p ≥  and if p <  and ξ = . In other words, Rp : (ξ , η) → Rp(ξ , η) is well defined on R × R if p ≥  and on (R × R) \ {(, )} if  < p < . For p with  ≤ p ≤ ∞, the Banach space which consists of pth integrable Lebesgue measurable functions is denoted by Lp( ). The norm of it is also denoted by · Lp or · p if it does not cause confusion. The Sobolev space of order one introduced by Lp is denoted Wp = Wp(Rn) for  ≤ p < ∞. The basic properties of Rp are summarized in the following proposition. () Rp satisfies the estimates ⎧⎨ p– (|ξ | ∨ |η|)p– if p ≥ , ⎩ p– (|ξ | ∧ |η|)p– if p < , ⎧⎨ p– (|ξ | ∧ |η|)p– if p ≥ , ⎩ p– (|ξ | ∨ |η|)p– if p < , We now state our main results. Theorem  Let n and p satisfy  < p < n. Then the equality holds for all f ∈ Wp(Rn). If the second term on the right hand side of (.) vanishes, then the left-hand side of (.) is finite if and only if f = . Remark  In fact, we prove that if the second term on the right hand side of (.) vanishes, then there exists a function ϕ : Sn– → R on the unit sphere Sn– such that f (x) = |x|– n–pp ϕ almost everywhere in Rn \ {}. In this case, Theorem  Let p and r satisfy  < p < ∞ and r > . Then: () The equality ∞ xp–r– f (x) p dx f (x) = cx pr – f (x) = cx– pr – ∞ xp+r– f (x) p dx holds for all real-valued measurable functions on (, ∞) with xf ∈ Lp(, ∞; x–r– dx). Moreover, there exists c ∈ R which satisfies, for almost everywhere x ∈ (, ∞), when the last term in the right hand side of (.) equals zero. In this case, and the left-hand side of (.) is finite if and only if c = . () The equality holds for all real-valued measurable functions on (, ∞) with xf ∈ Lp(, ∞; xr– dx). Moreover, there exists c ∈ R which satisfies, for almost everywhere x ∈ (, ∞), provided that the last term in the right hand side of (.) vanishes. In this case, We prove the theorems in subsequent sections. The first step of the proof is the same as the standard one. We need the following identity: |f|(xx|)p|p dx = – which holds for all f ∈ C∞(Rn), provided  < p < n. It can be obtained expressing the integral on the left-hand side by means of the spherical coordinates and using the integration by parts (cf. [], Proof of Theorem .). Equation (.) together with the Hölder inequality with /p + /p = ,  ≤ p < ∞, implies (.). In this sense, the standard method depends upon duality. In this paper, we adopt a different view. We rewrite (.) in the form |u|p dx = |u|p – |u|p–uv dx = . Now the equality (.) can be understood as representing a cancelation as well as an oscillation or an orthogonality. This point of view for equation (.) can be stated in the following way. u pp = () We have () We have u pp = v pp – u pp = v pp – p |v|p + (p – )|u|p – p|u|p–uv dμ = p immediately yields the conclusion. The subsequent sections are organized as follows. Proposition  will be proved in Section . Section  is devoted to the verification of Theorem . The proof of Theorem  is given in Section . There is a large literature on Hardy type inequalities and related subjects. See [–] and the references therein for instance. 2 Proof of Proposition 1 First of all, we remark that R(ξ , η) = /, by definition. This proves Part () as well as Parts (), (), and () for p = . By a direct calculation, (.) holds if ξ = η. Let ξ = η. We obtain = (p – ) – (p – ) = (p – ) which yields (.). Then Part () follows immediately from Part (). If p >  and Rp(ξ , η) = , then by the integral representation (.) we have θ ξ + ( – θ )η =  for any θ with  < θ < . This implies ξ = η = . If p <  and Rp(ξ , η) = , then |θ ξ + ( – θ )η| = ∞ for any θ with  < θ < , which is absurd. This proves Proposition . 3 Proof of Theorem 1 By a standard density argument, it is enough to prove Theorem  for f ∈ C∞(Rn). Applying (.), (.) is then a direct consequence of Lemma  with u = |xf| and v = – n–pp |x| · ∇f . Now x suppose that the second term on the right hand side of (.) vanishes. Then by Parts () and () of Proposition , we easily see that f satisfies the equation which is equivalent to This implies (.), which in turn implies the rest of the statements of the theorem. 4 Proof of Theorem 2 By integration by parts, we have dx = the statements of Part () follow if we notice that f = p x+ pr d Part () follows by the same argument. 5 Conclusions In this paper, we examined the sharp remainder terms of the Hardy inequality for Lpfunctions. From these sharp remainder terms, we can derive several consequences including the explicit form of the extremal function for the inequality which reveals the nature of the nonexistence of extremals in the Lp-setting. Our analysis only requires some elementary calculus with some insight in the structure of the remainder term and is also applicable to other critical type inequalities such as the Hardy inequalities in Ln. Competing interests The authors declare that they have no competing interests. Authors’ contributions TO prepared the manuscript, MI typeset it and NI helped to revise it. TO, MI, and NI have agreed to its contents and are responsible for all aspects of the accuracy and integrity of the manuscript. Acknowledgements The authors wish to extend their gratitude to the anonymous referees for valuable comments. 1. Folland , GB: Real Analysis, 2nd edn . Pure and Applied Math. Wiley, New York ( 1999 ) 2. Reed , M, Simon, B : Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness . Academic Press, San Diego ( 1975 ) 3. Machihara , S, Ozawa, T, Wadade , H: On the Hardy type inequalities . Preprint ( 2015 ) 4. 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Norisuke Ioku, Michinori Ishiwata, Tohru Ozawa. Hardy type inequalities in L p with sharp remainders, Journal of Inequalities and Applications, 2017, 5, DOI: 10.1186/s13660-016-1271-1