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

Rocha, Pablo, and Urciuolo, Marta. "On the $H^{p}$-$L^{q}$ boundedness of some fractional integral operators." Czechoslovak Mathematical Journal 62.3 (2012): 625-635. <http://eudml.org/doc/246304>.

@article{Rocha2012,
abstract = {Let $A_\{1\},\dots ,A_\{m\}$ be $n\times n$ real matrices such that for each $1\le i\le m,$$A_\{i\}$ is invertible and $A_\{i\}-A_\{j\}$ is invertible for $i\ne j$. 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\le q_\{i\}<\infty ,$$1/\{q_\{1\}\}+1/\{q_\{2\}\}+\dots +1/\{q_\{m\}\}=1-r,$$0\le r<1,$ 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<p<1/\{r\}$ 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.},
author = {Rocha, Pablo, Urciuolo, Marta},
journal = {Czechoslovak Mathematical Journal},
keywords = {integral operator; Hardy space; integral operator; Hardy space},
language = {eng},
number = {3},
pages = {625-635},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the $H^\{p\}$-$L^\{q\}$ boundedness of some fractional integral operators},
url = {http://eudml.org/doc/246304},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Rocha, Pablo
AU - Urciuolo, Marta
TI - On the $H^{p}$-$L^{q}$ boundedness of some fractional integral operators
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 625
EP - 635
AB - Let $A_{1},\dots ,A_{m}$ be $n\times n$ real matrices such that for each $1\le i\le m,$$A_{i}$ is invertible and $A_{i}-A_{j}$ is invertible for $i\ne j$. 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\le q_{i}<\infty ,$$1/{q_{1}}+1/{q_{2}}+\dots +1/{q_{m}}=1-r,$$0\le r<1,$ 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<p<1/{r}$ 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.
LA - eng
KW - integral operator; Hardy space; integral operator; Hardy space
UR - http://eudml.org/doc/246304
ER -