A generalization of the Littlewood-Paley inequality for the fractional Laplacian (Q662060)

The authors prove a parabolic-type Littlewood-Paley inequality for the fractional Laplacian \((-\Delta)^{\frac\alpha2}\), where \(0<\alpha<2\). Let \(S_{\alpha,t}\) denote the semigroup corresponding to the equation \(u_t=-(-\Delta)^{\frac\alpha2}u\), \(H\) a Hilbert space, \(2\leq p<\infty\), \(-\infty\leq a<b\leq\infty\), and \(f\) a measurable \(H\)-valued function of \((t,x)\). The main result of the paper states that then \[ \int_{\mathbb R^d}\int_a^b\left[ \int_a^t |\partial_x^{\frac\alpha2} S_{\alpha,t-s} f(s,\cdot)(x)|^2_H ds\right]^\frac{p}{2} dtdx\leq N(\alpha, p)\int_{\mathbb R^d}\int_a^b|f|^p_H dtdx. \]