Chapter 6

(A Generalized Hölder Inequality) Suppose that \(1 \leq p_{j} \leq \infty\) and \(\sum_{1}^{n} p_{j}^{-1}=\) \(r^{-1} \leq 1 .\) If \(f_{j} \in L^{p_{2}}\) for \(j=1, \ldots, n\), then \(\prod_{1}^{n} f_{j} \in L^{r}\) and \(\left\|\prod_{1}^{n} f_{j}\right\|_{r} \leq\) \(\prod_{1}^{n}\left\|f_{j}\right\|_{P_{j}}\). (First do the case \(n=2 .\) )

Given \(1

If \(f\) is absolutely continuous on \([\epsilon, 1]\) for \(0<\epsilon<1\) and \(\int_{0}^{1} x\left|f^{\prime}(x)\right|^{p} d x<\infty\), then \(\lim _{x \rightarrow 0} f(x)\) exists (and is finite) if \(p>2,|f(x)| /|\log x|^{1 / 2} \rightarrow 0\) as \(x \rightarrow 0\) if \(p=2\), and \(|f(x)| / x^{1-(2 / p)} \rightarrow 0\) as \(x \rightarrow 0\) if \(p<2\).

If \(f \in\) weak \(L^{p}\) and \(\mu(\\{x: f(x) \neq 0\\})<\infty\), then \(f \in L^{q}\) for all \(qp\).

Suppose \(1

Let \(H\) be the Hardy-Littlewood maximal operator on \(\mathbb{R}\). Compute \(H \chi(0,1)\) explicitly. Show that it is in \(L^{p}\) for all \(p>1\) and in weak \(L^{1}\) but not in \(L^{1}\), and that its \(L^{p}\) norm tends to \(\infty\) like \((p-1)^{-1}\) as \(p \rightarrow 1\), although \(\left\|\chi_{(0,1)}\right\|_{p}=1\) for all \(p\).

If \(0<\alpha