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I would like to add that perhaps in the following books / theses you find some interesting applications of Hardy's inequality (or Hardy-type inequalities), e.g., to financial analysis:

  1. B. Opic, A. Kufner. Hardy-type inequalities. Pitman Research Notes in Mathematics 219, 1990.

  2. A. Wedestig. Weighted Inequalities of Hardy-Type and their Limiting Inequalities, Ph.D. thesis, Department of Mathematics, Lulea University of Technology, 2003.

  3. A. Kufner, L. E. Persson. Weighted inequalities of Hardy type. World Scientific. 2003

  4. This book here (several applications are mentioned, e.g., to differential geometry)