On the $H^{p}$-$L^{q}$ boundedness of some fractional integral operators

Rocha, Pablo ; Urciuolo, M.

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 625-635

Résumé

Let $A_{1},\dots ,A_{m}$ be $n\times n$ real matrices such that for each $1\leq i\leq m,$ $A_{i}$ is invertible and $A_{i}-A_{j}$ is invertible for $i\neq j$. In this paper we study integral operators of the form $$ Tf( x) =\int k_{1}( x-A_{1}y) k_{2}( x-A_{2}y) \dots k_{m}( x-A_{m}y) f( y) {\rm d} y, $$ $k_{i}( y) =\sum _{j\in \mathbb Z}2^{jn/{q_{i}}}\varphi _{i,j}( 2^{j}y) $, $1\leq q_{i}\infty ,$ $1/{q_{1}}+1/{q_{2}}+\dots +1/{q_{m}}=1-r,$ $0\leq r1,$ and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T\colon H^{p}( \mathbb {R} ^{n}) \rightarrow L^{q}( \mathbb {R}^{n}) $ for $ 0$ and $1/{q}=1/{p}-r.$ We also show that we can not expect the $H^{p}$-$H^{q}$ boundedness of this kind of operators.

Détail

DOI : 10.1007/s10587-012-0054-1

Classification : 42B20, 42B30
Keywords: integral operator; Hardy space