Real interpolation of generalized Besov-Hardy spaces and applications (Q636815)

The authors introduce the generalized homogeneous Hardy spaces \(H_q(\phi)\) and the generalized non-homogeneous Hardy spaces \(h_q(\phi),\) \(0<q<\infty,\) where \(\phi\in {\mathcal B},\) i.e., \(\phi: (0,\infty)\to (0,\infty)\) is continuous, \(\phi(1)=1\) and \(\sup_{s>0}\frac{\phi(st)}{\phi(s)}<\infty,\) \(0<t<\infty\). These generalizations correspond to the classical Hardy spaces \(H_p\) and \(h_p\), replacing the Lebesgue space \(L_p\) by the Lorentz space \(\Lambda_q(\phi).\) The Besov spaces \(B^s_{p,q}, s\in {\mathbb{R}}\), \(0<p,q\leq \infty\), are defined via the Fourier transform. More general Besov spaces \(B^\phi_{p,q}\) are introduced just replacing the power function \(t^s\) by \(\phi.\) Even more general Besov spaces \(B^\phi_{\psi,q,r}\) are defined, replacing \(L_p\) by the Hardy space \(h_q(\psi)\), \(\psi\in {\mathcal B}.\) The authors present results about the interpolation of generalized Hardy and Besov spaces using the \(K-\)method with a parameter \(L_p(1/\gamma),\) \(\gamma\in {\mathcal B}\). In particular, the following formula is proved: \[ (B^\phi_{p_0,q}, B^\phi_{p_1,q})_{\gamma,q}=B^\phi_{\psi,q,q},\; p_0\neq p_1,\; \phi, \gamma\in {\mathcal B}, \] \(1/\psi(t)=t^{-1/p_0} \gamma(t^{1/p_0-1/p_1})\) if the Boyd indices of \(\gamma\) are in the interval \((0,1)\). As an application, the boundedness of pseudodifferential operators in the generalized Besov and Hardy spaces is investigated, and a wavelet decomposition of \(h_q(\phi)\) is provided.