Math 720: Homework. Assignment 2: Assigned Wed 09/04. Due Wed 09/11. Assignment 1: Assigned Wed 08/28. Due Wed 09/04

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1 Math 720: Homework Do, but don t turn in optional problems There is a firm no late homework policy Assignment : Assigned Wed 08/28 Due Wed 09/04 Keep in mind there is a firm no late homework policy Starred problems are optional; but I d recommend looking at them They often involve results I will use later in class Let µ be a positive measure on (, Σ) (a) If A i Σ are such that A i A i+, show that µ( i= A i) = lim i µ(a i ) (b) If A i Σ are such that A i A i+, show that µ( i= A i) = lim i µ(a i ), provided µ(a ) < Show by example this is false true if µ(a ) = 2 Prove any open subset of R d is a countable union of cells 3 For each of the following sets, compute the Lebesgue outer measure (a) Any countable set (b) The Cantor set (c) {x [0, ] x Q} 4 (a) If V R d is a subspace with dim(v ) < d, then show that λ (V ) = 0 (b) If P R 2 is a polygon show that area(p ) = λ (P ) 5 Does there exist a σ-algebra whose cardinality is countably infinite? Disprove, or find an example Define µ(a) to be the number of elements in A Show that µ is a measure on (, P()) (This is called the counting measure) Let x 0 be fixed Define δ x0 (A) = if x 0 A and 0 otherwise Show that δ x0 is a measure on (, P()) (This is called the delta measure at x 0 ) Show that λ (a + E) = λ (E) for all a R d, E R d Show that λ (I) = l(i) for all cells (I only proved it for closed cells in class) Show that B(R) has the same cardinality as R (Challenge) Suppose f n : [0, ] [0, ] are all Riemann integrable, 0 f n and (f n ) 0 pointwise Show that lim 0 f n = 0, using only standard tools from Riemann integration n Assignment 2: Assigned Wed 09/04 Due Wed 09/ (a) Say µ is a translation invariant measure on (R d, L) (ie µ(x + A) = µ(a) for all A L, x R d ) which is finite on bounded sets Show that c 0 such that µ(a) = cλ(a) (b) Let T : R d R d be a linear transformation, and A L Show that T (A) L and λ(t (A)) = det(t ) λ(a) [Hint: Express T in terms of elementary transformations] 2 (a) Let E P(), and ρ : E [0, ] be such that E, E and ρ( ) = 0 For any A define { µ (A) = inf ρ(e i ) E i E, and A E j } Show that µ is an outer measure (b) Let (, d) be any metric space, δ > 0, α 0 and define E δ = {A ( diam(a) < δ} and ρα (A) = πα/2 diam(a) ) α Γ( + α 2 ) 2 Let H α,δ be the outer measure obtained with ρ = ρ α and the collection of sets E δ Define H α = lim δ 0 H α,δ Show H α is an outer measure and restricts to a measure H α on a σ-algebra that contains all Borel sets The measure H α is called the Hausdorff measure of dimension α (c) If = R d, and α = d show that H d is a non-zero, finite constant multiple of the Lebesgue measure [In fact H d = λ because of our choice of normalization constant, but the proof is much harder] (d) Let S B() Show that there exists (a unique) d [0, ] such that H α (S) = for all α (0, d), and H α (S) = 0 for all α (d, ) This number is called the Hausdorff dimension of the set S (e) Compute the Hausdorff dimension of the Cantor set 3 Using notation from the previous question, let S δ = {B(x, r) x, r (0, δ)} Using the collection of sets S δ and the function ρ = ρ α, we obtain an outer measure Sα,δ As before one can show that S α = lim δ 0 Sα,δ is an outer measure, and gives a Borel measure S α (a) Show by example S α H α in general (b) If = R d with the standard metric show that S d = λ [You may assume ρ d (B r) = λ(b r)] Details in class I left for you to check (Do it, but don t turn it in) Using notation from the proof of Caratheodory, show that µ (A ( E i )) = µ (A E i ) [We only proved it for A = in class]

2 Assignment 3: Assigned Wed 09/ Due Wed 09/8 Let µ, ν be two measures on (, Σ) Suppose C Σ is a π-system such that µ = ν on C (a) Suppose C i C such that C i = and µ(c i ) = ν(c i ) < Show that µ = ν on σ(c) (b) If we drop the finiteness condition µ(c i ) < is the previous subpart still true? Prove or find a counter example 2 (a) Let be a metric space and µ a Borel measure on Suppose there exists a sequence of sets B n such that B n B n+, Bn is compact, = B n and µ(b n ) < Show that µ is regular (b) Show directly that for all A L, λ(a) = sup{λ(k)} where K A is compact, and λ(a) = inf{λ(u)} where U A is open [Note: The previous subpart will only show this for all A B(R d )] 3 (a) Find E B(R) so that for all a < b, we have 0 < λ(e (a, b)) < b a (b) Let κ (0, /2) Does there exist E B(R) such that for all a < b R, we have κ(b a) λ(i (a, b)) ( κ)(b a)? Prove it 4 Let A L(R d ) Prove every subset of A is Lebesgue measurable λ(a) = 0 5 (a) Prove B(R m+n ) = σ({a B A B(R m ) & B B(R n )}) (b) Prove L(R m+n ) σ({a B A L(R m ) & B L(R n )}) (c) Show L(R 2 ) B(R 2 ) Let µ be a finite Borel measure on a compact metric space Let C = {A sup µ(k) = µ(a) = inf µ(u)} K A U A K compact U open We saw in class that C is closed under countable increasing unions Show C is closed under relative compliments Is any σ-finite Borel measure on R d regular? Show that there exists A R such that if B A and B L then λ(b) = 0, and further, if B A c and B L then λ(b) = 0 We say A P() is an algebra if A, and A is closed under complements and finite unions We say µ 0 : A [0, ] is a (positive) pre-measure on A if µ 0 ( ) = 0, and for any countable disjoint sequence of sets sequence A i A such that A i A, we have µ 0 ( A i) = µ 0(A i ) Namely, a pre-measure is a finitely additive measure on an algebra A, which is also countably additive for disjoint unions that belong to the algebra (Caratheodory extension) If A is an algebra, and µ 0 is a pre-measure on A, show that there exists a measure µ defined on σ(a) that extends µ 0 Assignment 4: Assigned Wed 09/8 Due Wed 09/25 Let C R d be convex Must C be Lebesgue measurable? Must C be Borel measurable? Prove or find counter examples [The cases d = and d > are different] 2 Let (, Σ, µ) be a measure space For A P () define µ (A) = inf{µ(e) E A & E Σ}, and µ (A) = sup{µ(e) E A & E Σ} (a) Show that µ is an outer measure (b) Let A, A 2, P() be disjoint Show that µ ( A i) µ (A i ) [The set function µ is called an inner measure] (c) Show that for all A, µ (A) + µ (A c ) = µ() (d) Let A P() with µ (A) < Show that A Σ µ µ (A) = µ (A) 3 Let f : R be measurable, and g : R R be Lebesgue measurable True or false: g f : R is measurable? Prove or find a counter example 4 Let (, Σ) be a measure space, and f, g : [, ] be measurable Suppose whenever g = 0, f 0, and whenever f = ±, g (, ) Show that f : [, ] is measurable [ Note that by the given data you will never get a g meaningless quotient of the form 0 0 defined in the natural manner] ± or ± The remainder of the quotients (eg ) can be 5 Let f n : R be a sequence of measurable functions such that (f n ) f almost everywhere (ae) Let g : R R be a Borel function (a) If for ae x, g is continuous at f(x), then show (g f n ) g f ae (b) Is the previous part true without the continuity assumption on g? (An alternate approach to λ-systems) Let M P () We say M is a Monotone Class, if whenever A i, B i M with A i A i+ and B i B i+ then A i M and B i M If A P () is an algebra, then show that the smallest monotone class containing A is exactly σ(a) [You should also address existence of a smallest monotone class containing A] Prove that the completion Σ µ we defined in class is the smallest µ-complete σ-algebra that contains Σ Show that f : [, ] is measurable if and only if any of the following conditions hold (a) {f < a} Σ for all a R (c) {f a} Σ for all a R (b) {f > a} Σ for all a R (d) {f a} Σ for all a R Let (f n ) is a sequence of real valued measurable functions Define f(x) = lim f n (x) if the limit exists, and f(x) = otherwise Show that f is measurable

3 Assignment 5: Assigned Wed 09/25 Due Wed 0/02 Let (, Σ, µ) be a measure space, and (, Σ µ, µ) it s completion Show that g : [, ] is Σ µ -measurable if and only if there exists two Σ-measurable functions f, h : [, ] such that f = h µ-almost everywhere, and f g h everywhere 2 Let µ be a regular (but not necessarily finite) Borel measure on a metric space (a) True or false: For any f : R measurable and ε > 0 there exists g : R continuous such that µ{f g} < ε? Prove it or find a counter example (b) Do the previous subpart when = R d 3 Let for n N define A n = k Z [ 2k 2, 2k+ n 2 ) If E B(R) does lim λ(a n n E) n exist? Prove it 4 If f 0 is measurable show that f dµ = 0 f = 0 almost everywhere 5 (a) Suppose I R d is a cell, and f : I R is Riemann integrable Show that f is measurable, Lebesgue integrable and that the Lebesgue integral of f equals the Riemann integral (b) Is the previous subpart true if we only assume that an improper (Riemann) integral of f exists? Prove or find a counter example Let f : [0, ] [0, ] be the Cantor function, and g(x) = inf{f = x} Show that f is (Hölder) continuous, and the range of g is the Cantor set What is the largest exponent α for which f is Hölder-α continuous? Let µ be the counting measure on N, and f : N R a function (a) If f(n) <, then show that n= f(n) = N f dµ (b) If the series n= f(n) is conditionally convergent, show that f dµ is N not defined Let be a metric space C be closed and f : C R be continuous (a) If 0 f, then show that there exists F : R continuous such that F (c) = f(c) for all c C [Hint: Let F (x) = f(x) for all x C, and F (x) = inf{f(c) + d(x,c) c C} for x C] d(x,c) (b) (Tietze extension theorem in metric spaces) Do the previous subpart without assuming 0 f [Hint: Put g = tan (f), construct G by the previous subpart and set F = tan(g)] Finish the proof of Lusin s theorem functions in class) (I only proved it for bounded positive Find a Borel measurable function f : [0, ] R which is not continuous almost everywhere Let 0 s t be two simple functions Show s t Show directly αf = α f for any α R and integrable function f Assignment 6: Assigned Wed 0/02 Due Never In light of your MIDTERM this homework is optional (a) If f is a bounded measurable function and µ() <, then show fdµ = inf{ t dµ t f is simple} (b) If f, g are bounded measurable functions and µ() < show directly that (f + g) dµ = f dµ + g dµ 2 Let f : [0, ) R be a measurable function We define the Laplace Transform of f to be the function F (s) = exp( st)f(t) dt wherever defined 0 (a) If f(t) dt <, show that F : [0, ) R is continuous 0 (b) If t f(t) dt <, show that F : [0, ) R is differentiable 0 (c) If f is continuous and bounded, compute lim s sf (s) sin(xy) 3 For p R define define F (y) = 0 + x p dx (a) For what p R is F defined? When defined, is F continuous? Prove it (b) Show that F is differentiable for p > 2, and not differentiable when p = 2 4 (Push forward measures) Let µ be a measure on (, Σ), and f : Y be any function Define τ P(Y ) by τ = {A Y f (A) Σ} For A τ define ν(a) = µ(f (A)) (a) Show that τ is a σ-algebra, and ν is a measure on (Y, τ) [The measure ν is called the push-forward of µ under f, and often denoted by µ f ] (b) If g L (Y, ν), then show that g f L (, µ) and g f dµ = Y g dν 5 (Pull back measures) Say ν is a measure on (Y, τ) and f : Y is surjective (a) Show that Σ = {A f(a) τ} need not be a σ-algebra If Σ is a σ-algebra, show that µ(a) = ν(f(a)) need not be a measure on (, Σ) (b) Define instead Σ = {A f (f(a)) = A, &f(a) τ}, and µ(a) = ν(f(a)) Show that Σ is a σ-algebra and µ is a measure (c) If g L (Y, ν), then show that g f L (, µ) and g f dµ = Y g dν 6 (Linear change of variable) Let f : R d R be integrable (a) For any y R d show that R d f(x + y) dλ(x) = R d f(x) dλ(x) (b) If T : R d R d an invertible linear transformation, and E L(R d ) Show that (f T ) det T dλ = f dλ T (E) Details in class I left for you to check Check that if s, t are non-negative simple functions then (s+t) = s+ t Show that there exists f : R [0, ) Borel measurable such that b a f dλ = for all a, b R with a < b R [Hint: Let g(x) = χ { x <} x /2, and define h(x) = m= n= 2 m n g(x m/n)] E

4 Assignment 7: Assigned Wed 0/09 Due Wed 0/6 Do questions 3, 5, and 6 from HW6 2 (a) (Jensen s inequality) Let a, b [, ] with a < b and ϕ : (a, b) R be a convex function If µ() = and f : (a, b) is integrable then show ( ) ϕ f dµ ϕ f dµ (b) If ϕ above is strictly convex, when can you have equality? 3 (a) Suppose p, q, r [, ] with p < q < r Prove that for all f L p L r, f L q Further, find θ (0, ) such that f q f θ p f θ r (b) If for some p [, ), f L p () L () show that lim q f q = f [This sort of justifies the notation ] (c) Let p 0 (0, ], µ() = and f L p0 () Prove lim p 0 + f p = exp ( ln f dµ) Let g 0 be measurable, and define ν(a) = g dµ Show that ν is a measure, A and E f dν = fg dµ E Prove Hölder s inequality for p = and q = If p i, q [, ] with N p i = q, show that n f i q f i pi Show that L is a Banach space For p [0, ) show that you need not have f + g p f p + g p Let p, q (, ), p + q =, f Lp and g L q Show that fg dµ = f p g q if and only if there exists constants α, β 0 such that αf p = βg q (a) If is σ-finite, then show f = sup g fg dµ g L {0} (b) Show that the previous subpart is false if is not σ-finite Assignment 8: Assigned Wed 0/6 Due Wed 0/23 (a) If µ() <, p < q, show L q () L p () and the inclusion map from L q () L p () is continuous Find an example where L q () L p () [Hint: Show f p µ() p q f q ] (b) Let l p = L p (N) with respect to the counting measure If p < q show that l p l q Is the inclusion map l p l q continuous? Prove your answer 2 (a) Suppose p [, ), and f L p (R d, λ) For y R d, let τ y f : R d R be defined by τ y f(x) = f(x y) Show that (τ y f) f in L p as y 0 (b) What happens for p =? 3 Suppose Σ = σ(c), where C P() is countable If µ is a σ-finite measure and p <, show that L p () is seperable (ie has a countable dense subset) 4 (a) Suppose lim λ sup n f n >λ f n dµ = 0 Show that there exists an increasing funciton ϕ with ϕ(λ)/λ as λ, such that sup n ϕ( f n ) < (b) Suppose {f n } is uniformly integrable, and sup n fn < lim λ sup n f f n >λ n = 0 Show that (c) Show that the previous part fails without the assumption sup n fn < 5 Let e n (x) = e 2πinx, = [0, ] For what p [, ] does {e n } have a convergent subsequence in L p (, λ)? Prove it In Vitali s convergence theorem prove that the assumption f L is unnecessary If (f n ) f in L, show that {f n } is uniformly integrable [This is part of Vitali s theorem which I didn t have time to prove in class] Show that if (f n ) f in measure, then (f n ) need not converge to f in L p Finish the proof that C c () is dense in L p [I only did the case when is compact in class] Show that simple functions are dense in L Show that C c (R) is not dense in L (R) Show that L (R) is not separable

5 Assignment 9: Assigned Wed 0/23 Due Wed 0/30 Recall we defined the variation of µ by µ = µ + + µ, and the total variation by µ = µ () (You should check that these are well defined) (a) Let M be the space of all finite signed measures on (, Σ) Show that M with total variation norm (ie with µ = µ ()) is a Banach space (b) Show that (µ n ) µ if and only if (µ n (A)) µ(a) uniformly in A, A Σ 2 (a) For a signed measure, we define f dµ = f dµ+ f dµ Suppose (f n ) f, (g n ) g, and f n g n almost everywhere with respect to µ If lim g n d µ = g d µ <, show that lim f n dµ = f dµ (b) Suppose f, f n L, and (f n ) f almost everywhere Show that lim f n f d µ = 0 if and only if lim f n d µ = f d µ 3 (a) If µ is a positive σ-finite measure, and ν is a finite signed measure such that ν µ, show that there exists f L (, µ) such that dν = f dµ (b) Compute dν d ν in terms of the Hanh decomposition of ν [Notation: We say g = dν if dν = g dµ] dµ 4 (a) Let ν and ν 2 be two finite signed measures on Show that there exists a finite signed measure ν ν 2 such that ν ν 2 (A) ν (A) ν 2 (A), and for any other finite signed measure ν such that ν(a) ν (A) ν 2 (A) we ust have ν ν 2 ν (b) If ν, ν 2 above are absolutely continuous with respect to a positive σ-finite measure µ, prove ν ν 2 µ and express d(ν ν2) dµ in terms of dν dν2 dµ and dµ 5 Let (Ω, F, P ) be a measure space with P (Ω) =, and L (Ω, F, P ) [The probabilistic interpretation is that Ω is the sample space, A F is an event, is a random variable, and P ( B) is the chance that B, where B B(R)] (a) Suppose G F is a σ-sub-algebra of F Show that there exists a unique G-measurable function Y such that A Y dp = dp for all A G [Y is A called the conditional expectation of given G, and denoted by E( G)] (b) (Tower property) If H G is a σ-sub-algebra, show that E( H) = E ( E( G) H) almost everywhere (c) (Conditional Jensen) If ϕ : R R is convex, show that ϕ(e( G)) E(ϕ() G) almost everywhere (d) Suppose L 2 (Ω, F, P ) Show that E( G) is the L 2 -orthogonal projection of onto the subspace L 2 (Ω, G) [Namely show E( G) L 2 (Ω, G), and Ω ( E( G))Y dp = 0 for all Y L2 (Ω, G)] In the proof of the Hanh decomposition, prove the following: Say µ() >, and α = inf{µ(b)} Let B n be a sequence of negative sets such that µ(b n) α Let N = B n Show µ(n) = α Prove the Hanh decomposition is unique up to null sets Prove uniqueness of the Jordan decomposition Show that the Radon-Nikodym theorem need not hold if µ, ν are not σ-finite Assignment 0: Assigned Wed 0/30 Due Wed /06 (a) Let p [, ) and q be conjugate Hölder exponent If is σ-finite, show that there exists a bijective linear isometry between (L p ) and L q (b) The above result is false for p = even when µ() < Find where our proof from class (when µ() < ) fails when p = (c) We can (partially) construct a counter example on l as follows The Hanh-Banach theorem shows that there exists exists T (l ) such that T a = lim a n, for all a = (a n ) l such that lim a n exists and is finite Show that there does not exist b l such that T a = a n b n for all a l 2 (a) Suppose m= ( n= a m,n ) < Show that m= ( n= a m,n) = n= ( m= a m,n) (b) Give a counter example to (a) if we only assume m n a m,n < Find a counter example where both iterated sums are finite 3 (a) If and Y are not σ-finite, show that Fubini s theorem need not hold (b) If [,] 2 f dλ is not assumed to exist (in the extended sense), show that both iterated integrals can exist, be finite, but need not be equal 4 (Fubini for completions) Suppose (, Σ, µ) and (Y, τ, ν) are two σ-finite, complete measure spaces Let ϖ = (Σ τ) π denote the completion of Σ τ with respect to the product measure π = µ ν (a) Show that Σ τ need not be π-complete (ie ϖ Σ τ in general) (b) Suppose f : Y [, ] is F-measurable Define as usual the slices ϕ f,x : Y [0, ] by ϕ f,x (y) = f(x, y), and similarly ψ f,y (x) = f(x, y) Show that for µ-almost all x, ϕ f,x is an τ-measurable, and for ν- almost all y, ψ f,y is an Σ-measurable (c) Suppose f is integrable on Y in the extended sense Define F (x) = Y f(x, y) dν(y) and G(y) = f(x, y) dµ(x) Show F is defined µ-ae and Σ-measurable Similarly show G is defined ν-ae, and τ-measurable Further, show and that F dµ = Y G dν = fd(µ ν) Y 5 Let (, Σ, µ), (Y, τ, ν) be two σ-finite measure spaces, p [, ], and f : Y R is Σ τ measurable Let F (x) = Y f(x, y) dν(y), and ψ y,f be the slice of f defined by ψ y,f (x) = f(x, y) Show that F L p () Y ψ y,f L p () dν(y) [When Y = {, 2} with the counting measure, this is exactly Minkowski s triangle inequality] Let µ() <, p [, ) and T (L p ) Let ν(a) = T (χ A ) We ve seen in class that ν µ and so dν = g dµ for some g L (µ) (a) Show that T f = fg dµ for all f simple (b) If p + q = show g q = sup{ sg}, where the supremum runs over all simple functions s such that s p Conclude g L q and g q T (c) Show that T f = fg dµ for all f Lp, to conclude the proof Show that the Lebesgue measure on R m+n is the product of the Lebesgue measures on R m and R n respectively

6 Assignment : Assigned Wed /06 Due Wed /3 If p + q =, f Lp, g L q show that f g is bounded and continuous If p, q <, show further f g(x) 0 as x 2 Let {ϕ n } be an approximate identity (a) If f C(R d ) L, show f ϕ n f pointwise (b) For α (0, ) define f C α f(x) f(y) = f +sup x y x y α, and C α = {f : R d R f C α < } If f C α, show that f ϕ n C α and f ϕ n f in C α 3 Define S(R d ) = {f C (R d ) m, α, sup x ( + x m ) D α f(x) < } Here m N {0}, and α = (α,, α d ) (N {0}) d is a multi-index, and D α f = α α d d f The space S is called the Schwartz Space (a) If p [, ), f L p (R d ), g S(R d ), show that f g C (R d ), and further D α (f g) = f (D α g) (b) For p [, ), show that Cc and S are dense subsets of L p 4 Let A, B L(R) be measurable, and define A + B = {a + b a A, b B} If λ(a) > 0 and λ(b) > 0 show A + B contains an interval Though I encourage you to check the properties on the Dirichlet and Fejér kernels stated in the optional problems, you may assume them here without proof Let C per = {f C(R) τ f = f} denote all continuous functions with period Since the Fejér kernels are an approximate identity, it immediately follows that the Cesàro sums σ N f f uniformly, for any f C per For general f C per, however, the partial sums S N f need not converge to f even pointwise (In fact, there exist many f C per such that S N f is divergent on a dense G δ ) If, however, f is a little bit better than continuous, then the Fourier series of f converges to f pointwise 5 Let α (0, ) and f C α per Show that (S N f) f uniformly, as N If f L p, g L q with p, q [, ] and /p + /q, show that f g = g f If f L p, g L q, h L r with p, q, r [, ] and /p + /q + /r 2, show that (f g) h = f (g h) Define the Derichlet kernel by D N (x) = N N exp(2πinx) (a) Show that S N f(x) = D N f(x) def = 0 f(y)d N(x y) dy [Recall, S N f = N ˆf(n)e N n, where e n(x) = e 2πinx, and ˆf(n) = f, e n = 0 f(y)ēn(y) dy] (b) Show that D N (x) = sin((2n+)πx) ε sin(πx) Further show lim N D ε N = N Define Fejér kernel by F N = N 0 D n (a) Show that σ N f def = N N 0 S n f = F N f (b) Show that F N (x) = sin2 (Nπx) N sin 2 (πx), and that {F N} is an approximate identity Assignment 2: Assigned Wed /3 Due Wed /20 Let µ be a finite signed Borel measure on [0, ] If n Z ˆµ(n) = 0, show µ = 0 2 Let 0 r < s Show that any bounded sequence in H s per has a subsequence that is convergent in H r per 3 Let f L 2 ([0, ]) Show that there exists a unique u C (R (0, )) such that u(x +, t) = u(x, t), lim t 0 + u(, t) f( ) L 2 per = 0, and t u 2 xu = 0 [Hint: You may assume the result of the optional problems] 4 Let s (0, ] and f L 2 per Prove f H s per sup 0<h [Update: The converse is false A correction with solution will be posted] h α τ h f f L 2 < 5 (a) Let n N be even, n + n = If ˆf l n (Z), show that f L n per([0, ]) and f L n ˆf l n [Hint: Let n = 2m Then f n L n = (f m ) 2 l 2 ] (b) Let s > 2 p 0, and p + q = If f Hs per show ˆf l q (Z) Further show that the map f ˆf is continuous from H s per l q (c) If n N is even, s > 2 n then show that Hs per L n ([0, ]) and that the inclusion map is continuous [This is one of the Sobolev embedding theorems] (a) If f, g L 2 per([0, ]), show that (f g) (n) = ˆf(n)ĝ(n) (b) If f, g L 2 per([0, ]), show that (fg) (n) = ˆf ĝ(n) def = m Z ˆf(m)ĝ(n m) (a) If α (0, ), f C α per([0, ]), show that lim n n α ˆf(n) = 0 (b) Show by example that the converse of the previous part is false For any s 0 show that H s per is a closed subspace of L 2 Let s, and f Hper s Show that f has a weak derivative Df, and Df H s Further, show that the map f Df : H s H s is linear and continuous Let s > 3/2 and f, g H s per Show that fg H, and further D(fg) = (Df)g + f(dg) Find a function f H /2 per L [So the Sobolev embedding theorem is false for s = /2] Let n N {0}, α [0, ) s > /2 + n + α Show that Hper s Cper n,α [0, ] and the inclusion map is continuous [Recall Cper n,α [0, ] is the set of all C n periodic functions on R (ie τ f = f) whose n th derivative is Hölder continuous with exponent α] Show that f H s for all s 0 f C per

7 Assignment 3: Assigned Wed /20 Due Wed /27 Let s > 3/2 and f, g H s per Show that fg H per, and further D(fg) = (Df)g + f(dg) [This was optional on last times homework] 2 (a) If f L (R d ) and f is not identically 0 (ae), then show that Mf L (R d ) The next few subparts outline a proof that for any p >, the maximal function is an L p bounded sublinear operator Let p (, ), f L p (R d ) and f 0 (b) Show that λ{mf > α} f 0 measurable 3d ( δ)α {f>δα} f, for any t > 0, δ (0, ) and (c) Let p (, ], and d N Show that there exists a constant c = c(p, d) such that Mf p c f p for all f L p (R d ) [Hint: For p <, use the previous part, the identity Mf p p = 0 pαp λ{mf > α} dα and optimise in δ] 3 Let µ be a finite signed Borel measure on R d such that µ λ Show that D µ =, µ-almost everywhere 4 Let α [0, d], and A B(R d ) If H α (A) <, show lim H α -almost all x A H α(a B(x,r)) r 0 c αr α = 0 for 5 (a) Suppose f : [a, b] R is a right continuous increasing function Show that there exists a finite Borel measure µ such that µ((x, y]) = f(y) f(x) for every x, y [a, b] Show further that µ = µ ac + µ s + i α iδ ai, where µ ac λ, α i > 0, a i [a, b), i α i <, and µ sc λ is such that µ sc ({x}) = 0 for all x R [Hint: If f is strictly increasing and continuous, define µ(a) = λ(f(a)), and consider its Lebesgue decomposition] (b) Let f : [a, b] R be monotone Show that f is differentiable almost everywhere, f L ([a, b]) and that b a f f(b) f(a) Let c α = πα/2 Γ(+ α 2 ) be the normalization constant from the definition of H α, the Hausdorff measure of dimension α (a) If 0 < H α (A) <, show lim sup r 0 H α(a B(x,r)) c αr α [2 α, ] for H α -ae x A (b) Show that there exists α < d and A R d with H α (A) (0, ) such that lim inf r 0 H α (A B(x, r)) H α (A B(x, r)) c α r α = 0 and lim sup r 0 c α r α <, for H α -almost every x R d (c) If C is the Cantor set, and α = log 2/ log 3, compute lim sup r 0 H α(c B(x,r)) c αr α (Infinite version of Vitali) Suppose A B α, where {B α } α A is an infinite collection of balls such that sup λ(b α ) < Show that there exists A A such that the sub-collection {B α } α A is disjoint and A 5B α If f L (R d ), show that Mf(x) f(x) at all Lebesgue points of f Assignment 4: Assigned Wed /27 Due Wed 2/04 Let µ be a positive finite Borel measure on R d, and α > 0 Show that for every A {Dµ > α}, we must have µ(a) αλ(a) 2 (a) (Polar Coordinates) Let f L (R 2 ) Show that f(x, y) dx dy = f(r cos θ, r sin θ) r dr dθ R 2 [0,) [0,2π) (b) (Higher dimensional version) Let f L (R d ) Let S = {y R d y = } be the d dimensional sphere of radius Show that there exists a unique measure σ on S such that f(x) dx = f(ry) r d dσ(y) dλ(r) R d r [0,) y S [Hint: For A B(S ) define σ(a) = λ(a ) where A = {rx x A, r [0, ]} Now for any B B(S ) prove the desired equality when f = χ A where A = {rx a < r < b, x B} ] Show that the arbitrary union of closed (non-degenerate) cells is Lebesgue measurable Find an example of E L(R d ) and x R d λ(e B(x,r)) such that lim r 0 λ(b(x,r)) does not exist Suppose f : R R is measurable Let α, β > 0 with α/β Q If f has period α, and also has period β (ie for all x R, f(x) = f(x + α) = f(x + β)), then show that f is constant almost everywhere (But f need not be constant everywhere!) We say the family {E r } shrinks nicely to x R d if there exists δ > 0 such that for all r, E r B(x, r) and λ(e r ) > δλ(b(x, r)) If {E r } shrinks nicely to x, show that lim λ(e r) E r f = f(x) for all Lebesgue points of f If f L (R d ), show that Mf(x) f(x) at all Lebesgue points of f If f : [a, b] R is absolutely continuous, then show that f is of bounded variation, and that the variation is absolutely continuous Conclude f can be written as the difference of two monotone absolutely continuous functions Let U, V R d be open and ϕ : U V be C and injective If x 0 U and ϕ(x 0 ) is not invertible, show that λ(ϕ(b(x 0, r))) lim = 0 r 0 λ(b(x 0, r))