# Article

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Keywords:
singular measures; convolution operators
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.
References:
[1] M. Christ: Endpoint bounds for singular fractional integral operators. UCLA Preprint (1988). MR 0951506
[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. DOI 10.1017/S0305004100070201 | MR 1104610
[3] E. Ferreyra, T.  Godoy and M.  Urciuolo: Convolution operators with fractional measures associated to holomorphic functions. Acta Math. Hungar 92 (2001), 27–38. DOI 10.1023/A:1013795825882 | MR 1924246
[4] D.  Oberlin: Convolution estimates for some measures on curves. Proc. Amer. Math. Soc. 99 (1987), 56–60. DOI 10.1090/S0002-9939-1987-0866429-6 | MR 0866429 | Zbl 0613.43002
[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.
[6] S.  Secco: Fractional integration along homogeneous curves in $R^3$. Math. Scand. 85 (1999), 259–270. MR 1724238
[7] E. M.  Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1970. MR 0290095 | Zbl 0207.13501
[8] E. M.  Stein: Harmonic Analysis. Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. MR 1232192 | Zbl 0821.42001

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