Herz-type Hardy spaces for conjugate systems of temperatures (Q2767361)

The authors consider the conjugate systems of temperatures as follows: A \(C^2\)-valued vector field in \(\mathbb R_+^{n+1}\), \(F(t,x_1,\ldots,x_n)=(u_0,u_1,\ldots, u_n)\) is said to belong to \(\mathcal{AT}\), if (i) \(\sum_{j=1}^{n}D_{x_j}u_j=iD_t^{1/2}u_{0}\), (ii) \(D_{x_k}u_j=D_{x_j}u_k\), \(j,k=1,2,\ldots,n\), (iii) \(D_{x_j}u_{0}=-iD_t^{1/2}u_{j}\), \(j=1,2,\ldots,n\), where \(D_t^{1/2}\) is Weyl's fractional derivative of order \(1/2\). For \(0<p<q<\infty\) and \(\alpha=n(1/p-1/q)\) they define the homogeneous Herz-type Hardy space of conjugate systems of temperatures by NEWLINE\[NEWLINETH\dot K_q^{\alpha,p}(\mathbb R^n)= \Bigl\{F\in \mathcal{AT};\;\sup_{t>0}\||F|\|_ {\dot K_q^{\alpha,p}(\mathbb R^n)}<\infty \Bigr\},NEWLINE\]NEWLINE where \(\dot K_q^{\alpha,p}(\mathbb R^n)\) is the homogeneous Herz space. They show that for \((n-1)/n<p<\infty\), these spaces are equivalent to the homogeneous Herz-type Hardy spaces of conjugate systems of harmonic functions. The spaces of boundary distributions of \(0\)-th component of these Hardy spaces coincide with the Herz-type Hardy spaces. Their work is a Herz-type version of the usual Hardy space case, a work by \textit{M. Guzmán-Partida} and \textit{S. Pérez-Esteva} [``Hardy spaces of conjugate systems of temperatures'', Can. J. Math. 50, No. 3, 605-619 (1998; Zbl 0923.42015)].