Weighted Inequalities and Stein-weiss Potentials

Sharp extensions of Pitt’s inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include HardyRellich inequalities. Weighted inequalities provide quantitative information to characterize integrability for differential and integral operators and intrinsically are determined by their dilation character. In the classical context, weighted inequalities for the Fourier transform provide a natural measure of uncertainty. For functions on R the issue is the balance between the relative size of a function and its Fourier transform at infinity. An inequality that illustrates this principle at the spectral level is Pitt’s inequality: (1) ∫ Rn Φ(1/|x|)|f(x)| dx ≤ CΦ ∫ Rn Φ(|y|)|f̂(y)| dy where Φ is an increasing function, the function f is in the Schwartz class S(Rn) and the Fourier transform is defined by (Ff)(y) = f̂(y) = ∫ Rn ef(x) dx . Such inequalities may be fully determined by dilation invariance, and some cases may be realized with explicit gradient forms as Hardy-Rellich inequalities. In earlier work (see [3]) the effective calculation for the constant in Pitt’s inequality was reduced to Young’s inequality for convolution on a non-compact unimodular group. The objective here will be to study more general forms of Pitt’s inequality (2) ∫ Rn Φ(1/|x|)|∇f | dx ≤ 4πDΦ ∫ Rn Φ(|y|)|y||f̂(y)| dy using the structure of Stein-Weiss potentials and convolution estimates in concert with the Hecke-Bochner representation for L(R). The previous work is described by the following three theorems. Theorem 1 (Pitt’s inequality). For f ∈ S(Rn) and 0 ≤ α < n ∫ Rn |x||f(x)| dx ≤ Cα ∫ Rn |y||f̂(y)| dy (3) Cα = π α [ Γ (n− α 4 )/ Γ (n+ α 4 )]2 . Since the above inequality becomes an identity for α = 0, a differentiation argument provides a logarithmic form that controls the uncertainty principle by using dimensional asymptotics. Theorem 2 (logarithmic uncertainty). For f ∈ S(Rn) ∫ Rn ln |x| |f(x)| dx+ ∫ Rn ln |y| |f̂(y)| dy ≥ D ∫ |f(x)| dx (4) D = ψ(n/4)− lnπ , ψ(t) = d dt ln Γ(t) . Logarithmic integrals are indeterminate so D may take negative values. The proof of Pitt’s inequality (3) follows from a sharp estimate for an equivalent integral realization as a Stein-Weiss fractional integral on R.