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For every $r\in(0,+\infty)$, we define $f^r:\mathbb R\to\mathbb R$ to be

$$f^r(x)=\begin{cases}\sqrt{r^2-x^2}&\text{if } |x|\leq r\\ 0 & \text{otherwise} \end{cases}$$

a)Find all $p\geq 1$ such that the map $r\to f^r$ from $(0,+\infty)$ to $L^p(\mathbb R)$ is continuous

b)Find all $p\geq 1$ such that the map $r\to f^r$ from $(0,+\infty)$ to $L^p(\mathbb R)$ is differentiable

-Mario-

Edit

sorry everybody... i miscopied the text... i edited.. really i apologize..

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    In the questions a) and b), is it $f$ instead of $f^r$ ? And what did you try ?2011-08-29
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    I am somewhat tempted to replace $]p,q[$ with $(p,q)$...2011-08-29
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    i've edited thanks davide. I've tried to calculate the $L^p$ norm and find a lipschitz constant. I've plugged the integral into mathematica but the expression is in terms of the gamma function so i was looking for something more manageable2011-08-29
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    Did you compute $\lVert f^{r+h}-f^r\lVert_{L^p}^p$ for $h>0$? Maybe you don't need to compute the integrals, but only to use some inequalities.2011-08-29
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    I am trying to understand how $f^r$ is defined: maybe you wanted to say that for every $r \in (0, \infty)$, $f^r$ is in $L^p(\mathbb{R})$. That's different from saying that for every $r$ we have $f^r : (0,\infty) \to L^p(\mathbb{R})$.2011-08-29
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    Sorry Angelo you were right... i miscopied.. now i've fixed the text... thanks for pointing it out.. -Mario-2011-08-29
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    @Angelo: You are correct: For each $r \in (0,\infty)$, we have $f^r : \mathbb R \to \mathbb R$ and $f^r \in L^p(\mathbb R)$. You need to prove **continuity** of a certain function specified in (a).2011-08-29
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    It's ok: it was probably the only possible interpretation, and in fact looks like nobody misunderstood what you meant.2011-08-30
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    I would like to put my own two cents regarding part a). Like Davide Giraudo suggested one calculates $\|f^{r+h}-f^r\|_{L^p}^p=2\int_{-r-h}^{-r}(\sqrt{(r+h)^2-x^2})^p\mathrm d x+\int_{-r}^{r}\left(\frac{2rh+h^2}{\sqrt{(r+h)^2-x^2}+\sqrt{r^2-x^2}}\right)^p\mathrm d x\leq 2h(r+h)^p+2rh^p$. Analogously $\|f^r-f^{r-h}\|_{L^p}^p\leq 2hr^p+2(r-h)h^p$. Hence continuity is ensured for any $p\geq 1$. Am I right?2011-08-30

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