L[p] measure of growth and higher order Hardy-Sobolev-Morrey inequalities on ℝ[N] <i>L</i><sup><i>p</i></sup> measure of growth and higher order Hardy–Sobolev–Morrey inequalities on ℝ<sup><i>N</i></sup>

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Abstract

<p>When the growth at infinity of a function <i>u</i> on ℝ<sup><i>N</i></sup> is compared with the growth of |<i>x</i>|<sup><i>s</i></sup> for some <i>s</i> ∈ ℝ, this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined <i>L</i><sup><i>p</i></sup> sense for every 1 ≤ <i>p</i> < ∞ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub |<i>x</i>|<sup>−<i>N/p</i></sup> growth of ∇<i>u</i> in the <i>L</i><sup><i>p</i></sup> sense implies sub |<i>x</i>|<sup>1−<i>N/p</i></sup> growth of <i>u</i> in the <i>L</i><sup><i>q</i></sup> sense for well chosen values of <i>q</i>.</p><p>By investigating how sub |<i>x</i>|<sup><i>s</i></sup> growth of ∇<sup><i>k</i></sup><i>u</i> in the <i>L</i><sup><i>p</i></sup> sense implies sub |<i>x</i>|<sup><i>s</i>+<i>j</i></sup> growth of ∇<sup><i>k−j</i></sup><i>u</i> in the <i>L</i><sup><i>q</i></sup> sense for (almost) arbitrary <i>s</i> ∈ ℝ and for <i>q</i> in a <i>p</i>-dependent range of values, a family of higher order Hardy/Sobolev/Morrey type inequalities is obtained, under optimal integrability assumptions.</p><p>These optimal inequalities take the form of estimates for ∇<sup><i>k−j</i></sup>(<i>u</i> − π<sub><i>u</i></sub>), 1 ≤ <i>j</i> ≤ <i>k</i>, where π<sub><i>u</i></sub> is a suitable polynomial of degree at most <i>k</i> − 1, which is unique if and only if <i>s</i> < −<i>k</i>. More generally, it can be chosen independent of (<i>s,p</i>) when <i>s</i> remains in the same connected component of ℝ\{−<i>k</i>,…,−1}.</p>

Journal

  • Journal of the Mathematical Society of Japan

    Journal of the Mathematical Society of Japan 69(1), 127-151, 2017

    The Mathematical Society of Japan

Codes

  • NII Article ID (NAID)
    130005310386
  • NII NACSIS-CAT ID (NCID)
    AA0070177X
  • Text Lang
    ENG
  • ISSN
    0025-5645
  • NDL Article ID
    027859030
  • NDL Call No.
    Z53-A209
  • Data Source
    NDL  J-STAGE 
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