Weighted hardy inequalities with sharp constants

Abstract

Using the Bessel functions we obtain several weighted Hardy inequalities with sharp constants. The following inequality for absolutely continuous functions is a simple example: If p and ν are positive numbers, and f: [0, 1] → ℝ satisfies the conditions f(0) = 0 and x1/2−p/2f′ ∈ L2(0, 1), then $$ \int\limits_0^1 {\frac{{f'^2 \left( x \right)}} {{\left( {F_\nu \left( x \right)} \right)^{p - 1} }}dx \geqslant \frac{{pj_{\nu - 1}^2 }} {{4\nu ^2 }}} \int\limits_0^1 {\frac{{f^2 \left( x \right)}} {{x^{{{2 - 1} \mathord{\left/ {\vphantom {{2 - 1} \nu }} \right. \kern-\nulldelimiterspace} \nu }} \left( {F_\nu \left( x \right)} \right)^{p - 1} }}dx} $$ , where Fν (x) = √xJν(jν−1x1/(2ν)), Jν is the Bessel function of order ν and jν−1 is the first positive zero of Jν−1. In the general case we have to introduce constants z = λν(2/q) as the first positive root of the Lamb equation Jν(z) + qzJ′ν (z) = 0 and the functions z = z(q) that may be found as the solution of the initial values problem $$ \frac{{dz}} {{dq}} = - \frac{z} {{1 - \nu ^2 q^2 + z^2 q^2 }}, z\left| {_{q = {1 \mathord{\left/ {\vphantom {1 \nu }} \right. \kern-\nulldelimiterspace} \nu }} : = \lambda _\nu \left( {2\nu } \right) = j_{\nu - 1} } \right. $$ .