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Keywords:
singular measures; convolution operators
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.
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