1. Riesz Representation for measures2. Paley-Wiener-Schwartz theorem3. Lojasiewicz-Hörmander4. deRham theorem with tempered coefficients5. Schwartz kernel theorem for tempered distibutions6. Gaglardo-Nirenberg7. Symbols and conormal distibutionsASSIGNMENT 1, 18.155DUE MIDNIGHT MONDAY 31 OCTOBER, 2011Date: Fri Oct 28 15:30:58 2011This assignment replaces Problem set 7 and should represent aboutas much work, maybe a little more, as required for one of the Problemsets. You can submit work at any time (electronically as usual) for meto review and return – the mark will be based on the final submissiononly. You can also ask for help etc, conditions as for a Problem set –I do not mind what or who you consult as long as the final product isyour own.In particular I do not mind if two, or three, people collaborate onone project subject to the conditions above.Here are the projects, with a list of who has asked to work on whatto date. If someone really wants to work on something else I am willingto negtotiate.So, what do I actually want. I will try to give you a guide in Prob-lems6 but basically I want you to state clearly the, or a, result corre-sponding to the given title and provide a proof – in some cases maybejust quoting exterior results – and some indication of the context ofthe result. As I say above, I am expecting a few pages, not a thesis.(1) Riesz representation for measures on locally compact spaces.(2) Paley-Wiener-Schwartz Theorem. Fourier transform and someholomorphic function theory. [Michael V.](3) Atiyah’s proof of Lojasiewicz-H¨ormander Theorem. Fouriertransform and some Algebraic Geometry.(4) Tempered distributional deRham cohomology of Euclidean space.Not much topology really.[Yasha, Michael A.](5) Tempered Schwartz Kernel Theorem. Functional Analysis, Fouriertransform.[Jelena, Jonathan](6) Gagliardo-Nirenberg, that the Sobolev spaces are rings for s >n/2 (or more generally).[Hans](7) Symbols and conormal distributions on Euclidean space. Fouriertransform.[Alex]1. Riesz Representation for measuresShow that every continuous linear functional on C00(M) for M a lo-cally compact metric space is given in terms of integration with respect12 ASSIGNMENT 1, 18.155 DUE MIDNIGHT MONDAY 31 OCTOBER, 2011to a measure on the space. This is basically the first part of the oldnotes, or you can get it from somewhere else. I’m assuming you wouldwant to give a brief account of the measure theory involved as well asthe theorem itself.2. Paley-Wiener-Schwartz theoremThis characterizes the Fourier(-Laplace) transform of distributionsof compact support in terms of entire functions on Cnwith prescribedgrowth properties. There are many variants of the theorem – I wouldlike in n dimensions – and you might like also to describe similar resultsfor smooth functions of compact support and for tempered distribu-tions, or L2functions, with support in a half-space (which is closer towhat Wiener did for the half-line).3. Lojasiewicz-H¨ormanderFirst get the accents correct on Lojasiewicz.A fundamental solution of a differential operator with constant co-efficients is a distribution – in this case E ∈ S0(Rn) such thatP (D)E(x) = δ(x).Taking the Fourier transform this reduces to the ‘division problem’ offindingˆE ∈ S0(Rn) satisfyingP (ξ)ˆE(ξ) = 1i.e. at least informallyˆE(ξ) = 1/P (ξ) where the zeros will be theproblem. This in fact has a solution for every non-zero polynomial,originally proved by Lojasiewicz then improved (at least the proof) byH¨ormander and then in the late 60s Atiyah gave a conceptually simple(but much more sophisticated) proof using resolution of singularitiesfollowing Hironaka.Look at the paper by Atiyah and give a discuss the analytic aspectsof his proof. Can you use his method to show solvability, that for anyf ∈ S0(Rn) there exists u ∈ S0(Rn) such thatP (ξ)u(ξ) = f(ξ) in S0(Rn)?4. deRham theorem with tempered coefficientsSince we can differentiate tempered distributions it is straight for-ward to define the deRham differential on distributional-formsdXIuIdxI=XIXj∂juIdxj∧ dI, uI∈ S0(Rn),ASSIGNMENT 1, 18.155 DUE MIDNIGHT MONDAY 31 OCTOBER, 2011 3assuming you can do this in the smooth case. Proving a Poincar´eLemma allows one to compute the tempered cohomology of Rn. If youthen want to push on further you can think about the same sort ofthing on manifolds (if you have the necessary background).5. Schwartz kernel theorem for tempered distibutionsShow that there is a natural bijection between the space of linearoperatorsT : S(Rm) −→ S0(Rn)which are continuous in the sense that of φj→ φ in S(Rm) thenT φj→ T φ weakly in S0(Rn), and the space S0(Rn× Rm). This canbe done using the Fourier transform and Schwartz Representation the-orem. You might like to discuss what this means in terms of tensorproducts (or not). Note that a ‘kernel’ A ∈ S0(Rn× Rm) gives riseto an operator TA(there is something to prove here of course) via theformula(1) (TAφ)(ψ) = A(ψ  φ)where the ‘exterior tensor product’ here just means ψ(x)φ(y) ∈ S(Rn×Rm).6. Gaglardo-NirenbergThe Sobolev spaces Hs(Rn) for s > n/2 are subspaces of C00(Rn),the bounded continuous functions on Rnwith supremum norm. In factthey are also subalgebras, i.e. are closed under pointwise products:Hs(Rn) · Hs(Rn) ⊂ Hs(Rn), s > n/2and indeed form a C∞-algebra, meaning that if f is a smooth functionon Rkthen f(u1, . . . , uk) ∈ Hs(Rn) if all the ui∈ Hs(Rn). This resultfollows from the Gagliardo-Nirenberg estimates.You maybe should start with the case s ∈ N when the estimates aremore straightforward.7. Symbols and conormal distibutionsThe standard space of symbols (with bounds) usually written Sm(Rn)consists of those smooth functions satisfying the estimates|∂αxa(x)| ≤ Cα(1 + |x|)m−|α|for all multiindices α. Describe in detail the Fourier transform of thesesymbols (conormal distributions at the origin) and how these are re-lated to Sobolev spaces. There is plenty more to do about these if you4 ASSIGNMENT 1, 18.155 DUE MIDNIGHT MONDAY 31 OCTOBER, 2011want (the beginning of microlocal analysis). It is difficult to get a pre-cise characterization of the Fourier transform of Smbut you should atleast characterize the union over m for instance in terms of regularity:If a ∈ Sm(Rn) there exists s = s(m) such thatξβDαξˆa ∈ Hs+|β|−|α|(Rn)and conclude that singsupp(ˆa) ⊂