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We prove an improved version of Poincar\'e-Hardy inequality in suitablesubspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As aconsequence, we obtain a new Hardy type inequality with an improved constant(than the usual Hardy constant). Furthermore, we derive a new kind of improvedCaffarelli-Kohn-Nirenberg inequality on the hyperbolic space.

Improved Poincar\'e-Hardy inequalities on certain subspaces of the Sobolev space