A logarithmic Hardy inequality

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved among radial functions, in some range of the parameters

The fractional Hardy inequality with a remainder term

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]Comment: Major change

Self-improvement of pointwise Hardy inequality

We prove the self-improvement of a pointwise $p$-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves