On the composition of the integral and derivative operators of functional order

Hartzstein, Silvia I., and Viviani, Beatriz E.. "On the composition of the integral and derivative operators of functional order." Commentationes Mathematicae Universitatis Carolinae 44.1 (2003): 99-120. <http://eudml.org/doc/249161>.

@article{Hartzstein2003,
abstract = {The Integral, $I_\{\phi \}$, and Derivative, $D_\{\phi \}$, operators of order $\phi $, with $\phi $ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha $, where $\phi (t)=t^\{\alpha \}$, given in [GSV]. In this work we show that the composition $T_\{\phi \}= D_\{\phi \}\circ I_\{\phi \}$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_\{\phi \}$ and $D_\{\phi \}$ or the $T1$-theorems proved in [HV1] yield the fact that $T_\{\phi \}$ is a Calder’on-Zygmund operator bounded on the generalized Besov, $\dot\{B\}_\{p\}^\{\psi ,q\}$, $1 \le p,q < \infty $, and Triebel-Lizorkin spaces, $\dot\{F\}_\{p\}^\{\psi ,q\}$, $1< p, q < \infty $, of order $\psi = \psi _1/\psi _2$, where $\psi _1$ and $\psi _2$ are two quasi-increasing functions of adequate upper types $s_1$ and $s_2$, respectively.},
author = {Hartzstein, Silvia I., Viviani, Beatriz E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {fractional integral operators; fractional derivative operators; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces; fractional derivative; fractional integral; space of homogeneous type; kernel operators; singular integral operators},
language = {eng},
number = {1},
pages = {99-120},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the composition of the integral and derivative operators of functional order},
url = {http://eudml.org/doc/249161},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Hartzstein, Silvia I.
AU - Viviani, Beatriz E.
TI - On the composition of the integral and derivative operators of functional order
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 1
SP - 99
EP - 120
AB - The Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi $, with $\phi $ a function of positive lower type and upper type less than $1$, were defined in [HV2] in the setting of spaces of homogeneous-type. These definitions generalize those of the fractional integral and derivative operators of order $\alpha $, where $\phi (t)=t^{\alpha }$, given in [GSV]. In this work we show that the composition $T_{\phi }= D_{\phi }\circ I_{\phi }$ is a singular integral operator. This result in addition with the results obtained in [HV2] of boundedness of $I_{\phi }$ and $D_{\phi }$ or the $T1$-theorems proved in [HV1] yield the fact that $T_{\phi }$ is a Calder’on-Zygmund operator bounded on the generalized Besov, $\dot{B}_{p}^{\psi ,q}$, $1 \le p,q < \infty $, and Triebel-Lizorkin spaces, $\dot{F}_{p}^{\psi ,q}$, $1< p, q < \infty $, of order $\psi = \psi _1/\psi _2$, where $\psi _1$ and $\psi _2$ are two quasi-increasing functions of adequate upper types $s_1$ and $s_2$, respectively.
LA - eng
KW - fractional integral operators; fractional derivative operators; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces; fractional derivative; fractional integral; space of homogeneous type; kernel operators; singular integral operators
UR - http://eudml.org/doc/249161
ER -