0 M ay 2 00 9 PITT ’ S INEQUALITY AND THE FRACTIONAL LAPLACIAN : SHARP ERROR ESTIMATES for

Considerable interest exists in understanding the framework of weighted inequalities for differential operators and the Fourier transform, and the application of quantitative information drawn from these inequalities to varied problems in analysis and mathematical physics, including nonlinear partial differential equations, spectral theory, fluid mechanics, stability of matter, stellar dynamics and uncertainty. Such inequalities provide both refined size estimates for differential operators and singular integrals and quantitative insight on symmetry invariance and geometric structure. The purpose of this note is to improve the sharp Pitt’s inequality at the spectral level by using the optimal Stein-Weiss inequality and a new representation formula derived by Frank, Lieb and Seiringer [17] which expresses fractional Sobolev embedding in terms of a Besov norm characteristic of the problem’s dilation invariance and extends with weights an earlier classical formula of Aronszajn and Smith using the L2 modulus of continuity (see [1], page 402). Moreover reflecting the natural duality, this formula can be combined with the Hardy-Littlewood-Sobolev inequality to provide new techniques to determine sharp embedding constants, including a sharp form of the Besov norm Sobolev embedding studied by Bourgain, Brezis and Mironescu [14]. A secondary bootstrap argument produces an improved Stein-Weiss inequality accompanied by an intriguing monotonicity property at the spectral level. A direct proof of the weighted representation formula for the fractional Laplacian starting from the classical formula of Aronszajn and Smith is given in the appendix below.