Math Problems Vault

Problem 3.8.1

(a)(b)(c)

The following simple properties of holomorphic functions of several variables were tacitly used in this chapter. Prove them.

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(a)

If $f$ is entire in $\mathbb {C}^{n}$ , if $w \in \mathbb {C}^{n}$ , and if $\phi (\lambda ) = f(\lambda w)$ , then $\phi$ is an entire function of one complex variable.

(b)

If $P$ is a polynomial in $\mathbb {C}^{n}$ and if

\int _{T^{n}}\left|P\right| d \sigma _{n} = 0

then $P$ is identically 0. Hint: Compute $\int_{T^{n}}\left|P\right|^{2} d \sigma_{n}$ .

(c)

If $P$ is a polynomial (not identically 0) and $g$ is an entire function in $\mathbb {C}^{n}$ , then there is at most one entire function $f$ that satisfies $P f = g$ . Find generalizations of these three properties.

Problem 3.8.2

Prove the statement about convex hulls made in the last sentence of the proof of Theorem 8.4.

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Problem 3.8.3

Find a fundamental solution for the operator $\partial^{2} / \partial x_{1} \partial x_{2}$ in $\mathbb {R}^{2}$ . (There is one that is the characteristic function of a certain subset of $\mathbb {R}^{2}$ ).

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Problem 3.8.4

Show that the equation

\frac{\partial ^{2} u}{\partial x_{1}^{2}}-\frac{\partial ^{2} u}{\partial x_{2}^{2}} = 0

is satisfied (in the distribution sense) by every locally integrable function $u$ of the form

u\left(x_{1}, x_{2}\right) = f\left(x_{1}+x_{2}\right) \quad \text{ or } \quad u\left(x_{1}, x_{2}\right) = f\left(x_{1}-x_{2}\right)

and that even classical solutions (i.e., twice continuously differentiable functions) need not be in $C^{\infty }$ . Note the contrast between this and the Laplace equation.

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Problem 3.8.5

(a)(b)(c)

For $x \in \mathbb {R}^{3}$ , define $f(x) = \left(1+\left|x\right|^{2}\right)^{-1}$ . Show that $f \in L^{2}\left(\mathbb {R}^{3}\right)$ and that $\hat{f}$ is a fundamental solution of the operator $I-\Delta$ in $\mathbb {R}^{3}$ . Find $\hat{f}$ , by direct computation and also by the following reasoning:

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(a)

Since $f$ is a radial function (i.e., one that depends only on the distance from the origin) the same is true of $\hat{f}$ ; see Exercise 1 of Chapter 7.

(b)

Away from the origin, $(I-\Delta ) \hat{f} = 0$ , and $\hat{f} \in C^{\infty }$ .

(c)

If $F(\left|y\right|) = \hat{f}(y)$ , (b) implies that $F$ satisfies an ordinary differential equation in $(0, \infty )$ that can easily be solved explicitly. Ans. $\hat{f}(y) = (\pi / 2)^{1 / 2}\left|y\right|^{-1} \exp (-\left|y\right|)$ . Do the same with $\mathbb {R}^{n}$ in place of $\mathbb {R}^{3}$ ; you will meet Bessel functions.

Problem 3.8.6

For $0 < \lambda < n$ and $x \in \mathbb {R}^{n}$ , define

K_{\lambda }(x) = \left|x\right|^{-\lambda }

Show that

\hat{K}_{\lambda }(y) = c(n, \lambda ) K_{n-\lambda }(y) \quad \left(y \in \mathbb {R}^{n}\right) \tag {a}

where

c(n, \lambda ) = \frac{2^{n / 2-\lambda } \Gamma [(n-\lambda ) / 2]}{\Gamma (\lambda / 2)}

Suggestion: If $n < 2 \lambda < 2 n, K_{\lambda }$ is the sum of an $L^{1}$ -function and an $L^{2}$ -function. For these $\lambda$ , Equation (a) can be deduced from the homogeneity condition

K_{\lambda }(t x) = t^{-\lambda } K_{\lambda }(x) \quad \left(x \in \mathbb {R}^{n}, t > 0\right)

The case $0 < 2 \lambda < n$ follows from the inversion theorem (for tempered distributions). A passage to the limit gives the case $2 \lambda = n$ . The constants $c(n, \lambda )$ can be computed from $\int f \hat{\phi } = \int \hat{\delta } \phi$ , with $\phi (x) = \exp \left(-\left|x\right|^{2} / 2\right)$ .

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Problem 3.8.7

Take $n \geq 3$ and $\lambda = 2$ in Exercise 6, and deduce that $-c(n, 2) K_{n-2}$ is a fundamental solution of the Laplacian $\Delta$ in $\mathbb {R}^{n}$ . For example, if $v$ has compact support in $\mathbb {R}^{3}$ , show that a solution of $\Delta u = v$ is given by

u(x) = -\frac{1}{4 \pi } \int _{\mathbb {R}^{3}}\left|x-y\right|^{-1} v(y) d y

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Problem 3.8.8

Identify $\mathbb {R}^{2}$ and $\mathbb {C}$ (so that $z = x_{1}+i x_{2}$ ); put

\partial = \frac{\partial }{\partial x_{1}}-i \frac{\partial }{\partial x_{2}}, \quad \bar{\partial } = \frac{\partial }{\partial x_{1}}+i \frac{\partial }{\partial x_{2}}

Show that the Fourier transform of $1 / z$ (regarded as a tempered distribution) is $-i / z$ . Show that this result is equivalent to the Cauchy formula

\phi (z) = \int _{\mathbb {R}^{2}}(\bar{\partial } \phi )(w) \frac{d m_{2}(w)}{w-z} \quad \left[\phi \in \mathscr {D}\left(\mathbb {R}^{2}\right)\right]

Since $\partial \log \left|w\right| = 1 / w$ and $\Delta = \partial \bar{\partial }$ , deduce that

\phi (z) = \int _{\mathbb {R}^{2}}(\Delta \phi )(w) \log \left|w-z\right| d m_{2}(w) \quad \left[\phi \in \mathscr {D}\left(\mathbb {R}^{2}\right)\right]

Thus $\log \left|z\right|$ is a fundamental solution of the Laplacian in $\mathbb {R}^{2}$ .

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Problem 3.8.9

Use Exercise 6 to compute that

\lim _{\varepsilon \rightarrow 0}\left[\varepsilon ^{-1}-b-\hat{K}_{2-\varepsilon }(y)\right] = \log \left|y\right| \quad \left(y \in \mathbb {R}^{2}\right)

where $b$ is a certain constant. Show that this leads to another proof of the last statement in Exercise 8.

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Problem 3.8.10

Suppose $P(D) = D^{2}+a D+b I$ . (We are now in the case $n = 1$ ). Let $f$ and $g$ be solutions of $P(D) u = 0$ which satisfy

f(0) = g(0) \quad \text{ and } \quad f^{\prime }(0)-g^{\prime }(0) = 1

Define

G(x) = \begin{cases} f(x) & \text{ if } x \leq 0 \\ g(x) & \text{ if } x > 0\end{cases}

and put

\Lambda \phi = -\int _{-\infty }^{\infty } \phi (x) G(x) d x \quad [\phi \in \mathscr {D}(R)]

Prove that $\Lambda$ is a fundamental solution of $P(D)$ .

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Problem 3.8.11

Suppose $u$ is a distribution in $\mathbb {R}^{n}$ whose first derivatives $D_{1} u, \ldots , D_{n} u$ are locally $L^{2}$ . Prove that $u$ is then locally $L^{2}$ . Hint: If $\psi \in \mathscr {D}\left(\mathbb {R}^{n}\right)$ is 1 in a neighborhood of the origin and if $\Delta E = \delta$ , then $\Delta (\psi E)-\delta \in \mathscr {D}\left(\mathbb {R}^{n}\right)$ . Hence

u-\sum _{i = 1}^{n}\left(D_{i} u\right) * D_{i}(\psi E)

is in $C^{\infty }\left(\mathbb {R}^{n}\right)$ . Each $D_{i}(\psi E)$ is an $L^{1}$ -function with compact support.

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Problem 3.8.12

Suppose $u$ is a distribution in $\mathbb {R}^{n}$ whose Laplacian $\Delta u$ is a continuous function. Prove that $u$ is then a continuous function. Hint: As in Exercise 11,

u-(\psi E) *(\Delta u) \in C^{\infty }\left(\mathbb {R}^{n}\right)

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Problem 3.8.13

Prove analogues of Exercises 11 and 12, with $\mathbb {R}^{n}$ replaced by an arbitrary open set $\Omega$ .

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Problem 3.8.14

(a)(b)

Show, under the hypotheses of Exercise 12, that

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(a)

$\partial^{2} u / \partial x_{1}^{2}$ is locally $L^{2}$ , but

(b)

$\partial^{2} u / \partial x_{1}^{2}$ need not be a continuous function.

Outline of $(b)$ for periodic distributions in $\mathbb {R}^{2}$ (Exercise 22, Chapter 7): If $g \in C\left(T^{2}\right)$ has Fourier coefficients $\hat{g}(m, n)$ and if $f$ is defined by

\hat{f}(m, n) = \left(1+m^{2}+n^{2}\right)^{-1} \hat{g}(m, n)

then $f \in C\left(T^{2}\right)$ and $\Delta f = f-g \in C\left(T^{2}\right)$ , since $\sum \left|\hat{f}(m, n)\right| < \infty$ . The Fourier coefficients of $\partial^{2} f / \partial x_{1}^{2}$ are $-m^{2} \hat{f}(m, n)$ . If $\partial^{2} f / \partial x_{1}^{2}$ were continuous for every $g \in C\left(T^{2}\right)$ , then $\left(\partial^{2} f / \partial x_{1}^{2}\right)(0,0)$ would be a continuous linear functional of $g$ . Hence there would be a complex Borel measure $\mu$ on $T^{2}$ , with Fourier coefficients

\hat{\mu }(m, n) = \frac{m^{2}}{1+m^{2}+n^{2}}

The next exercise shows that no such measure exists.

Problem 3.8.15

If $\mu$ is a complex Borel measure on $T^{2}$ , and if

\gamma (A, B) = \frac{1}{(2 A+1)(2 B+1)} \sum _{n = -A}^{A} \sum _{m = -B}^{B} \hat{\mu }(m, n)

prove that

\lim _{A \rightarrow \infty }\left[\lim _{B \rightarrow \infty } \gamma (A, B)\right] = \lim _{B \rightarrow \infty }\left[\lim _{A \rightarrow \infty } \gamma (A, B)\right]

Suggestion: If $D_{A}(t) = (2 A+1)^{-1} \sum_{-A}^{A} e^{i n t}$ , then $D_{A}(x) = 1$ if $x = 0$ , $D_{A}(x) \rightarrow 0$ otherwise, and

\gamma (A, B) = \int _{T^{2}} D_{A}(x) D_{B}(y) d \mu (x, y)

Conclude that each of the two iterated limits exists and that both are equal to $\mu (\left\{ 0,0\right\} )$ .

If $\mu$ were as in Exercise 14, one of the iterated limits would be 1, the other 0.

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Problem 3.8.16

(a)(b)

Suppose $L$ is an elliptic linear operator in some open set $\Omega \subset \mathbb {R}^{n}$ , and suppose that the order of $L$ is odd.

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(a)

Prove that then $n = 1$ or $n = 2$ .

(b)

If $n = 2$ , prove that the coefficients of the characteristic polynomial of $L$ cannot all be real.

In view of (a), the Cauchy-Riemann operator is not a very typical example of an elliptic operator.