Problem Sets

Continuous on compact implies uniformly continuous

Show that a continuous function on a compact set is uniformly continuous.

Use a stronger result: a continuous function on a compact metric is uniformly continuous. Fix \({\varepsilon}\). Suppose \(f\) is continuous, then for each \(z\in X\) choose \(\delta_z = \delta({\varepsilon}, z)\) to ensure \(B_\delta(z) \hookrightarrow B_{\varepsilon}(f(z))\) and form the cover \(\left\{{B_{\delta_z} (z)}\right\}_{x\in X}\rightrightarrows X\). By compactness, choose a finite subcover corresponding to \(\left\{{z_1, \cdots, z_m}\right\}\) and choose \(\delta = \min \left\{{\delta_1, \cdots, \delta_m}\right\}\). The claim is that this \(\delta\) works for uniform continuity: if \({\left\lvert {x-y} \right\rvert} < \delta\) then \({\left\lvert {x-y} \right\rvert} < \delta_i\) for all \(i\). Note that \(x\in B_{\delta_z}(z)\) for one of the finitely many \(z\) above, and if we adjust \(\delta\) to \(\delta/2\), we can arrange so that both \(x, y\in B_{\delta_z}(z)\) for some \(z\), since \begin{align*} {\left\lvert {x-y} \right\rvert} = {\left\lvert {x-z+z-y} \right\rvert} \leq {\left\lvert {x-z} \right\rvert} + {\left\lvert {z-y} \right\rvert} < {\delta \over 2 } + {\delta \over 2} = \delta < \delta_{z} ,\end{align*} and similarly \begin{align*} {\left\lvert {f(x)-f(y) } \right\rvert} \leq {\left\lvert {f(x)-f(z)} \right\rvert} + {\left\lvert {f(z)-f(y)} \right\rvert} < {\varepsilon}+ {\varepsilon} ,\end{align*} so just adjust the original \({\varepsilon}\) chosen by the continuity of \(f\) to \({\varepsilon}/2\).

2010 6.1

Show that \begin{align*} \int_{{\mathbb{B}}^n} {1 \over {\left\lvert {x} \right\rvert}^p } \,dx&< \infty \iff p < n \\ \\ \\ \int_{{\mathbb{R}}^n\setminus{\mathbb{B}}^n} {1 \over {\left\lvert {x} \right\rvert}^p } \,dx&< \infty \iff p > n .\end{align*}


2010 6.2

Show that \begin{align*} \int_{{\mathbb{R}}^n} {\left\lvert { f} \right\rvert} = \int_0^{\infty } m(A_t)\,dt&& A_t \coloneqq\left\{{x\in {\mathbb{R}}^n {~\mathrel{\Big\vert}~}{\left\lvert {f(x)} \right\rvert} > t}\right\} .\end{align*}


2010 6.5

Suppose \(F \subseteq {\mathbb{R}}\) with \(m(F^c) < \infty\) and let \(\delta(x) \coloneqq d(x, F)\) and \begin{align*} I_F(x) \coloneqq\int_{\mathbb{R}}{ \delta(y) \over {\left\lvert {x-y} \right\rvert}^2 } \,dy .\end{align*}

  • Show that \(\delta\) is continuous.

  • Show that if \(x\in F^c\) then \(I_F(x) = \infty\).

  • Show that \(I_F(x) < \infty\) for almost every \(x\)


2010 7.1

Let \((X, \mathcal{M}, \mu)\) be a measure space and prove the following properties of \(L^ \infty (X, \mathcal{M}, \mu)\):

  • If \(f, g\) are measurable on \(X\) then \begin{align*} {\left\lVert {fg} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_{\infty } .\end{align*}

  • \({\left\lVert {{-}} \right\rVert}_{\infty }\) is a norm on \(L^{\infty }\) making it a Banach space.

  • \({\left\lVert {f_n - f} \right\rVert}_{\infty } \overset{n\to \infty }\to 0 \iff\) there exists an \(E\in \mathcal{M}\) such that \(\mu(X\setminus E) = 0\) and \(f_n \to f\) uniformly on \(E\).

  • Simple functions are dense in \(L^{\infty }\).

2010 7.2

Show that for \(0 < p < q \leq \infty\), \({\left\lVert {a} \right\rVert}_{\ell^q} \leq {\left\lVert {a} \right\rVert}_{\ell^p}\) over \({\mathbb{C}}\), where \({\left\lVert {a} \right\rVert}_{\infty } \coloneqq\sup_j {\left\lvert {a_j} \right\rvert}\).

2010 7.3

Let \(f, g\) be non-negative measurable functions on \([0, \infty)\) with \begin{align*} A &\coloneqq\int_0^{\infty } f(y) y^{-1/2} \,dy< \infty \\ B &\coloneqq\qty{ \int_0^{\infty } {\left\lvert { g(y) } \right\rvert} }^2 \,dy< \infty .\end{align*}

Show that \begin{align*} \int_0^{\infty } \qty{ \int_0^{\infty } f(y) \,dy} {g(x) \over x} \,dx\leq AB .\end{align*}

2010 7.4

Let \((X, \mathcal{M}, \mu)\) be a measure space and \(0 < p < q< \infty\). Prove that if \(L^q(X) \subseteq L^p(X)\), then \(X\) does not contain sets of arbitrarily large finite measure.

2010 7.5

Suppose \(0 < a < b \leq \infty\), and find examples of functions \(f \in L^p((0, \infty ))\) if and only if:

  • \(a < p < b\)

  • \(a \leq p \leq b\)

  • \(p = a\)

Hint: consider functions of the following form: \begin{align*} f(x) \coloneqq x^{- \alpha} {\left\lvert { \log(x) } \right\rvert}^{ \beta} .\end{align*}

2010 7.6

Define \begin{align*} F(x) &\coloneqq\qty{ \sin(\pi x) \over \pi x}^2 \\ G(x) &\coloneqq \begin{cases} 1 - {\left\lvert {x} \right\rvert} & {\left\lvert {x} \right\rvert} \leq 1 \\ 0 & \text{else}. \end{cases} \end{align*}

  • Show that \(\widehat{G}(\xi) = F(\xi)\)

  • Compute \(\widehat{F}\).

  • Give an example of a function \(g\not \in L^1({\mathbb{R}})\) which is the Fourier transform of an \(L^1\) function.

Hint: write \(\widehat{G}(\xi) = H(\xi) + H(-\xi)\) where \begin{align*} H(\xi) \coloneqq e^{2\pi i \xi} \int_0^1 y e^{2\pi i y \xi }\,dy .\end{align*}

2010 7.7

Show that for each \(\epsilon>0\) the following function is the Fourier transform of an \(L^1({\mathbb{R}}^n)\) function: \begin{align*} F(\xi) \coloneqq\qty{1 \over 1 + {\left\lvert {\xi} \right\rvert}^2}^{\epsilon} .\end{align*}

Hint: show that

\begin{align*} K_\delta(x) &\coloneqq\delta^{-n/2} e^{-\pi {\left\lvert {x} \right\rvert}^2 \over \delta} \\ f(x) &\coloneqq\int_0^{\infty } K_{\delta}(x) e^{-\pi \delta} \delta^{\epsilon - 1} \,d \delta \\ \Gamma(s) &\coloneqq\int_0^{\infty } e^{-t} t^{s-1} \,dt\\ \implies \widehat{f}(\xi) &= \int_0^{\infty } e^{- \pi \delta {\left\lvert {\xi} \right\rvert}^2} e^{ -\pi \delta} \delta^{\epsilon - 1} = \pi^{-s} \Gamma(\epsilon) F(\xi) .\end{align*}

2010 7 Challenge 1: Generalized Holder

Suppose that \begin{align*} 1\leq p_j \leq \infty, && \sum_{j=1}^n {1\over p_j} = {1\over r} \leq 1 .\end{align*}

Show that if \(f_j \in L^{p_j}\) for each \(1\leq j \leq n\), then \begin{align*} \prod f_j \in L^r, && {\left\lVert { \prod f_j } \right\rVert}_r \leq \prod {\left\lVert {f_j} \right\rVert}_{p_j} .\end{align*}

2010 7 Challenge 2: Young’s Inequality

Suppose \(1\leq p,q,r \leq \infty\) with \begin{align*} {1\over p } + {1 \over q} = 1 + {1 \over r} .\end{align*}

Prove that \begin{align*} f \in L^p, g\in L^q \implies f \ast g \in L^r \text{ and } {\left\lVert {f \ast g} \right\rVert}_r \leq {\left\lVert {f} \right\rVert}_p {\left\lVert {g} \right\rVert}_q .\end{align*}

2010 9.1

Show that the set \(\left\{{ u_k(j) \coloneqq\delta_{ij} }\right\} \subseteq \ell^2({\mathbb{Z}})\) and forms an orthonormal system.

2010 9.2

Consider \(L^2([0, 1])\) and define \begin{align*} e_0(x) &= 1 \\ e_1(x) &= \sqrt{3}(2x-1) .\end{align*}

  • Show that \(\left\{{e_0, e_1}\right\}\) is an orthonormal system.

  • Show that the polynomial \(p(x)\) where \(\deg(p) = 1\) which is closest to \(f(x) = x^2\) in \(L^2([0, 1])\) is given by \begin{align*} h(x) = x - {1\over 6} .\end{align*}

Compute \({\left\lVert {f - g} \right\rVert}_2\).

2010 9.3

Let \(E \subseteq H\) a Hilbert space.

  • Show that \(E\perp \subseteq H\) is a closed subspace.

  • Show that \((E^\perp)^\perp = { \operatorname{cl}} _H(E)\).

2010 9.5b

Let \(f\in L^1((0, 2\pi))\).

  • Show that for an \(\epsilon>0\) one can write \(f = g+h\) where \(g\in L^2((0, 2\pi))\) and \({\left\lVert {H} \right\rVert}_1 < \epsilon\).

2010 9.6

Prove that every closed convex \(K \subset H\) a Hilbert space has a unique element of minimal norm.

2010 9 Challenge

Let \(U\) be a unitary operator on \(H\) a Hilbert space, let \(M \coloneqq\left\{{x\in H {~\mathrel{\Big\vert}~}Ux = x}\right\}\), let \(P\) be the orthogonal projection onto \(M\), and define \begin{align*} S_N \coloneqq{1\over N} \sum_{n=0}^{N-1} U^n .\end{align*} Show that for all \(x\in H\), \begin{align*} {\left\lVert { S_N x - Px} \right\rVert}_H \overset{N\to \infty } \to 0 .\end{align*}

2010 10.1

Let \(\nu, \mu\) be signed measures, and show that \begin{align*} \nu \perp \mu \text{ and } \nu \ll {\left\lvert { \mu} \right\rvert} \implies \nu = 0 .\end{align*}

2010 10.2

Let \(f\in L^1({\mathbb{R}}^n)\) with \(f\neq 0\).

  • Prove that there exists a \(c>0\) such that \begin{align*} Hf(x) \geq {c \over (1 + {\left\lvert {x} \right\rvert})^n } .\end{align*}

2010 10.3

Consider the function \begin{align*} f(x) \coloneqq \begin{cases} {1\over {\left\lvert {x} \right\rvert} \qty{ \log\qty{1\over x}}^2 } & {\left\lvert {x} \right\rvert} \leq {1\over 2} \\ 0 & \text{else}. \end{cases} \end{align*}

  • Show that \(f \in L^1({\mathbb{R}})\).

  • Show that there exists a \(c>0\) such that for all \({\left\lvert {x} \right\rvert} \leq 1/2\), \begin{align*} Hf(x) \geq {c \over {\left\lvert {x} \right\rvert} \log\qty{1\over {\left\lvert {x} \right\rvert}} } .\end{align*} Conclude that \(Hf\) is not locally integrable.

2010 10.4

Let \(f\in L^1({\mathbb{R}})\) and let \(\mathcal{U}\coloneqq\left\{{(x, y) \in {\mathbb{R}}^2 {~\mathrel{\Big\vert}~}y > 0}\right\}\) denote the upper half plane. For \((x, y) \in \mathcal{U}\) define \begin{align*} u(x, y) \coloneqq f \ast P_y(x) && \text{where } P_y(x) \coloneqq{1\over \pi}\qty{y \over t^2 + y^2} .\end{align*}

  • Prove that there exists a constant \(C\) independent of \(f\) such that for all \(x\in {\mathbb{R}}\), \begin{align*} \sup_{y > 0} {\left\lvert { u(x, y) } \right\rvert} \leq C\cdot Hf(x) .\end{align*}

    Hint: write the following and try to estimate each term: \begin{align*} u(x, y) = \int_{{\left\lvert {t} \right\rvert} < y} f(x - t) P_y(t) \,dt+ \sum_{k=0}^{\infty } \int_{A_k} f(x-t) P_y(t)\,dt&& A_k \coloneqq\left\{{2^ky \leq {\left\lvert {t} \right\rvert} < 2^{k+1}y}\right\} .\end{align*}

  • Following the proof of the Lebesgue differentiation theorem, show that for \(f\in L^1({\mathbb{R}})\) and for almost every \(x\in {\mathbb{R}}\), \begin{align*} u(x, y) \overset{y\to 0} \to f(x) .\end{align*}