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- Jun 20, 2014

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Find the norm of the linear operator $T: \mathscr{L}^p(0, \infty) \to \mathscr{L}^p(0, \infty)$ defined by the equation

$$(Tf)(x) = \frac{1}{x}\int_0^x f(t)\, dt$$

Here it is assumed that $1 < p < \infty$.

Note: The space $\mathscr{L}^p(0,\infty)$ consists of all Lebesgue integrable functions $f : (0,\infty) \to \Bbb R$ such that $\|f\|_p < \infty$. For $p < \infty$, $\|f\|_p := \left(\int_0^\infty \lvert f(x)\rvert^p\, dx\right)^{1/p}$.

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