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

Authors:

Affiliation:

(1) Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

Abstract:

Let

be realhomogeneous functions in $C^infty (mathbb{R}^n - ;{ 0} )$ ofdegree $k geqslant 2,{text{ let }}varphi {text{(}}x{text{)}}; = (varphi _1 (x),;.,.;.;,;varphi _n (x))$ and let mgr

bethe Borel measure on $mathbb{R}^{2n}$ given by

$mu (E) = int_{mathbb{R}^n } {chi _E } (x,alpha (x));left| x right|^{gamma - n} {text{d}}x$

where dx denotes theLebesgue measure on $mathbb{R}^n$ and gamma

> 0. Let T be the convolution operator $$T_mu f(x) = (mu * f)(x)$$

and let

$E_mu = { ({1 mathord{left/ {vphantom {1 {p,;{1 mathord{left/ {vphantom {1 q}} right. kern-nulldelimiterspace} q}}}} right. kern-nulldelimiterspace} {p,;{1 mathord{left/ {vphantom {1 q}} right. kern-nulldelimiterspace} q}}}):;;parallel {kern 1pt} T_mu {kern 1pt} parallel _{p,q} ; < infty ,;;1 leqslant p,q leqslant infty } .$

Assume that, for x

0, the followingtwo conditions hold: $det ({text{d}}^{text{2}} varphi (x)h)$ vanishes only at h = 0 and $$det ({text{d}}varphi (x)) ne 0$$

. In this paper we show that if $$gamma >n(k + 1)/3$$

then E is the empty set and if $$gamma leqslant n(k + 1)/3$$

then E is the closed segment withendpoints $D = (1 - tfrac{gamma }{{n(k + 1)}},;1 - tfrac{{2gamma }}{{n(k + 1)}})$ and $D' = (tfrac{{2gamma }}{{n(1 + k)}},;tfrac{gamma }{{n(1 + k)}})$ . Also, we give some examples.

Keywords:

singular measures convolution operators

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