On the compactness of the commutator on the real axis (Q1382690)

The authors establish the compactness of the commutator \[ (T \varphi)(x)={1\over\pi i}\int_\mathbb{R}{a(t)-a(x)\over t-x}\varphi(t)dt,\quad x\in\mathbb{R}, \] in certain functional spaces on \(\mathbb{R}\); here \(a(x)\) is a function of a certain class and the function \[ R(x,t)=(t-x)^{-1}(a(t)-a(x)) \] is the kernel of the commutator. Then, if \(a\in H_\alpha^{(r+1)}\), \(r\geq 0\), the following five conditions are sufficient for the compactness of the operator \(T\). 1. \(0<\alpha\leq 1\); \(T:L\to L_{q\rho}^{(r)}\), \(1\leq q<+\infty\), \(\rho(x)=(1+x^2)^{-1}\), 2. \(2^{-1}<\alpha\leq 1\); \(T:L_p\to L_{q\rho}^{(r)}\), \((2\alpha-1)^{-1}\leq p\leq 2(2\alpha-1)^{-1}\), \(1 \leq q<+\infty\), \(\rho(x)=(1+x^2)^{-1}\), 3. \(2^{-1}<\alpha\leq 1\); \(T:L_p\to C_0^{(r)}\), \((2\alpha-1)^{-1}<p<2(2\alpha-1)^{-1}\), 4. \(2^{-1}<\alpha\leq 1\); \(T:L_p\to C_{01}^{(r)}\), \((2\alpha-1)^{-1}<p<2(2\alpha-1)^{-1}\), 5. \(2^{-1}<\alpha\leq 1\): \(T:C_{01}\to C_{01}^{(r)}\).