Analysis Of Functions

• Paper 1, Section II, $22 F$

  Analysis of Functions

  Part II, 2017

  Consider a sequence $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ of measurable functions converging pointwise to a function $f: \mathbb{R} \rightarrow \mathbb{R}$. The Lebesgue measure is denoted by $\lambda$.

  (a) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Define for $k, n \geqslant 1$ the sets

  $$E_{n}^{(k)}:=\bigcap_{m \geqslant n}\left\{x \in A|| f_{m}(x)-f(x) \mid \leqslant \frac{1}{k}\right\}$$

  Prove that for any $k, n \geqslant 1$, one has $E_{n}^{(k)} \subset E_{n+1}^{(k)}$ and $E_{n}^{(k+1)} \subset E_{n}^{(k)}$. Prove that for any $k \geqslant 1, A=\cup_{n \geqslant 1} E_{n}^{(k)}$.

  (b) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that for any $\varepsilon>0$, there is a Borel set $A_{\varepsilon} \subset A$ for which $\lambda\left(A \backslash A_{\varepsilon}\right) \leqslant \varepsilon$ and such that $f_{n}$ converges to $f$ uniformly on $A_{\varepsilon}$ as $n \rightarrow+\infty$. Is the latter still true when $\lambda(A)=+\infty$ ?

  (c) Assume additionally that $f_{n} \in L^{p}(\mathbb{R})$ for some $p \in(1,+\infty]$, and there exists an $M \geqslant 0$ for which $\left\|f_{n}\right\|_{L^{p}(\mathbb{R})} \leqslant M$ for all $n \geqslant 1$. Prove that $f \in L^{p}(\mathbb{R})$.

  (d) Let $f_{n}$ and $f$ be as in part (c). Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that $f_{n}, f$ are integrable on $A$ and $\int_{A} f_{n} d \lambda \rightarrow \int_{A} f d \lambda$ as $n \rightarrow \infty$. Deduce that $f_{n}$ converges weakly to $f$ in $L^{p}(\mathbb{R})$ when $p<+\infty$. Does the convergence have to be strong?

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• Paper 3, Section II, F

  Analysis of Functions

  Part II, 2017

  Denote by $C_{0}\left(\mathbb{R}^{n}\right)$ the space of continuous complex-valued functions on $\mathbb{R}^{n}$ converging to zero at infinity. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$.

  (i) Prove that the image of $L^{1}\left(\mathbb{R}^{n}\right)$ under $\mathcal{F}$ is included and dense in $C_{0}\left(\mathbb{R}^{n}\right)$, and that $\mathcal{F}: L^{1}\left(\mathbb{R}^{n}\right) \rightarrow C_{0}\left(\mathbb{R}^{n}\right)$ is injective. [Fourier inversion can be used without proof when properly stated.]

  (ii) Calculate the Fourier transform of $\chi_{[a, b]}$, the characteristic function of $[a, b] \subset \mathbb{R}$.

  (iii) Prove that $g_{n}:=\chi_{[-n, n]} * \chi_{[-1,1]}$ belongs to $C_{0}(\mathbb{R})$ and is the Fourier transform of a function $h_{n} \in L^{1}(\mathbb{R})$, which you should determine.

  (iv) Using the functions $h_{n}, g_{n}$ and the open mapping theorem, deduce that the Fourier transform is not surjective from $L^{1}(\mathbb{R})$ to $C_{0}(\mathbb{R})$.

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• Paper 4, Section II, $22 F$

  Analysis of Functions

  Part II, 2017

  Consider $\mathbb{R}^{n}$ with the Lebesgue measure. Denote by $\mathcal{F} f(\xi)=\int_{\mathbb{R}^{n}} e^{-2 i \pi x \cdot \xi} f(x) d x$ the Fourier transform of $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and by $\hat{f}$ the Fourier-Plancherel transform of $f \in L^{2}\left(\mathbb{R}^{n}\right)$. Let $\chi_{R}(\xi):=\left(1-\frac{|\xi|}{R}\right) \chi_{|\xi| \leqslant R}$ for $R>0$ and define for $s \in \mathbb{R}_{+}$

  $$H^{s}\left(\mathbb{R}^{n}\right):=\left\{f \in L^{2}\left(\mathbb{R}^{n}\right) \mid\left(1+|\cdot|^{2}\right)^{s / 2} \hat{f}(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)\right\}$$

  (i) Prove that $H^{s}\left(\mathbb{R}^{n}\right)$ is a vector subspace of $L^{2}\left(\mathbb{R}^{n}\right)$, and is a Hilbert space for the inner product $\langle f, g\rangle:=\int_{\mathbb{R}^{n}}\left(1+|\xi|^{2}\right)^{s} \hat{f}(\xi) \overline{\hat{g}(\xi)} d \xi$, where $\bar{z}$ denotes the complex conjugate of $z \in \mathbb{C}$.

  (ii) Construct a function $f \in H^{s}(\mathbb{R}), s \in(0,1 / 2)$, that is not almost everywhere equal to a continuous function.

  (iii) For $f \in L^{1}\left(\mathbb{R}^{n}\right)$, prove that $F_{R}: x \mapsto \int_{\mathbb{R}^{n}} \mathcal{F} f(\xi) \chi_{R}(\xi) e^{2 i \pi x \cdot \xi} d \xi$ is a well-defined function and that $F_{R} \in L^{1}\left(\mathbb{R}^{n}\right)$ converges to $f$ in $L^{1}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.

  [Hint: Prove that $F_{R}=K_{R} * f$ where $K_{R}$ is an approximation of the unit as $R \rightarrow+\infty .]$

  (iv) Deduce that if $f \in L^{1}\left(\mathbb{R}^{n}\right)$ and $\left(1+|\cdot|^{2}\right)^{s / 2} \mathcal{F} f(\cdot) \in L^{2}\left(\mathbb{R}^{n}\right)$ then $f \in H^{s}\left(\mathbb{R}^{n}\right)$.

  [Hint: Prove that: (1) there is a sequence $R_{k} \rightarrow+\infty$ such that $K_{R_{k}} * f$ converges to $f$ almost everywhere; (2) $K_{R} * f$ is uniformly bounded in $L^{2}\left(\mathbb{R}^{n}\right)$ as $R \rightarrow+\infty$.]

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• Paper 1, Section II, F

  Analysis of Functions

  Part II, 2018

  (a) Consider a measure space $(X, \mathcal{A}, \mu)$ and a complex-valued measurable function $F$ on $X$. Prove that for any $\varphi:[0,+\infty) \rightarrow[0,+\infty)$ differentiable and increasing such that $\varphi(0)=0$, then

  $$\int_{X} \varphi(|F(x)|) \mathrm{d} \mu(x)=\int_{0}^{+\infty} \varphi^{\prime}(s) \mu(\{|F|>s\}) \mathrm{d} \lambda(s)$$

  where $\lambda$ is the Lebesgue measure.

  (b) Consider a complex-valued measurable function $f \in L^{1}\left(\mathbb{R}^{n}\right) \cap L^{\infty}\left(\mathbb{R}^{n}\right)$ and its maximal function $M f(x)=\sup _{r>0} \frac{1}{|B(x, r)|} \int_{B(x, r)}|f| \mathrm{d} \lambda$. Prove that for $p \in(1,+\infty)$ there is a constant $c_{p}>0$ such that $\|M f\|_{L^{p}\left(\mathbb{R}^{n}\right)} \leqslant c_{p}\|f\|_{L^{p}\left(\mathbb{R}^{n}\right)}$.

  [Hint: Split $f=f_{0}+f_{1}$ with $f_{0}=f \chi_{\{|f|>s / 2\}}$ and $f_{1}=f \chi_{\{|f| \leqslant s / 2\}}$ and prove that $\lambda(\{M f>s\}) \leqslant \lambda\left(\left\{M f_{0}>s / 2\right\}\right)$. Then use the maximal inequality $\lambda(\{M f>s\}) \leqslant$ $\frac{C_{1}}{s}\|f\|_{L^{1}\left(\mathbb{R}^{n}\right)}$ for some constant $\left.C_{1}>0 .\right]$

  (c) Consider $p, q \in(1,+\infty)$ with $p<q$ and $\alpha \in(0, n)$ such that $1 / q=1 / p-\alpha / n$. Define $I_{\alpha}|f|(x):=\int_{\mathbb{R}^{n}} \frac{|f(y)|}{|x-y|^{n-\alpha}} \mathrm{d} \lambda(y)$ and prove $I_{\alpha}|f|(x) \leqslant\|f\|_{L^{p}\left(\mathbb{R}^{n}\right)}^{\alpha p / n} M f(x)^{1-\alpha p / n}$.

  [Hint: Split the integral into $|x-y| \geqslant r$ and $|x-y| \in\left[2^{-k-1} r, 2^{-k} r\right)$ for all $k \geqslant 0$, given some suitable $r>0 .]$

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• Paper 3, Section II, $22 F$

  Analysis of Functions

  Part II, 2018

  (a) Let $(X, \mathcal{A}, \mu)$ be a measure space. Define the spaces $L^{p}(X)$ for $p \in[1, \infty]$. Prove that if $\mu(X)<\infty$ then $L^{q}(X) \subset L^{p}(X)$ for all $1 \leqslant p<q \leqslant \infty$.

  (b) Now let $X=\mathbb{R}^{n}$ endowed with Borel sets and Lebesgue measure. Describe the dual spaces of $L^{p}(X)$ for $p \in[1, \infty)$. Define reflexivity and say which $L^{p}(X)$ are reflexive. Prove that $L^{1}(X)$ is not the dual space of $L^{\infty}(X)$

  (c) Now let $X \subset \mathbb{R}^{n}$ be a Borel subset and consider the measure space $(X, \mathcal{A}, \mu)$ induced from Borel sets and Lebesgue measure on $\mathbb{R}^{n}$.

  (i) Given any $p \in[1, \infty]$, prove that any sequence $\left(f_{n}\right)$ in $L^{p}(X)$ converging in $L^{p}(X)$ to some $f \in L^{p}(X)$ admits a subsequence converging almost everywhere to $f$.

  (ii) Prove that if $L^{q}(X) \subset L^{p}(X)$ for $1 \leqslant p<q \leqslant \infty$ then $\mu(X)<\infty$. [Hint: You might want to prove first that the inclusion is continuous with the help of one of the corollaries of Baire's category theorem.]

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• Paper 4, Section II, 23F

  Analysis of Functions

  Part II, 2018

  Here and below, $\Phi: \mathbb{R} \rightarrow \mathbb{R}$ is smooth such that $\int_{\mathbb{R}} e^{-\Phi(x)} \mathrm{d} x=1$ and

  $$\lim _{|x| \rightarrow+\infty}\left(\frac{\left|\Phi^{\prime}(x)\right|^{2}}{4}-\frac{\Phi^{\prime \prime}(x)}{2}\right)=\ell \in(0,+\infty)$$

  $C_{c}^{1}(\mathbb{R})$ denotes the set of continuously differentiable complex-valued functions with compact support on $\mathbb{R}$.

  (a) Prove that there are constants $R_{0}>0, \lambda_{1}>0$ and $K_{1}>0$ so that for any $R \geqslant R_{0}$ and $h \in C_{c}^{1}(\mathbb{R})$ :

  $$\int_{\mathbb{R}}\left|h^{\prime}(x)\right|^{2} e^{-\Phi(x)} d x \geqslant \lambda_{1} \int_{\{|x| \geqslant R\}}|h(x)|^{2} e^{-\Phi(x)} d x-K_{1} \int_{\{|x| \leqslant R\}}|h(x)|^{2} e^{-\Phi(x)} d x$$

  [Hint: Denote $g:=h e^{-\Phi / 2}$, expand the square and integrate by parts.]

  (b) Prove that, given any $R>0$, there is a $C_{R}>0$ so that for any $h \in C^{1}([-R, R])$ with $\int_{-R}^{+R} h(x) e^{-\Phi(x)} d x=0$ :

  $$\max _{x \in[-R, R]}|h(x)|+\operatorname{sip}_{\{x, y \in[-R, R], x \neq y\}} \frac{|h(x)-h(y)|}{|x-y|^{1 / 2}} \leqslant C_{R}\left(\int_{-R}^{+R}\left|h^{\prime}(x)\right|^{2} e^{-\Phi(x)} d x\right)^{1 / 2}$$

  [Hint: Use the fundamental theorem of calculus to control the second term of the left-hand side, and then compare $h$ to its weighted mean to control the first term of the left-hand side.]

  (c) Prove that, given any $R>0$, there is a $\lambda_{R}>0$ so that for any $h \in C^{1}([-R, R])$ :

  $$\int_{-R}^{+R}\left|h^{\prime}(x)\right|^{2} e^{-\Phi(x)} d x \geqslant \lambda_{R} \int_{-R}^{+R}\left|h(x)-\frac{\int_{-R}^{+R} h(y) e^{-\Phi(y)} d y}{\int_{-R}^{+R} e^{-\Phi(y)} d y}\right|^{2} e^{-\Phi(x)} d x$$

  [Hint: Show first that one can reduce to the case $\int_{-R}^{+R} h e^{-\Phi}=0$. Then argue by contradiction with the help of the Arzelà-Ascoli theorem and part (b).]

  (d) Deduce that there is a $\lambda_{0}>0$ so that for any $h \in C_{c}^{1}(\mathbb{R})$ :

  $$\int_{\mathbb{R}}\left|h^{\prime}(x)\right|^{2} e^{-\Phi(x)} d x \geqslant \lambda_{0} \int_{\mathbb{R}}\left|h(x)-\left(\int_{\mathbb{R}} h(y) e^{-\Phi(y)} d y\right)\right|^{2} e^{-\Phi(x)} d x$$

  [Hint: Show first that one can reduce to the case $\int_{\mathbb{R}} h e^{-\Phi}=0$. Then combine the inequality (a), multiplied by a constant of the form $\epsilon=\epsilon_{0} \lambda_{R}$ (where $\epsilon_{0}>0$ is chosen so that $\epsilon$ be sufficiently small), and the inequality (c).]

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• Paper 1, Section II, H

  Analysis of Functions

  Part II, 2019

  (a) Consider the topology $\mathcal{T}$ on the natural numbers $\mathbb{N} \subset \mathbb{R}$ induced by the standard topology on $\mathbb{R}$. Prove it is the discrete topology; i.e. $\mathcal{T}=\mathcal{P}(\mathbb{N})$ is the power set of $\mathbb{N}$.

  (b) Describe the corresponding Borel sets on $\mathbb{N}$ and prove that any function $f: \mathbb{N} \rightarrow \mathbb{R}$ or $f: \mathbb{N} \rightarrow[0,+\infty]$ is measurable.

  (c) Using Lebesgue integration theory, define $\sum_{n \geqslant 1} f(n) \in[0,+\infty]$ for a function $f: \mathbb{N} \rightarrow[0,+\infty]$ and then $\sum_{n \geqslant 1} f(n) \in \mathbb{C}$ for $f: \mathbb{N} \rightarrow \mathbb{C}$. State any condition needed for the sum of the latter series to be defined. What is a simple function in this setting, and which simple functions have finite sum?

  (d) State and prove the Beppo Levi theorem (also known as the monotone convergence theorem).

  (e) Consider $f: \mathbb{R} \times \mathbb{N} \rightarrow[0,+\infty]$ such that for any $n \in \mathbb{N}$, the function $t \mapsto f(t, n)$ is non-decreasing. Prove that

  $$\lim _{t \rightarrow \infty} \sum_{n \geqslant 1} f(t, n)=\sum_{n \geqslant 1} \lim _{t \rightarrow \infty} f(t, n) .$$

  Show that this need not be the case if we drop the hypothesis that $t \mapsto f(t, n)$ is nondecreasing, even if all the relevant limits exist.

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• Paper 3, Section II, H

  Analysis of Functions

  Part II, 2019

  (a) Prove that in a finite-dimensional normed vector space the weak and strong topologies coincide.

  (b) Prove that in a normed vector space $X$, a weakly convergent sequence is bounded. [Any form of the Banach-Steinhaus theorem may be used, as long as you state it clearly.]

  (c) Let $\ell^{1}$ be the space of real-valued absolutely summable sequences. Suppose $\left(a^{k}\right)$ is a weakly convergent sequence in $\ell^{1}$ which does not converge strongly. Show there is a constant $\varepsilon>0$ and a sequence $\left(x^{k}\right)$ in $\ell^{1}$ which satisfies $x^{k} \rightarrow 0$ and $\left\|x^{k}\right\|_{\ell^{1}} \geqslant \varepsilon$ for all $k \geqslant 1$.

  With $\left(x^{k}\right)$ as above, show there is some $y \in \ell^{\infty}$ and a subsequence $\left(x^{k_{n}}\right)$ of $\left(x^{k}\right)$ with $\left\langle x^{k_{n}}, y\right\rangle \geqslant \varepsilon / 3$ for all $n$. Deduce that every weakly convergent sequence in $\ell^{1}$ is strongly convergent.

  [Hint: Define $y$ so that $y_{i}=\operatorname{sign} x_{i}^{k_{n}}$ for $b_{n-1}<i \leqslant b_{n}$, where the sequence of integers $b_{n}$ should be defined inductively along with $\left.x^{k_{n}} .\right]$

  (d) Is the conclusion of part (c) still true if we replace $\ell^{1}$ by $L^{1}([0,2 \pi]) ?$

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• Paper 4, Section II, H

  Analysis of Functions

  Part II, 2019

  (a) Let $(\mathcal{H},\langle\cdot, \cdot\rangle)$ be a real Hilbert space and let $B: \mathcal{H} \times \mathcal{H} \rightarrow \mathbb{R}$ be a bilinear map. If $B$ is continuous prove that there is an $M>0$ such that $|B(u, v)| \leqslant M\|u\|\|v\|$ for all $u, v \in \mathcal{H}$. [You may use any form of the Banach-Steinhaus theorem as long as you state it clearly.]

  (b) Now suppose that $B$ defined as above is bilinear and continuous, and assume also that it is coercive: i.e. there is a $C>0$ such that $B(u, u) \geqslant C\|u\|^{2}$ for all $u \in \mathcal{H}$. Prove that for any $f \in \mathcal{H}$, there exists a unique $v_{f} \in \mathcal{H}$ such that $B\left(u, v_{f}\right)=\langle u, f\rangle$ for all $u \in \mathcal{H}$.

  [Hint: show that there is a bounded invertible linear operator $L$ with bounded inverse so that $B(u, v)=\langle u, L v\rangle$ for all $u, v \in \mathcal{H}$. You may use any form of the Riesz representation theorem as long as you state it clearly.]

  (c) Define the Sobolev space $H_{0}^{1}(\Omega)$, where $\Omega \subset \mathbb{R}^{d}$ is open and bounded.

  (d) Suppose $f \in L^{2}(\Omega)$ and $A \in \mathbb{R}^{d}$ with $|A|_{2}<2$, where $|\cdot|_{2}$ is the Euclidean norm on $\mathbb{R}^{d}$. Consider the Dirichlet problem

  $$-\Delta v+v+A \cdot \nabla v=f \quad \text { in } \Omega, \quad v=0 \quad \text { in } \partial \Omega$$

  Using the result of part (b), prove there is a unique weak solution $v \in H_{0}^{1}(\Omega)$.

  (e) Now assume that $\Omega$ is the open unit disk in $\mathbb{R}^{2}$ and $g$ is a smooth function on $\mathbb{S}^{1}$. Sketch how you would solve the following variant:

  $$-\Delta v+v+A \cdot \nabla v=0 \quad \text { in } \Omega, \quad v=g \quad \text { in } \partial \Omega .$$

  [Hint: Reduce to the result of part (d).]

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• Paper 4, Section II, 23I

  Analysis of Functions

  Part II, 2020

  (a) Define the Sobolev space $H^{s}\left(\mathbb{R}^{n}\right)$ for $s \in \mathbb{R}$.

  (b) Let $k$ be a non-negative integer and let $s>k+\frac{n}{2}$. Show that if $u \in H^{s}\left(\mathbb{R}^{n}\right)$ then there exists $u^{*} \in C^{k}\left(\mathbb{R}^{n}\right)$ with $u=u^{*}$ almost everywhere.

  (c) Show that if $f \in H^{s}\left(\mathbb{R}^{n}\right)$ for some $s \in \mathbb{R}$, there exists a unique $u \in H^{s+4}\left(\mathbb{R}^{n}\right)$ which solves:

  $$\Delta \Delta u+\Delta u+u=f$$

  in a distributional sense. Prove that there exists a constant $C>0$, independent of $f$, such that:

  $$\|u\|_{H^{s+4}} \leqslant C\|f\|_{H^{s}}$$

  For which $s$ will $u$ be a classical solution?

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• Paper 1, Section II, I

  Analysis of Functions

  Part II, 2020

  Let $\mathbb{R}^{n}$ be equipped with the $\sigma$-algebra of Lebesgue measurable sets, and Lebesgue measure.

  (a) Given $f \in L^{\infty}\left(\mathbb{R}^{n}\right), g \in L^{1}\left(\mathbb{R}^{n}\right)$, define the convolution $f \star g$, and show that it is a bounded, continuous function. [You may use without proof continuity of translation on $L^{p}\left(\mathbb{R}^{n}\right)$ for $\left.1 \leqslant p<\infty .\right]$

  Suppose $A \subset \mathbb{R}^{n}$ is a measurable set with $0<|A|<\infty$ where $|A|$ denotes the Lebesgue measure of $A$. By considering the convolution of $f(x)=\mathbb{1}_{A}(x)$ and $g(x)=\mathbb{1}_{A}(-x)$, or otherwise, show that the set $A-A=\{x-y: x, y \in A\}$ contains an open neighbourhood of 0 . Does this still hold if $|A|=\infty$ ?

  (b) Suppose that $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is a measurable function satisfying

  $$f(x+y)=f(x)+f(y), \quad \text { for all } x, y \in \mathbb{R}^{n}$$

  Let $B_{r}=\left\{y \in \mathbb{R}^{m}:|y|<r\right\}$. Show that for any $\epsilon>0$ :

  (i) $f^{-1}\left(B_{\epsilon}\right)-f^{-1}\left(B_{\epsilon}\right) \subset f^{-1}\left(B_{2 \epsilon}\right)$,

  (ii) $f^{-1}\left(B_{k \epsilon}\right)=k f^{-1}\left(B_{\epsilon}\right)$ for all $k \in \mathbb{N}$, where for $\lambda>0$ and $A \subset \mathbb{R}^{n}, \lambda A$ denotes the set $\{\lambda x: x \in A\}$.

  Show that $f$ is continuous at 0 and hence deduce that $f$ is continuous everywhere.

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• Paper 3, Section II, 22I

  Analysis of Functions

  Part II, 2020

  Let $X$ be a Banach space.

  (a) Define the dual space $X^{\prime}$, giving an expression for $\|\Lambda\|_{X^{\prime}}$ for $\Lambda \in X^{\prime}$. If $Y=L^{p}\left(\mathbb{R}^{n}\right)$ for some $1 \leqslant p<\infty$, identify $Y^{\prime}$ giving an expression for a general element of $Y^{\prime}$. [You need not prove your assertion.]

  (b) For a sequence $\left(\Lambda_{i}\right)_{i=1}^{\infty}$ with $\Lambda_{i} \in X^{\prime}$, what is meant by: (i) $\Lambda_{i} \rightarrow \Lambda$, (ii) $\Lambda_{i} \rightarrow \Lambda$ (iii) $\Lambda_{i} \stackrel{*}{\rightarrow} \Lambda$ ? Show that (i) $\Longrightarrow$ (ii) $\Longrightarrow$ (iii). Find a sequence $\left(f_{i}\right)_{i=1}^{\infty}$ with $f_{i} \in$ $L^{\infty}(\mathbb{R})=\left(L^{1}(\mathbb{R})\right)^{\prime}$ such that, for some $f, g \in L^{\infty}\left(\mathbb{R}^{n}\right)$ :

  $$f_{i} \stackrel{*}{\rightarrow} f, \quad f_{i}^{2} \stackrel{*}{\rightarrow} g, \quad g \neq f^{2} .$$

  (c) For $f \in C_{c}^{0}\left(\mathbb{R}^{n}\right)$, let $\Lambda: C_{c}^{0}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$ be the map $\Lambda f=f(0)$. Show that $\Lambda$ may be extended to a continuous linear map $\tilde{\Lambda}: L^{\infty}\left(\mathbb{R}^{n}\right) \rightarrow \mathbb{C}$, and deduce that $\left(L^{\infty}\left(\mathbb{R}^{n}\right)\right)^{\prime} \neq L^{1}\left(\mathbb{R}^{n}\right)$. For which $1 \leqslant p \leqslant \infty$ is $L^{p}\left(\mathbb{R}^{n}\right)$ reflexive? [You may use without proof the Hahn-Banach theorem].

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• Paper 3, Section II, H

  Analysis of Functions

  Part II, 2021

  (a) State the Riemann-Lebesgue lemma. Show that the Fourier transform maps $\mathscr{S}\left(\mathbb{R}^{n}\right)$ to itself continuously.

  (b) For some $s \geqslant 0$, let $f \in L^{1}\left(\mathbb{R}^{3}\right) \cap H^{s}\left(\mathbb{R}^{3}\right)$. Consider the following system of equations for $\mathbf{B}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$

  $$\nabla \cdot \mathbf{B}=f, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{0}$$

  Show that there exists a unique $\mathbf{B}=\left(B_{1}, B_{2}, B_{3}\right)$ solving the equations with $B_{j} \in$ $H^{s+1}\left(\mathbb{R}^{3}\right)$ for $j=1,2,3$. You need not find $\mathbf{B}$ explicitly, but should give an expression for the Fourier transform of $B_{j}$. Show that there exists a constant $C>0$ such that

  $$\left\|B_{j}\right\|_{H^{s+1}} \leqslant C\left(\|f\|_{L^{1}}+\|f\|_{H^{s}}\right), \quad j=1,2,3$$

  For what values of $s$ can we conclude that $B_{j} \in C^{1}\left(\mathbb{R}^{n}\right)$ ?

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• Paper 4, Section II, $23 \mathrm{H}$

  Analysis of Functions

  Part II, 2021

  Fix $1<p<\infty$ and let $q$ satisfy $p^{-1}+q^{-1}=1$

  (a) Let $\left(f_{j}\right)$ be a sequence of functions in $L^{p}\left(\mathbb{R}^{n}\right)$. For $f \in L^{p}\left(\mathbb{R}^{n}\right)$, what is meant by (i) $f_{j} \rightarrow f$ in $L^{p}\left(\mathbb{R}^{n}\right)$ and (ii) $f_{j} \rightarrow f$ in $L^{p}\left(\mathbb{R}^{n}\right)$ ? Show that if $f_{j} \rightarrow f$, then

  $$\|f\|_{L^{p}} \leqslant \liminf _{j \rightarrow \infty}\left\|f_{j}\right\|_{L^{p}}$$

  (b) Suppose that $\left(g_{j}\right)$ is a sequence with $g_{j} \in L^{p}\left(\mathbb{R}^{n}\right)$, and that there exists $K>0$ such that $\left\|g_{j}\right\|_{L^{p}} \leqslant K$ for all $j$. Show that there exists $g \in L^{p}\left(\mathbb{R}^{n}\right)$ and a subsequence $\left(g_{j_{k}}\right)_{k=1}^{\infty}$, such that for any sequence $\left(h_{k}\right)$ with $h_{k} \in L^{q}\left(\mathbb{R}^{n}\right)$ and $h_{k} \rightarrow h \in L^{q}\left(\mathbb{R}^{n}\right)$, we have

  $$\lim _{k \rightarrow \infty} \int_{\mathbb{R}^{n}} g_{j_{k}} h_{k} d x=\int_{\mathbb{R}^{n}} g h d x .$$

  Give an example to show that the result need not hold if the condition $h_{k} \rightarrow h$ is replaced by $h_{k} \rightarrow h$ in $L^{q}\left(\mathbb{R}^{n}\right)$.

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• Paper 1, Section II, $23 \mathrm{H}$

  Analysis of Functions

  Part II, 2021

  Below, $\mathcal{M}$ is the $\sigma$-algebra of Lebesgue measurable sets and $\lambda$ is Lebesgue measure.

  (a) State the Lebesgue differentiation theorem for an integrable function $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$. Let $g: \mathbb{R} \rightarrow \mathbb{C}$ be integrable and define $G: \mathbb{R} \rightarrow \mathbb{C}$ by $G(x):=\int_{[a, x]} g d \lambda$ for some $a \in \mathbb{R}$. Show that $G$ is differentiable $\lambda$-almost everywhere.

  (b) Suppose $h: \mathbb{R} \rightarrow \mathbb{R}$ is strictly increasing, continuous, and maps sets of $\lambda$-measure zero to sets of $\lambda$-measure zero. Show that we can define a measure $\nu$ on $\mathcal{M}$ by setting $\nu(A):=\lambda(h(A))$ for $A \in \mathcal{M}$, and establish that $\nu \ll \lambda$. Deduce that $h$ is differentiable $\lambda$-almost everywhere. Does the result continue to hold if $h$ is assumed to be non-decreasing rather than strictly increasing?

  [You may assume without proof that a strictly increasing, continuous, function $w: \mathbb{R} \rightarrow \mathbb{R}$ is injective, and $w^{-1}: w(\mathbb{R}) \rightarrow \mathbb{R}$ is continuous.]

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• Paper 2, Section II, H

  Analysis of Functions

  Part II, 2021

  Define the Schwartz space, $\mathscr{S}\left(\mathbb{R}^{n}\right)$, and the space of tempered distributions, $\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$, stating what it means for a sequence to converge in each space.

  For a $C^{k}$ function $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$, and non-negative integers $N, k$, we say $f \in X_{N, k}$ if

  $$\|f\|_{N, k}:=\sup _{x \in \mathbb{R}^{n} ;|\alpha| \leqslant k}\left|\left(1+|x|^{2}\right)^{\frac{N}{2}} D^{\alpha} f(x)\right|<\infty$$

  You may assume that $X_{N, k}$ equipped with $\|\cdot\|_{N, k}$ is a Banach space in which $\mathscr{S}\left(\mathbb{R}^{n}\right)$ is dense.

  (a) Show that if $u \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ there exist $N, k \in \mathbb{Z}_{\geqslant 0}$ and $C>0$ such that

  $$|u[\phi]| \leqslant C\|\phi\|_{N, k} \text { for all } \phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$$

  Deduce that there exists a unique $\tilde{u} \in X_{N, k}^{\prime}$ such that $\tilde{u}[\phi]=u[\phi]$ for all $\phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$.

  (b) Recall that $v \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is positive if $v[\phi] \geqslant 0$ for all $\phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$ satisfying $\phi \geqslant 0$. Show that if $v \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is positive, then there exist $M \in \mathbb{Z}_{\geqslant 0}$ and $K>0$ such that

  $$|v[\phi]| \leqslant K\|\phi\|_{M, 0}, \quad \text { for all } \phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$$

  $\left[\right.$ Hint: Note that $\left.|\phi(x)| \leqslant\|\phi\|_{M, 0}\left(1+|x|^{2}\right)^{-\frac{M}{2}} \cdot\right]$

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