### Bibliography [1] M. Abramowitz, and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Washington,NationalBureauofStandardsAppliedMathematics,1964.

"... [5] Adimurthi, Maria J. Esteban, An improved Hardy-Sobolev inequality in W 1,p and its application to Schrödinger operators, NoDEANonlinearDi↵erentialEquationsAppl.12(2005),no.2,243-263. [6] Adimurthi, M. Grossi, S. Santra, Optimal Hardy-Rellich inequalities, maximum principles and related eigenvalu ..."

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[5] Adimurthi, Maria J. Esteban, An improved Hardy-Sobolev inequality in W 1,p and its application to Schrödinger operators, NoDEANonlinearDi↵erentialEquationsAppl.12(2005),no.2,243-263. [6] Adimurthi, M. Grossi, S. Santra, Optimal Hardy-Rellich inequalities, maximum principles and related eigenvalue problems, J.Funct.Anal.240(2006),36-83. [7] Adimurthi, A. Sekar, Role of the fundamental solution in Hardy-Sobolev-type inequalities,Proceedings

### HARDY AND RELLICH TYPE INEQUALITIES WITH REMAINDERS FOR BAOUENDI-GRUSHIN VECTOR FIELDS

, 704

"... Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields: ∇γ = (∇x, |x | 2γ ∇y) where γ> 0, ∇x and ∇y are usual gradient operators in the variables x ∈ R m and y ∈ R k, respectively. In the first part of the paper, we prove some weighted Hardy type i ..."

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Abstract. In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields: ∇γ = (∇x, |x | 2γ ∇y) where γ> 0, ∇x and ∇y are usual gradient operators in the variables x ∈ R m and y ∈ R k, respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains. 1.

### SHARP WEIGHTED RELLICH AND UNCERTAINTY PRINCIPLE INEQUALITIES ON CARNOT GROUPS

, 2010

"... In this work we prove sharp weighted Rellich-type inequalities and their improved versions for general Carnot groups. To derive the improved Rellich-type inequalities we have established new weighted Hardy-type inequalities with remainder terms. We also prove new sharp forms of the weighted Hardy-Po ..."

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In this work we prove sharp weighted Rellich-type inequalities and their improved versions for general Carnot groups. To derive the improved Rellich-type inequalities we have established new weighted Hardy-type inequalities with remainder terms. We also prove new sharp forms of the weighted Hardy-Poincaré and uncertainty principle inequalities for polarizable Carnot groups.