Parabolic Besov regularity for the heat equation (Q443890)

Let \(D\) be a bounded Lipschitz domain in \(\mathbb{R}^d(d\geq 2)\), \(T>0\), and \(\Omega=D\times(0,T)\). Let \(\Theta(\Omega)\) denote the space of all solutions of the equation \(u_t=\Delta u\) in \(\Omega\). In this paper the authors proved the following result on Besov smoothness of functions \(\Theta(\Omega)\): Let \(1<p<\infty\), \(\lambda >0\), \(l\) the lagest integer less than \(\lambda+d\), \(0<\alpha<\min\{l,\frac{\lambda d}{d-1}\}\) and \(\frac{1}{\tau}=\frac{1}{p}+\frac{\alpha}{d}\). Then \[ ||u||_{L_\tau((0,T);B_\tau^\alpha(D))}\leq C||u||_{L_p((0,T);B_p^\lambda(D))} \] for all \(u\in \Theta(\Omega)\) and for some constant \(C\) depending on \(\Omega,d,p\), and \(\lambda\). In particular \[ \Theta(\Omega)\cap L_p((0,T);B_p^\lambda(D))\subset L_\tau((0,T);B_\tau^\alpha(D)). \] A similar result for harmonic functions was proved by \textit{S. Dahlke} and \textit{R. A. DeVore} [Commun. Partial Differ. Equations 22, No. 1--2, 1--16 (1997; Zbl 0883.35018)]. As in the elliptic case, a central tool in this paper is the use of the characterization of the Besov spaces through Daubichies wavelet bases.