The type set for some measures on ${ℝ}^{2n}$ with $n$-dimensional support

Ferreyra, E., Godoy, T., and Urciuolo, Marta. "The type set for some measures on $\mathbb {R}^{2n}$ with $n$-dimensional support." Czechoslovak Mathematical Journal 52.3 (2002): 575-583. <http://eudml.org/doc/30726>.

@article{Ferreyra2002,
abstract = {Let $\varphi _1,\dots ,\varphi _n$ be real homogeneous functions in $C^\infty (\mathbb \{R\}^n-\lbrace 0\rbrace )$ of degree $k\ge 2$, let $\varphi (x) =(\varphi _1(x),\dots ,\varphi _n(x))$ and let $\mu $ be the Borel measure on $\mathbb \{R\}^\{2n\}$ given by \[ \mu (E) =\int \_\{\mathbb \{R\}^n\}\chi \_E(x,\varphi (x))\, |x|^\{\gamma -n\}\mathrm \{d\}x \] where $\mathrm \{d\}x$ denotes the Lebesgue measure on $\mathbb \{R\}^n$ and $\gamma >0$. Let $T_\mu $ be the convolution operator $T_\mu f(x)=(\mu *f)(x)$ and let \[ E\_\mu =\lbrace (1/p,1/q)\:\Vert T\_\mu \Vert \_\{p,q\}<\infty ,\hspace\{5.0pt\}1\le p, \,q\le \infty \rbrace . \] Assume that, for $x\ne 0$, the following two conditions hold: $\det (\{\mathrm \{d\}\}^2\varphi (x) h)$ vanishes only at $h=0$ and $\det (\{\mathrm \{d\}\} \varphi (x)) \ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_\mu $ is the empty set and if $\gamma \le n(k+1)/3$ then $E_\mu $ is the closed segment with endpoints $D=\bigl (1-\frac\{\gamma \}\{n(k+1)\},1-\frac\{2\gamma \}\{n(k+1)\}\bigr )$ and $D^\{\prime \}=\bigl (\frac\{2\gamma \}\{n(1+k)\},\frac\{\gamma \}\{n(1+k)\}\bigr )$. Also, we give some examples.},
author = {Ferreyra, E., Godoy, T., Urciuolo, Marta},
journal = {Czechoslovak Mathematical Journal},
keywords = {singular measures; convolution operators; singular measures; convolution operators},
language = {eng},
number = {3},
pages = {575-583},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The type set for some measures on $\mathbb \{R\}^\{2n\}$ with $n$-dimensional support},
url = {http://eudml.org/doc/30726},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Ferreyra, E.
AU - Godoy, T.
AU - Urciuolo, Marta
TI - The type set for some measures on $\mathbb {R}^{2n}$ with $n$-dimensional support
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 3
SP - 575
EP - 583
AB - Let $\varphi _1,\dots ,\varphi _n$ be real homogeneous functions in $C^\infty (\mathbb {R}^n-\lbrace 0\rbrace )$ of degree $k\ge 2$, let $\varphi (x) =(\varphi _1(x),\dots ,\varphi _n(x))$ and let $\mu $ be the Borel measure on $\mathbb {R}^{2n}$ given by \[ \mu (E) =\int _{\mathbb {R}^n}\chi _E(x,\varphi (x))\, |x|^{\gamma -n}\mathrm {d}x \] where $\mathrm {d}x$ denotes the Lebesgue measure on $\mathbb {R}^n$ and $\gamma >0$. Let $T_\mu $ be the convolution operator $T_\mu f(x)=(\mu *f)(x)$ and let \[ E_\mu =\lbrace (1/p,1/q)\:\Vert T_\mu \Vert _{p,q}<\infty ,\hspace{5.0pt}1\le p, \,q\le \infty \rbrace . \] Assume that, for $x\ne 0$, the following two conditions hold: $\det ({\mathrm {d}}^2\varphi (x) h)$ vanishes only at $h=0$ and $\det ({\mathrm {d}} \varphi (x)) \ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_\mu $ is the empty set and if $\gamma \le n(k+1)/3$ then $E_\mu $ is the closed segment with endpoints $D=\bigl (1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\bigr )$ and $D^{\prime }=\bigl (\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\bigr )$. Also, we give some examples.
LA - eng
KW - singular measures; convolution operators; singular measures; convolution operators
UR - http://eudml.org/doc/30726
ER -