用户: Solution/ 解答: GTM249 Chapter 2 Part I
2.1 Maximal Functions
2.1.1 | (a) Similar to the proof of Theorem 2.1.6, except for the only difference (b) |
2.1.2 | (a) Select a subset of with the minimal cardinality such that . Assume that is connected, otherwise consider every connected component. |
2.1.3 | (a) Obviously, For ,So . |
2.1.4 | (a) Write . (c) From Theorem 2.1.20, there exists a collection of disjoint open cubes such that and , soFrom Exercise 2.1.3,(d) Let for some . For , we haveConsequently, for , , and thuswhich implies that Using polar coordiantes, the determinant of the map is . Thus we haveFrom Exercise 2.1.3, we conclude that (e) For ,Therefore we havewhere the second inequality is from part (c). |
2.1.5 | We haveThe second result follows from taking , then . |
2.1.6 | (a) Define It is obvious that . |
2.1.7 | Write , and . |
2.1.8 | Let be the characteristic function of the unit ball in . Note thatwhere for . |
2.1.9 | Consider the Besicovitch Set (see Elias M. Stein, Harmonic Analysis, P435), which satisfies |
2.1.10 | (a) Define , then for , We claim Note that So we have Let , we have proved our assertion, which implies that . (b) DefineBy Minkowski’s inequality, we have So we have |
2.1.11 | (a) It’s easy to see is open. (b)Here we use Hölder’s inequality for the last inequality. Then we get(c) Consider We deduce from the three inequalities that . |
2.1.12 | (a) Firstly notice that , defined to be the set of all dyadic cubes, is countable. So any cover of dyadic cubes has a countable subcover, which is pairwise disjoint. |
2.1.13 | Use the result of Exercise 1.1.12 (a), we deduceTaking the supremum in yields |
2.1.14 | It suffices to prove the inequality for , as we may replace by . Define as in Theorem 2.1.10, and let Then for any , |
2.2 The Schwartz Class and the Fourier Transform
2.2.1 | (a) (b) (c) Take some such that is odd, real-valued and compactly supported. For instance, where and is defined as in part (b). Then takes purely imaginary values, so takes real values. Letthenis nonnegative and compactly supported. |
2.2.2 | For all multi-indices ,The last step is because in implies in , and subsequently in , hence in by Proposition 2.2.6. And gives . |
2.2.3 | If there exists nonzero functions such that , thenso , then . Indeed is exactly the spectrum of the Fourier transform.shows that is an eigenvalue.shows that is an eigenvalue.shows that is an eigenvalue.shows that is an eigenvalue. |
2.2.4 | Without loss of generality, assume , , then the proposition becomes This is because |
2.2.5 | Let . When , let where , then is a smooth approximate identity all the supports of which are contained in . For , we have by the Lebesgue dominated convergence theorem, and by Theorem 1.2.19 (a), which shows that is dense in , hence dense in . When , by the case is dense in with respect to the norm and therefore also with respect to the norm because for we haveMoreover, is dense in since for we have , hence dense in . When , any sequence of functions converges with respect to the norm to a continuous function, so is not dense in . |
2.2.6 | (a) by Proposition 2.2.11 (6) as by the Lebesgue dominated convergence theorem. (b)by Fubini’s theorem. (c) Use (2.2.15). But now the left-hand side of (2.2.15) may not converge to pointwise; only is known that the left-hand side converges to with respect to the norm. Yet by the Riesz–Fischer theorem, there is a sequence of as such that the left-hand side converges to a.e. if is taken to be , and the right-hand side still converges to after the restriction of , so we obtain a.e. |
2.2.7 | (a) Let in Exercise 2.2.6 (b), then , which is an approximate identity, so because is continuous at and is an approximate identity. (b) Suppose that , then for some bounded area , so contradicting the result of part (a). So , and it follows that by the Lebesgue dominated convergence theorem. |
2.2.8 | Let , then becauseby Theorem 1.2.12,where (5) and (4) stands for Proposition 2.2.11 (5) and (4), and is continuous at becauseExercise 2.2.7 (b) yields |
2.2.9 | (a) For where ,And for similar reasons, So (b) is odd implies that , so by part (a) (c) Suppose that such exists, then by part (b)a contradiction. |
2.2.10 | By substituting with which increases from 0 to then from to as ranges from to , we get Then since increases from to twice as ranges from to , |
2.2.11 | (a) Use Exercise 2.2.10 with to obtain that (b) Set in part (a) to get that For all , use a rectangle contour to lift the -axis and get thatwhere and , plugging into above |
2.2.12 | (a) For all , (b) For , write . We have |
2.2.13 | Let . Integrate by parts and apply the Cauchy–Schwarz inequality and Plancherel’s identity, we getThen replace by in the inequality above and the desired inequality follows from |
2.2.14 | Let be the unit ball and its complement. By the Cauchy–Schwartz inequality,Then repalce by for and get |
2.3 The Class of Tempered Distributions
2.3.1 | Such can be identified with the functionalwhich is a tempered distribution because the condition of Proposition 2.3.4 (b) is satisfied: in fact,where the last inequality is due to (2.2.3). The Lebesgue measure as a tempered distribution coincides with since |
2.3.2 | Pick any nonzero vector , and it is easily shown that is times an approximate identity for any multi-index as . Then for all multi-indices ,In the last step and is hence uniformly continuous on . |
2.3.3 | We have Or check that the two distributions have identical Fourier transforms: |
2.3.4 | (a)(b) To simplify notations, let . Sure enough, everything is essentially the same for .(c)(d) |
2.3.5 | (a) For all multi-indices and ,where the last step is because are both dominated by a finite sum of some ’s. In part (b) and (c), assume is identically equal to in . (b) For all multi-indices ,Noticing that for some when , we havewhere is the multi-index whose -th component is and the others , so as . (c) By part (b) in since the Fourier transform is a homeomorphism (Corollary 2.2.15), which on one hand reduces the proposition to in , and on the other hand implies that for all and multi-indices , Then for all multi-indices , For the same reason as part (b), so as . |
2.3.6 | The Fourier transform of a nonzero function, as a distribution with compact support, is a real analytic function by Theorem 2.3.21, which cannot be a function for it would otherwise be identically due to the uniqueness of real analytic functions (the uniqueness is obtained by showing that the interior of the zero set of a real analytic function is both open and closed). |
2.3.7 | For all , |
2.3.8 | LetThen in because for all ,where as and in the last inequality (2.2.3) is used. Since (see Example 2.3.9), for all , |
2.3.9 | (a) The usual definition for functions (still denoted by ) homogeneous of degree is for all , which agrees with the one given in the problem because for functions , (b) This is because (c) Since (d) Since The converse follows similarly. |
2.3.10 | (a) and converge to in and as as a consequence of Proposition 2.2.17, from which we know that the multiplication of distributions is not a continuous operation even when it is defined, for in yet . (b) in . This is because for all , by the Lebesgue dominated convergence theorem, |
2.3.11 | Let be a Schwartz function in whose Fourier transform is equal to on the ball and vanishes outside the ball (which can be constructed as the inverse Fourier transform of a Schwartz function with the properties that the desired Fourier transform of is supposed to have). Sincewe establish the identity . Then |
2.3.12 | (a) in for as because for all multi-indices ,and for the same reason as Exercise 2.3.5, Thus for all , (b) Suppose that is equal to in , then is equal to in . Apply part (a), which is also true when is replaced by , and getnamely, |
2.3.13 | Assume the converse. Then for all , is locally integrable and hence the map is a well defined operator from to for all (i.e. for all , since is covered by a finite number of neighborhoods of on which is integrable). Then we shall use the closed graph theorem to deduce that the operator is bounded, or in other words, . Suppose in , then by Proposition 2.3.22, in , so in . Note that in implies convergence in , so we can conclude that is indeed once we check that: if two functions are identical as tempered distributions, then they are the same as functions. In fact, if for all , , then so will be true for , which is justified by finding a sequence of Schwartz functions all bounded by converging pointwise to and making use of the Lebesgue dominated convergence theorem — such Schwartz functions can be taken as a sequence of step functions mollified at the endpoints approximating , where the ’s and ’s are each a finite combination of intervals contained in and with and . Then we get in the usual sense of functions. To violate the previous inequality whenever , take where . Then for , And using Proposition 2.2.11 and Example 2.2.9, so is a constant. Therefore the inequality cannot be true, for as . |