Multipliers from \(H^1\) to \(L^p\)
The author proves a theorem which combines the Hörmander-Mikhlin multiplier theorem for Hardy spaces in [\textit{L. Hörmander}, Acta Math. 104, 93-140 (1960; Zbl 0093.11402)] with a result of \textit{R. Johnson} in [Ark. Mat. 16, 235-249 (1978; Zbl 0402.46024)]. Let \(\Delta_j=\{2^{j-1}\leq|\xi|\leq 2^{j+1}\}\). Assume that \(m\in C^k(\mathbb{R}^n\setminus\{0\})\) and that \[ \int_{\Delta_j}\sum_{|\alpha|\leq k} \left|2^{j|\alpha|}D_\alpha m(\xi)\right|^2 d\xi\leq 2^{nj(2-q)/q}, \quad j\in \mathbb{Z}, \] where \(k\) is the least integer \(>n(2-q)/2q\) and \(1\leq q\leq 2\). Then the author proves that the convolution operator defined by \(K=\widehat m\) continuously maps \(H^1(\mathbb{R}^n)\) to \(L^q(\mathbb{R}^n)\).
scientific article; zbMATH DE number 1987637