An example of relative Calderón couples of Hardy \(p\)-spaces with \(0<p<1\) (Q1924962)

The first example of relative Calderón couples of Hardy spaces was given in 1984 by P. W. Jones who showed that \((H^1,H^\infty)\) is a Calderón couple. Some years later, in 1992 Q. Xu proved that if \(X_i\), \(Y_i\), \(i=1,2\) are rearrangement invariant spaces and if \((X_1,X_2)\), \((Y_1,Y_2)\) are relative Calderón couples, then \((H(X_1),H(X_2))\), \((H(Y_1),H(Y_2))\) are relative Calderón couples as well. Therefore, there are many such couples of Hardy spaces generated by Banach function spaces. This is not the case when we work with Hardy spaces generated by quasi-Banach function spaces. However, it seems that there are no examples of relative Calderón couples of Hardy spaces generated by rearrangement invariant \(p\)-spaces with \(0<p<1\). The aim of the paper is to give such an example. It is obtained in fact by an analytic version of a recent result of J. Bastero and F. Ruiz. More precisely, the main result says that if \(X\) is a rearrangement invariant space and \(M^*(X)\) is its associate Marcinkiewicz space then, the couples \((H(X),H^\infty)\), \((H(M^*(X)),H^\infty)\) are relative Calderón couples. In particular, when \(X=L^1\) it is obtained that the couples \((H^1,H^\infty)\), \((H^{1,\infty},H^\infty)\) are relative Calderón couples.