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We consider a general class of sharp $L^p$ Hardy inequalities in $R^N$ involving distance from a surface of general codimension $1leq kleq N$. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite series improvement of $L^p$ Hardy inequalities.
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For a bounded convex domain Omega in R^N we prove refined Hardy inequalities that involve the Hardy potential corresponding to the distance to the boundary of Omega, the volume of $Omega$, as well as a finite number of sharp logarithmic corrections. We also discuss the best constant of these inequalities.
We present a unified approach to improved $L^p$ Hardy inequalities in $R^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where distance is taken from a surface of codimension $1<k<N$. In our main result we add to the right hand side of the classical Hardy inequality, a weighted $L^p$ norm with optimal weight and best constant. We also prove non-homogeneous improved Hardy inequalities, where the right hand side involves weighted L^q norms, q eq p.
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