Ngô Quốc Anh

June 9, 2010

Invariance under fractional linear transformations

Filed under: Giải Tích 5, Giải Tích 6 (MA5205) — Tags: — Ngô Quốc Anh @ 5:02

I just read the following result due to Loss-Sloane published in J. Funct. Anal. this year [here]. This is just a lemma in their paper that I found very interesting.

Let f be any function in C_0^\infty(\mathbb R \setminus \{0\}). Consider the inversion x \mapsto \frac{1}{x} and set

\displaystyle g(x) = {\left| x \right|^{\alpha - 1}}f\left( {\frac{1}{x}} \right).

Then g \in C_0^\infty(\mathbb R) and

\displaystyle\iint\limits_{{\mathbb{R}^2}} {\frac{{{{\left| {g(x) - g(y)} \right|}^2}}}{{{{\left| {x - y} \right|}^{\alpha + 1}}}}dxdy} = \iint\limits_{{\mathbb{R}^2}} {\frac{{{{\left| {f(x) - f(y)} \right|}^2}}}{{{{\left| {x - y} \right|}^{\alpha + 1}}}}dxdy}.

Proof. For fixed \varepsilon consider the regions

\displaystyle {R_1}: = \left\{ {(x,y) \in {\mathbb{R}^2}:\left| {\frac{x}{y}} \right| > 1 + \varepsilon } \right\}

and likewise,

\displaystyle {R_2}: = \left\{ {(x,y) \in {\mathbb{R}^2}:\left| {\frac{y}{x}} \right| > 1 + \varepsilon } \right\}.

(more…)

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