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Chapter 6: Problem 4

Use H枚lder's inequality to prove $$ \|u\|_{q} \leq\|u\|_{p}^{\lambda}\|u\|_{r}^{1-\lambda} \quad \text { for } u \in L^{r}(\Omega), $$ where \(p \leq q \leq r\) and \(q^{-1}=\lambda p^{-1}+(1-\lambda) r^{-1}\).

Short Answer

Expert verified
The proof of the inequality \(\|u\|_{q} \leq \|u\|_{p}^{\lambda} \|u\|_{r}^{1-\lambda}\) for a function \(u\) in \(L^{r}(\Omega)\), where \(p \leq q \leq r\) and \(q^{-1}=\lambda p^{-1}+(1-\lambda) r^{-1}\), is obtained through the application of H枚lder's inequality and the manipulation of the given relationship between p, q, and r.

Step by step solution

01

Understand the problem and Identify the given information

We are given that \(p \leq q \leq r\), the function \(u\) belongs to \(L^{r}(\Omega)\), and \(q^{-1}=\lambda p^{-1}+(1-\lambda) r^{-1}\). We need to prove that \(\|u\|_{q} \leq\|u\|_{p}^{\lambda}\|u\|_{r}^{1-\lambda}\). The symbols \(\|\cdot\|_{p}\), \(\|\cdot\|_{q}\), and \(\|\cdot\|_{r}\) represent the Lp, Lq, and Lr norms of a function, respectively.
02

Apply H枚lder's Inequality

Firstly, by H枚lder's inequality for two functions \(f\) and \(g\) in \(L^{p}\) and \(L^{p'}\) spaces respectively, where \(\frac{1}{p} + \frac{1}{p'} = 1\), we have \(\int |fg| \leq \left(\int |f|^{p}\right)^{\frac{1}{p}} \left(\int |g|^{p'} \right)^{\frac{1}{p'}}.\) On applying H枚lder's inequality to the function u with \(f=u^{q/p}\) and \(g=u^{q/r}\) with \(p'\) replaced by \(r\) (since \(p\leq r\)), we obtain \(\|u\|_{q}=\left( \int |u|^{q} \right)^{\frac{1}{q}} \leq \|u\|_{p}^{\frac{q}{p}} \|u\|_{r}^{\frac{q}{r}},\)
03

Apply the given identity

Finally, by the given identity \(q^{-1} = \lambda p^{-1} + (1-\lambda) r^{-1},\) we can rearrange terms to get \(\frac{q}{p} = \lambda\) and \(\frac{q}{r} = 1-\lambda.\) Substituting these into the inequality obtained in step 2, we get \(\|u\|_{q} \leq \|u\|_{p}^{\lambda} \|u\|_{r}^{1-\lambda},\) as required.

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Most popular questions from this chapter

Show that if \(u \in L^{p}(\Omega)\) for \(1 \leq p<\infty\) and \(z \in \mathbf{R}^{n}\), then the shift \(S_{z} u(r)=u(x-z)\) converges in \(L^{p}(\Omega)\) to \(u(x)\) uniformly as \(|z| \rightarrow 0\).

Use H枚lder's inequality to prove $$ \|u\|_{q} \leq\|u\|_{p}^{\lambda}\|u\|_{r}^{1-\lambda} \quad \text { for } u \in L^{r}(\Omega), $$ where \(p \leq q \leq r\) and \(q^{-1}=\lambda p^{-1}+(1-\lambda) r^{-1}\).

If \(T: X \rightarrow X\) is a compact linear operator on a complex Hilbert space \(X\), and \(\lambda \notin \sigma_{p}(T) \cup\\{0\\}\), then \(\lambda \in \rho(T)\).

If \(X\) is a Hilbert space and \(T: X \rightarrow X\) is a bounded linear operator, the adjoint of \(T\) is an operator \(T^{*}: X \rightarrow X\) defined as follows: (a) For \(y \in X\), use the Riesz representation theorem to define \(T^{*} y \in X\) satisfying \(\langle T x, y\rangle=\left\langle x, T^{*} y\right\rangle\) for all \(x, y \in X\). (b) Show that \(T^{*}: X \rightarrow X\) is a bounded linear operator with \(\left\|T^{*}\right\|=\) \(\|T\| .\)

(a) If \(X\) is a normed vector space, prove that \(|\|x\|-\|y\|| \leq\|x-y\|\). (b) Show that the norm defines a continuous function on \(X\). (c) If \(X\) is a real inner product space, prove (1).
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