Remarks on Some Higher Dimensional Hardy Inequalities

Main Article Content

Zraiqat Amjad, Jebril Iqbal, Hawawsheh Laith, Abudayah Mohmammad


In this note, we give an elementary proof of Hardy inequality in higher dimensions introduced by Christ and Grafakos. The advantage of our approach is that it uses the one-dimensional Hardy inequality to obtain higher dimensional versions. We go further and get some well-known weighted estimates using the same approach.

Article Details


  1. M. Christ, L. Grafakos, Best Constants for Two Nonconvolution Inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687–1693.
  2. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambridge, 1952.
  3. P. Drábek, H.P. Heinig, A. Kufner, Higher Dimensional Hardy Inequality, in: C. Bandle, W.N. Everitt, L. Losonczi, W. Walter (Eds.), General Inequalities 7, Birkhäuser Basel, Basel, 1997: pp. 3–16.
  4. B. Opic, A. Kufner, Hardy-Type Inequalities, Volume 219, Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990.
  5. A. Kufner, L. Maligranda, L.E. Persson, The Hardy Inequality: About Its History and Some Related Results, Vydavatelsky Servis Publishing House, Pilsen, 2007.
  6. A. Kufner, L.E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co. Inc., River Edge, 2003.
  7. G. Sinnamon, One-Dimensional Hardy-Type Inequalities in Many Dimensions, Proc. Royal Soc. Edinburgh: Sect. A Math. 128 (1998), 833–848.
  8. W.G. Alshanti, Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives, Abstr. Appl. Anal. 2019 (2019), 5648095.