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

Authors: Ferreyra E.¹; Godoy T.²; Urciuolo M.³

Source: Czechoslovak Mathematical Journal, Volume 52, Number 3, September 2002 , pp. 575-583(9)

Publisher: Springer

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Abstract:

Let \varphi_1, \ldots , \varphi_n be real homogeneous functions in C^\infty({\Bbb R}^n-\{0\}) of degree k\geq 2, let \varphi(x) =(\varphi_1(x),\ldots,\varphi_n(x)) and let \mu be the Borel measure on {\Bbb R}^{2n} given by \mu(E)= \int_{{\bf R}^n}\chi_ E(x,\varphi(x))\vert x \vert^{\gamma-n}{\rm d}x where {\rm d}x denotes the Lebesgue measure on {\Bbb R}^{n} and \gamma\gt 0. Let T_\mu be the convolution operator T_\mu f(x)=(\mu\ast f)(x) and let E\mu=\{(1/p,1/q): \ \parallel T_\mu\parallel_{p,q}\thinspace \lt \infty, \ 1 \leq p, q \leq \infty\}. Assume that, for x\neq 0, the following two conditions hold: {\rm \det(d^2}\varphi(x) h)vanishes only at h=0 and {\rm \det(d}\varphi(x))\neq 0. In this paper we show that if \gamma\gt n(k+1)/3then E_\mu is the empty set and if \gamma\leq n(k+1)/3 then E_\mu is the closed segment with endpoints D=(1-{\gamma \over {n(k+1)}},1-{2\gamma \over {n(k+1)}}) and D^\prime=({2\gamma \over {n(1+k)}},{\gamma \over {n(1+k)}}). Also, we give some examples.

Keywords: singular measures; convolution operators

Language: English

Document Type: Research article

Affiliations: 1: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina, eferrey@mate.uncor.edu 2: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina, godoy@mate.uncor.edu 3: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina, urciuolo@mate.uncor.edu

Publication date: 2002-09-01