Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type

@article{Hartzstein2002,
abstract = {In the setting of spaces of homogeneous-type, we define the Integral, $I_\{\phi \}$, and Derivative, $D_\{\phi \}$, operators of order $\phi $, where $\phi $ is a function of positive lower type and upper type less than $1$, and show that $I_\{\phi \}$ and $D_\{\phi \}$ are bounded from Lipschitz spaces $\Lambda ^\{\xi \}$ to $\Lambda ^\{\xi \phi \}$ and $\Lambda ^\{\xi /\phi \}$ respectively, with suitable restrictions on the quasi-increasing function $\xi $ in each case. We also prove that $I_\{\phi \}$ and $D_\{\phi \}$ are bounded from the generalized Besov $\dot\{B\}_\{p\}^\{\psi , q\}$, with $1 \le p, q < \infty $, and Triebel-Lizorkin spaces $\dot\{F\}_\{p\}^\{\psi , q\}$, with $1 < p, q < \infty $, of order $\psi $ to those of order $\phi \psi $ and $\psi /\phi $ respectively, where $\psi $ is the quotient of two quasi-increasing functions of adequate upper types.},
author = {Hartzstein, Silvia I., Viviani, Beatriz E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {integral and derivative operators of functional order; fractional integral operator; fractional derivative operator; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces; Besov spaces; Triebel-Lizorkin spaces; integral; derivative; functional order},
language = {eng},
number = {4},
pages = {723-754},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type},
url = {http://eudml.org/doc/248968},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Hartzstein, Silvia I.
AU - Viviani, Beatriz E.
TI - Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 4
SP - 723
EP - 754
AB - In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi $, where $\phi $ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi }$ and $D_{\phi }$ are bounded from Lipschitz spaces $\Lambda ^{\xi }$ to $\Lambda ^{\xi \phi }$ and $\Lambda ^{\xi /\phi }$ respectively, with suitable restrictions on the quasi-increasing function $\xi $ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov $\dot{B}_{p}^{\psi , q}$, with $1 \le p, q < \infty $, and Triebel-Lizorkin spaces $\dot{F}_{p}^{\psi , q}$, with $1 < p, q < \infty $, of order $\psi $ to those of order $\phi \psi $ and $\psi /\phi $ respectively, where $\psi $ is the quotient of two quasi-increasing functions of adequate upper types.
LA - eng
KW - integral and derivative operators of functional order; fractional integral operator; fractional derivative operator; spaces of homogeneous type; Besov spaces; Triebel-Lizorkin spaces; Besov spaces; Triebel-Lizorkin spaces; integral; derivative; functional order
UR - http://eudml.org/doc/248968
ER -