Hunter Liu's Website

Revisiting Qual Attempt #1

Date Written: July 28, 2023; Last Modified: July 28, 2023

I previously attempted to work on a past qualifying exam in analysis with rather dismal results. In the previous post, I only gave solutions for 6 of the 12 problems (2 real analysis, 4 complex analysis). Over the past few days, I have been able to put together solutions for 5 of the remaining problems, with only one problem from the real analysis section left unsolved. I do have some thoughts about it, but I’m happy with solving 11 of the 12 problems (albeit taking far longer than should be necessary).

Problem 2.

(a) Let dσd\sigma denote surface measure on the unit sphere S2R3\mathbb S^2\subset\mathbb R^3. Note dσ(x)=4π\int d\sigma(x)=4\pi. For ξR3\xi\in\mathbb R^3, compute S2eixξdσ(x),\int_{\mathbb S^2} e^{ix\cdot\xi} d\sigma(x), where \cdot denotes the usual inner product on R3\mathbb R^3. (b) Using this, or otherwise, show that the mapping fS2S2f(x+y)dσ(x)dσ(y)f\mapsto\int_{\mathbb S^2}\int_{\mathbb S^2} f(x+y) d\sigma(x) d\sigma(y) extends uniquely from the space of all CC^\infty functions on R3\mathbb R^3 with compact support to a bounded linear functional on L2(R3)L^2\left(\mathbb R^3\right).

This is the problem that I had to leave completely unsolved. It looks like it’s related to Fourier analysis or harmonic analysis, which I have yet to read up on (I plan to do so later this summer). I haven’t attempted to compute the first integral; things like the divergence theorem come to mind, but it seems difficult to apply to this situation. For the second part, this looks like a routine application of Hahn-Banach, but upon closer inspection, one may instead use the fact that CC^\infty is dense in L2(R3)L^2\left(\R^3\right) together with continuity of this map (with respect to the L2L^2 norm). However, the details are hazy at best, and I’m not clear on how the first integral is relevant. Oh well.

Problem 3.

Let 1<p,q<1\lt p, q\lt\infty satisfy 1p+1q=1\frac 1p+\frac 1q=1. Fix fLp(R3)f\in L^p\left(\mathbb R^3\right) and gLq(R3)g\in L^q\left(\mathbb R^3\right). (a) Show that [fg](x):=R3f(xy)g(y)dy[f*g](x):=\int_{\mathbb R^3}f(x-y)g(y)dy defines a continuous function on R3\mathbb R^3. (b) Moreover, show that [fg](x)0[f*g](x)\to 0 as x|x|\to\infty.

For the first part, we may consider that for any hR3h\in\mathbb R^3, fg(x+h)fg(x)R3(f(x+hy)f(xy))g(y)dyf(x+h)f(x)pgq.\begin{align*} \left\lvert f*g(x+h)-f*g(x)\right\rvert &\leq \int_{\mathbb R^3}\left\lvert\left(f(x+h-y)-f(x-y)\right) g(y)\right\rvert dy \\ &\leq \left\lVert f(x+h-\cdot)-f(x-\cdot)\right\rVert_p \left\lVert g\right\rVert_q. \end{align*} By the continuity of translations on Lp(R3)L^p\left(\mathbb R^3\right), it follows that the first multiplicand vanishes as h0h\to 0, as desired.

Frankly, the second part still eludes me. I believe there’s an approximation argument to be made — it seems that the claim should be pretty clear when either ff or gg is compactly supported. However, the specifics of this type of argument seem difficult to justify, as there will be two limits to deal with: one limit for approximating ff or gg, and another for x\lvert x\rvert\to\infty.

Problem 4.

Let fC([0,)×[0,1])f\in C^\infty\left([0,\infty)\times [0, 1]\right) such that 001tf(t,x)2(1+t2)dxdt<.\int_0^\infty \int_0^1 \lvert\partial_t f(t, x)\rvert^2\left(1+t^2\right) dx dt < \infty. Prove there exists a function gL2([0,1])g\in L^2([0, 1]) such that f(t,)f(t,\cdot) converges to g()g(\cdot) in L2([0,1])L^2([0, 1]) as tt\to\infty.

It turns out that my intuition about this problem was correct. By Tonelli’s theorem, we have that 010tf(t,x)2(1+t2)dtdx<.\int_0^1\int_0^\infty \lvert\partial_t f(t, x)\rvert^2 \left(1+t^2\right) dt dx < \infty. In particular, the inner integral converges for almost every xx. Then, by Hölder’s inequality, we have: 0tf(t,x)dttf(t,x)1+t2L2([0,))(1+t2)12L2([0,)).\int_0^\infty \lvert\partial_t f(t, x)\rvert dt \leq \left\lVert \partial_t f(t, x) \sqrt{1+t^2}\right\rVert_{L^2([0,\infty))}\cdot \left\lVert \left(1+t^2\right)^{-\frac 12}\right\rVert_{L^2([0,\infty))}. Both of the norms on the right are finite: the first multiplicand is finite by the first remark, and the second is just the squareroot of 011+t2dt=π2\int_0^\infty \frac{1}{1+t^2} dt =\frac\pi 2. Hence, the following function is well-defined for almost every xx: g(x)=f(0,x)+0tf(t,x)dt,g(x)=f(0, x)+\int_0^\infty \partial_t f(t, x) dt, and this is equal to limtf(t)\lim_{t\to\infty} f(t) by the fundamental theorem of calculus. Moreover, we have by Minkowski’s inequality that g()2f(0,)2+0tf(t,)dt2.\lVert g(\cdot) \rVert_2\leq \lVert f(0, \cdot)\rVert_2 + \left\lVert\int_0^\infty \partial_t f(t, \cdot) dt\right\rVert_2. f(0,x)f(0, x) is continuous on a compact set, so it’s in L2L^2. For the second term in the bound, we have 0tf(t,)dt22=010tf(t,x)2dtdx010tf(t,x)2(1+t2)dtdx<.\left\lVert \int_0^\infty \partial_t f(t,\cdot) dt\right\rVert_2^2 = \int_0^1 \int_0^\infty \lvert\partial_t f(t, x)\rvert^2 dt dx \leq \int_0^1 \int_0^\infty \lvert\partial_t f(t, x)\rvert^2 \left(1+t^2\right) dt dx < \infty. Hence both terms in the estimate are finite, and we conclude gL2g\in L^2. Finally, it’s not hard to show that g(x)f(t,x)=tτf(τ,x)dτ,g(x) - f(t, x) = \int_t^\infty \partial_\tau f(\tau, x) d\tau, whose L2L^2 norm necessarily must vanish as tt\to\infty due to the convergence of the first integral.

Problem 5.

For a function f:RRf:\mathbb R\to\mathbb R belonging to L1(R)L^1(\mathbb R), we define the Hardy-Littlewood maximal function as follows: (Mf)(x):=suph>012hxhx+hf(y)dy.(Mf)(x) := \sup_{h>0} \frac{1}{2h} \int_{x-h}^{x+h}\left| f(y)\right| dy. Prove that it has the following property: There exists a constant AA such that for any λ>0\lambda>0, {xR:(Mf)(x)>λ}Aλf1,\left|\left\lbrace x\in\mathbb R : (Mf)(x) > \lambda\right\rbrace\right| \leq \frac A\lambda \lVert f\rVert_1, where E\left| E\right| denotes the Lebesgue measure of EE. If you use a covering lemma, you should prove it.

In the previous post, I thought of a possible solution that could utilise a covering lemma. Upon some further research, I found the Vitali Covering Lemma. I will probably revisit the topic of covering theorems later this summer, since they appear to have useful applications (such as this problem). This lemma makes the problem pretty straight-forward, so I’ll omit the solution.

Problem 10.

Let ΩC\Omega\subsetneq \mathbb C be a simply connected domain, and f:ΩΩf:\Omega\to\Omega be a holomorphic mapping. Suppose that there exist points z1,z2Ωz_1,z_2\in\Omega, z1z2z_1\neq z_2, with f(z1)=z1f\left(z_1\right)=z_1 and f(z2)=z2f\left(z_2\right)=z_2. Show that ff is the identity mapping on Ω\Omega.

Last time, I was able to reduce the problem to the case where ff was a holomorphic mapping of the unit disc to itself with two fixed points. If z1z_1 was a fixed point, one could move it to the origin via the biholomorphic map wwz11z1ww\mapsto\frac{w-z_1}{1-\overline{z_1}w}, so ff fixed both the origin and some other nonzero z2Dz_2\in\mathbb D. Thus, by the Schwarz lemma, ff must be a rotation, and since nontrivial rotations have exactly one fixed point, ff is the identity.

I guess I was nearly there, I just forgot about the Schwarz lemma.

Problem 11.

Let f:CCf:\mathbb C\to\mathbb C be a holomorphic function with f(z)0f(z)\neq 0 for all zCz\in\mathbb C. Define U={zC:f(z)<1}U=\left\lbrace z\in\mathbb C: \lvert f(z)\rvert < 1\right\rbrace. Show that all connected components of UU are unbounded.

Let VUV\subset U be one of its connected compnonents, and suppose that it’s bounded. We claim that f(V)=D{0}f(V)=\mathbb D\setminus\lbrace 0\rbrace. First, f(V)f(V) is open in the punctured unit disc by the open mapping theorem. It is also closed in the punctured disc’s relative topology: if z0z_0 is in the interior of the punctured disc and znz_n is a sequence in f(V)f(V) approaching z0z_0, then we may construct a sequence wnw_n in VV such that f(wn)=znf\left(w_n\right)=z_n for all nn. Since the sequence wnw_n is bounded, we may assume without loss of generality that it’s convergent by passing to a subsequence. If w=limwnw=\lim w_n, then f(w)=z0f(w)=z_0. But wVw\in\overline V, and since VV is a connected component of UU and UU is open, ww should be in the same connected component as VV. Hence wVw\in V, and thus z0f(V)z_0\in f(V). We conclude that f(V)f(V) is closed and open in the punctured disc, so it’s the entire punctured disc (it’s obviously nonempty).

But we can apply the same argument to points on the boundary of the punctured disc. In particular, consider the sequence wnw_n in VV such that f(wn)=1n+1f\left(w_n\right)=\frac 1{n+1} for all nn. Refining to a convergent subsequence again, setting w=limwnw=\lim w_n yields f(w)=0f(w)=0, contradicting the assumption. We conclude that VV could not have been bounded.

The idea is that the only way ff can avoid 00 is if for every sequence znz_n such that f(zn)0f\left(z_n\right)\to 0, znz_n “escapes” to infinity. This can’t happen on a bounded set.