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Register a new userOn a class of Rellich inequalities
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We prove Rellich and improved Rellich inequalities that involve the distance function from a hypersurface of codimension $k$, under a certain geometric assumption. In case the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of these inequalities.
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We prove two improv
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.
We consider a Schrodinger hamiltonian $H(A,a)$ with scaling critical and time independent external electromagnetic potential, and assume that the angular operator $L$ associated to $H$ is positive definite. We prove the following: if $|e^{-itH(A,a)}|_{L^1to L^infty}lesssim t^{-n/2}$, then $ ||x|^{-g(n)}e^{-itH(A,a)}|x|^{-g(n)}|_{L^1to L^infty}lesssim t^{-n/2-g(n)}$, $g(n)$ being a positive number, explicitly depending on the ground level of $L$ and the space dimension $n$. We prove similar results also for the heat semi-group generated by $H(A,a)$.
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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