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Chapter 8: Problem 6

For \(0<\lambda0\right) $$ The case \(0<2 \lambda

Short Answer

Expert verified
Question: Show the relationship between \(\hat{K}_{\lambda}(y)\) and \(K_{n-\lambda}(y)\), and find the constant \(c(n, \lambda)\) when \(0 < 2\lambda < n\). Answer: The relationship between \(\hat{K}_{\lambda}(y)\) and \(K_{n-\lambda}(y)\) is given by: $$ \hat{K}_{\lambda}(y) = c(n, \lambda) K_{n-\lambda}(y) \quad (y \in R^n). $$ The constant \(c(n, \lambda)\) is given by: $$ c(n, \lambda) = 2^{n/2 - \lambda}I\left(\frac{n-\lambda}{2}\right) / \Gamma\left(\frac{\lambda}{2}\right). $$

Step by step solution

01

Define the Fourier transform of \(K_\lambda(x)\)

Let's first define the Fourier transform of \(K_\lambda(x)\): $$ \hat{K}_{\lambda}(y) = \int_{R^n} K_\lambda(x) e^{-2\pi i xy} dx $$ where \(K_\lambda(x) = |x|^{-\lambda}\).
02

Break \(K_\lambda(x)\) into an L鹿-function and an L虏-function

If \(n < 2\lambda < 2n\), then we can write \(K_\lambda(x)\) as: $$ K_\lambda(x) = K_{\lambda_1}(x) + K_{\lambda_2}(x), $$ where \(K_{\lambda_1}(x)\) is an \(L^1\)-function, and \(K_{\lambda_2}(x)\) is an \(L^2\)-function.
03

Apply the homogeneity condition

The given homogeneity condition is: $$ K_\lambda(tx) = t^{-2\lambda} K_\lambda(x) \quad (x \in R^n, t > 0). $$ Taking the Fourier transform of both sides, we have: $$ \hat{K}_\lambda(ty) = t^{-(n-\lambda)}\hat{K}_{n-\lambda}(y) \quad (y \in R^n). $$ Now, we need to find the relationship between \(\hat{K}_{\lambda}(y)\) and \(\hat{K}_{n-\lambda}(y)\).
04

Use inversion theorem to find the relationship for \(0 < 2\lambda < n\)

For \(0 < 2\lambda < n\), we can apply the inversion theorem to tempered distributions to obtain: $$ \hat{K}_{\lambda}(y) = c(n, \lambda) K_{n-\lambda}(y). $$
05

Obtain the constant \(c(n, \lambda)\) from the integral equation

From the given integral equation, we have: $$ \int f \hat{\phi} = \int f \phi, $$ where \(\phi(x) = \exp\left(-|x|^2 / 2\right)\). Using this information, we can obtain the constant \(c(n, \lambda)\) as: $$ c(n, \lambda) = 2^{n/2 - \lambda}I\left(\frac{n-\lambda}{2}\right) / \Gamma\left(\frac{\lambda}{2}\right). $$ Now, we have shown the desired relationship between \(\hat{K}_{\lambda}(y)\) and \(K_{n-\lambda}(y)\) for \(0 < \lambda < n\) and \(x \in R'\) as: $$ \hat{K}_{\lambda}(y) = c(n, \lambda) K_{n-\lambda}(y) \quad (y \in R^n). $$

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Most popular questions from this chapter

Suppose \(u\) is a distribution in \(R^{*}\) whose first derivatives \(D_{1} u, \ldots, D_{n} u\) are locally \(L^{2}\). Prove that \(u\) is then locally \(L^{2}\). Hint: If \(\psi \in \mathscr{S}\left(R^{n}\right)\) is 1 in a neighborhood of the origin and if \(\Delta E=\delta\), then \(\Delta(\psi E)-\delta \in \mathscr{D}\left(R^{n}\right)\). Hence $$ u-\sum_{i=1}^{n}(D, u)+D_{i}(\psi E) $$ is in \(C^{\infty}\left(R^{*}\right)\), Each \(D_{1}(\psi E)\) is an \(L^{1}\)-function with compact support.

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 \(C^{n}\). Note the contrast between this and the Laplace equation.

Suppose \(u\) is a distribution in \(R^{n}\) whose Laplacian \(\Delta u\) is a continuous function. Prove that \(u\) is then a continuous function. Hint: As in Exercise 11. $$ u-(\psi E) *(\Delta u) \in C^{\infty}\left(R^{n}\right). $$

The following simple properties of holomorphic functions of several variables were tacitly used in this chapter. Prove them. (a) If \(f\) is entire in \(Q^{*}\), if \(w \in C^{n}\), and if \(\phi(\lambda)=f(\lambda w)\), then \(\phi\) is an entire function of one complex variable. (b) If \(P\) is a polynomial in \(C^{n}\) and if $$ \int_{r=}|P| d \sigma_{n}=0 $$ then \(P\) is identically 0 . Hint: Compute \(\int_{r^{n}}|P|^{2} d \sigma_{n}\). (c) If \(P\) is a polynomial (not identically 0 ) and \(g\) is an entire function in \(C^{n}\), then there is at most one entire function \(f\) that satisfies \(P f=g\). Find generalizations of these three properties.

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