With the help of a radially invariant vector field, we derive inequalities of the Hardy kind, with no boundary terms, for W 1,p functions on bounded star domains. Our results are not obtainable from the classical inequalities for W1,p functions. Unlike in W1,p, our inequalities admit maximizers that we describe explicitly.
Ahmed A. ABDELHAKIM
. HARDY’S INEQUALITIES WITH MAXIMIZERS FOR W1,p FUNCTIONS ON BOUNDED STAR DOMAINS[J]. Acta mathematica scientia, Series B, 2019
, 39(1)
: 26
-36
.
DOI: 10.1007/s10473-019-0103-z
[1] Adams R A. Sobolev Spaces. New York:Academic Press, 1975
[2] Alvino A, Volpicelli R, Volzone B. On Hardy inequalities with a remainder term. Ric Mat, 2010, 59: 265-280
[3] Balinsky A, Evans W D, Lewis R T. The Analysis and Geometry of Hardy's Inequality. New York:Springer, 2015
[4] Barbatis G, Filippas S, Tertikas A. A unified approach to improved Lp Hardy inequalities with best constants. Trans Amer Math Soc, 2004, 356(6):2169-2196
[5] Brezis H, Vázquez J L. Blowup solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid, 1997, 10:443-469
[6] Brezis H, Marcus M. Hardy's inequality revisited. Ann Scuola Norm Sup Pisa, 1997, 25:217-237
[7] Brezis H, Marcus M, Shafrir I. Extremal functions for Hardy's inequality with weight. J Funct Anal, 2000, 171:177-191
[8] Cowan C. Optimal Hardy inequalities for general elliptic operators with improvements. Commun Pure Appl Anal, 2010, 9(1):109-140
[9] Cuomo S, Perrotta A. On best constants in Hardy inequalities with a remainder term. Nonlinear Analysis, 2011, 74:5784-5792
[10] Devyvera B, Fraasb M, Pinchovera Y. Optimal hardy weight for second-order elliptic operator:An answer to a problem of Agmon. J Funct Anal, 2014, 266(7):4422-4489
[11] Evans L C, Gariepy R F. Measure Theory and Fine Properties of Functions. New York:CRC Press, 1992
[12] Filippas S, Tertikas A. Optimizing improved Hardy inequalities. J Funct Anal, 2002, 192:186-233
[13] Gazzola F, Grunau H-C, Mitidieri E. Hardy inequalities with optimal constants and remainder terms. Trans Amer Math Soc, 2004, 356:2149-2168
[14] Ghoussoub N, Moradifam A. On the best possible remaining term in the Hardy inequality. Proc Natl Acad Sci USA, 2008, 105(37):13746-13751
[15] Ghoussoub N, Moradifam A. Functional Inequalities:New Perspectives and New Applications. Math Surveys Monogr, Vol 187. Providence, RI:Amer Math Soc, 2013
[16] Ioku N, Ishiwata M, Ozawa T. Sharp remainder of a critical Hardy inequality. Archiv der Mathematik, 2016, 106:65-71
[17] Ioku N, Ishiwata M, Ozawa T. Hardy type inequalities in Lp with sharp remainders. J Inequal Appl, 2017, 2017:5
[18] Ioku N, Ishiwata M. A scale invariant form of a critical Hardy inequality. Int Math Research Notices, 2015, 2015(18):8830-8846
[19] Lieb E H, Loss M. Analysis, Graduate Studies in Mathematics, Vol 14. 2nd ed. Providence, RI:American Mathematical Society, 2001
[20] Machihara S, Ozawa T, Wadade H. Hardy type inequalities on balls. Tohoku Math J, 2013, 65(3):321-330
[21] Machihara S, Ozawa T, Wadade H. Remarks on the Rellich inequality. Mathematische Zeitschrift, 2017, 286(3/4):1367-1373
[22] Machihara S, Ozawa T, Wadade H. Remarks on the Hardy type inequalities with remainder terms in the framework of equalities. arXiv:1611.03580
[23] Machihara S, Ozawa T, Wadade H. Scaling invariant Hardy inequalities of multiple logarithmic type on the whole space. Journal of Inequalities and Applications, 2015, 2015:281
[24] Ruzhansky M, Suragan D. Critical Hardy inequalities. arXiv:1602.04809
[25] Vazquez J L, Zuazua E. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential. J Funct Anal, 2000, 173:103-153
