| Title:
|
The type set for some measures on $\mathbb R^{2n}$ with $n$-dimensional support (English) |
| Author:
|
Ferreyra, E. |
| Author:
|
Godoy, T. |
| Author:
|
Urciuolo, M. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2002 |
| Pages:
|
575-583 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
singular measures |
| Keyword:
|
convolution operators |
| MSC:
|
28C10 |
| MSC:
|
42B15 |
| MSC:
|
42B20 |
| MSC:
|
47B38 |
| idZBL:
|
Zbl 1012.42012 |
| idMR:
|
MR1923263 |
| . |
| Date available:
|
2009-09-24T10:54:26Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127745 |
| . |
| Reference:
|
[1] M. Christ: Endpoint bounds for singular fractional integral operators.UCLA Preprint (1988). MR 0951506 |
| Reference:
|
[2] S. W. Drury and K. Guo: Convolution estimates related to surfaces of half the ambient dimension.Math. Proc. Camb. Phil. Soc. 110 (1991), 151–159. MR 1104610, 10.1017/S0305004100070201 |
| Reference:
|
[3] E. Ferreyra, T. Godoy and M. Urciuolo: Convolution operators with fractional measures associated to holomorphic functions.Acta Math. Hungar 92 (2001), 27–38. MR 1924246, 10.1023/A:1013795825882 |
| Reference:
|
[4] D. Oberlin: Convolution estimates for some measures on curves.Proc. Amer. Math. Soc. 99 (1987), 56–60. Zbl 0613.43002, MR 0866429, 10.1090/S0002-9939-1987-0866429-6 |
| Reference:
|
[5] F. Ricci: Limitatezza $L^p$-$L^q$ per operatori di convoluzione definiti da misure singolari in $R^n$.Bollettino U.M.I. 7 11-A (1997), 237–252. |
| Reference:
|
[6] S. Secco: Fractional integration along homogeneous curves in $R^3$.Math. Scand. 85 (1999), 259–270. MR 1724238, 10.7146/math.scand.a-18275 |
| Reference:
|
[7] E. M. Stein: Singular Integrals and Differentiability Properties of Functions.Princeton University Press, 1970. Zbl 0207.13501, MR 0290095 |
| Reference:
|
[8] E. M. Stein: Harmonic Analysis. Real Variable Methods, Orthogonality and Oscillatory Integrals.Princeton University Press, 1993. Zbl 0821.42001, MR 1232192 |
| . |
