A parabolic Littlewood-Paley inequality with applications to parabolic equations
The main result of this article is the following. Let \(K\in (0, \infty)\) and \(\psi(x)\) a \(C^\infty(\mathbb{R}^d)\) integrable function such that \[ \int_{\mathbb{R}^d} \psi dx= 0,\;|\psi|_{L_1(\mathbb{R}^d)}+ ||x|\psi|_{L_1(\mathbb{R}^d)}+ |\nabla \psi|_{L_1(\mathbb{R}^d)}+ |x\cdot \psi|_{L_1(\mathbb{R}^d)}\leq K. \] Define \(\Psi_t g(x)= t^{- d/2} \psi(x/\sqrt t)* g(x)\). Let \(H\) be a separable Hilbert space, \(p\in [2, \infty)\), \(- \infty\leq a< b\leq \infty\) and \(g\in L_p((a, b)\times \mathbb{R}^d, H)\). Then \[ \int_{\mathbb{R}^d} \int^b_a \Biggl[ \int^t_a |\Psi_{t- s} g(s, \cdot)(x)|^2_H {ds\over t- s}\Biggr]^{p/2} dt dx\leq N \int_{\mathbb{R}^d} \int^b_a|g(t, x)|^p_H dt dx, \] where the constant \(N\) depends only on \(d\), \(p\) and \(K\). The author also gives an extension of this result to parabolic equations with coefficients depending only on \(t\).
scientific article; zbMATH DE number 828288