Intereting Posts

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show that $^3\frac{\tan(10\pi/21)}{\tan(2\pi/21)}=4(23\sqrt7+35\sqrt3)$
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Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$
Why does drawing $\square$ mean the end of a proof?
Prove that in any GCD domain every irreducible element is prime
Embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.
What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?
Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$
Expectation of hitting time of a markov chain
Why is the collection of all groups considered a proper class rather than a set?
How many associative binary operations there are on a finite set?
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Gradients of marginal likelihood of Gaussian Process with squared exponential covariance, for learning hyper-parameters

Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$,

$$

\lVert f \rVert_{r} \leq

(\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)}

(\lVert f \rVert_{q})^{(1/r-1/q)/(1/p-1/q)}\:?

$$

I tried using Hölder to show that $f$ is in $L^{r}$ but I’m completely lost on how to go about it…

- A convex function is differentiable at all but countably many points
- Proving that $\lim\limits_{x\to 0}\frac{f(x)}{g(x)}=L$ implies $\lim\limits_{r\to 0}\frac {\int_R f(ry)h(y)\,dy}{\int_R g(ry)h(y)dy}=L$
- What does “sets of arbitrarily large measure” mean — question about $L_p$ embeddings
- An application of Jensen's Inequality
- Let $(s_n)$ be a sequence of nonnegative numbers, and $\sigma_n=\frac{1}{n}(s_1+s_2+\cdots +s_n)$. Show that $\liminf s_n \le \liminf \sigma_n$.
- Is $\sigma$-finiteness unnecessary for Radon Nikodym theorem?
- The distance function on a metric space
- An alternative proof of Cauchy's Mean Value Theorem
- Continuity/differentiability at a point and in some neighbourhood of the point
- Show that $f(x) = \log\log(\frac{e}{\Vert x \Vert_{2}})$ is in Sobolev space $W^{1,p}(B_1(0))$

If $p<r<q$ then $\frac{1}{q}<\frac{1}{r}<\frac{1}{p}$, hence there exists $\theta \in ]0,1[$ s.t. $\frac{1}{r}=\frac{\theta}{q}+\frac{1-\theta}{p}$; therefore:

$|f|^r=|f|^{r\theta}\ |f|^{r(1-\theta)} =|f|^{p\ \frac{r\theta}{p}}\ |f|^{q\ \frac{r(1-\theta)}{q}}$,

and you can apply *Hölder’s inequality*, because $|f|^{p \frac{r\theta}{p}} \in L^{\frac{p}{r\theta}}$ and $|f|^{q\ \frac{r(1-\theta)}{q}} \in L^{\frac{q}{r(1-\theta)}}$ and the exponents $\frac{p}{r\theta}, \frac{q}{r(1-\theta)}$ are conjugate.

Finally, to get your claim it suffices to compute explicitly the value of $\theta$.

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