Convolution multipliers on weighted Besov spaces
If \(0 < p,q \leq \infty\) and \(w: \mathbb Z \to (0,\infty)\) is a function such that \(w(k+1)/w(k)\) is uniformly bounded above and below, we can define the Besov space \(B^{p,q}_w(R^n)\) via the norm \[ \|f\|_{B^{p,q}_w} = \|w(k)^{-1} \|\Delta_k f \|_p \|_{l^q(Z)} \] where the \(l^q\) norm is taken over the \(k\) variable, and \(\Delta_k\) is a standard Littlewood-Paley projection to the frequency range \(|\xi|\sim 2^k\). The function \(w\) thus represents a weight in frequency space. Using Littlewood-Paley theory, the authors characterize the space of all convolution operators \(f \mapsto a * f\) which are bounded from \(B^{p_0,q_0}_{w_0}\) to \(B^{p_1,q_1}_{w_1}\), in terms of the \((L^{p_0},L^{p_1})\) operator norm. Indeed, they show that for any \(p_0,q_0,p_1,q_1,w_0,w_1\) with the above properties and \(p_0 \geq 1\), the Besov space operator norm is comparable to \[ \|w_0(k) w_1(k)^{-1} \|\Delta_k a \|_{(L^{p_0},L^{p_1})} \|_{l^s(Z)} \] where \(1/s = \max(1/q_1 - 1/q_0, 0)\), and \(\|\Delta_k a \|_{(L^{p_0},L^{p_1})}\) is the operator norm of \(f \mapsto \Delta_k a * f\) from \(L^{p_0}\) to \(L^{p_1}\). If \(p_0 < 1\) then the same statement holds, except that the space \(L^{p_0}\) is replaced by the stronger space \(L^{p_0}[2^{-k} B]\) of functions in \(L^{p_0}\) whose Fourier transform is supported in a ball of radius \(\sim 2^k\).
scientific article; zbMATH DE number 1187572