Humboldt-Universitat zu Berlin

Institut fur Mathematik

C. Wendl, S. Dwivedi,

L. Upmeier zu Belzen

FunktionalanalysisWiSe 2020–21

Problem Set 8

Due: Thursday, 14.01.2021 (21pts total)

Problems marked with p˚q will be graded. Solutions may be written up in German orEnglish and should be submitted electronically via the moodle before the Ubung on thedue date. For problems without p˚q, you do not need to write up your solutions, but it ishighly recommended that you think through them before the next Tuesday lecture. Youmay also use the results of those problems in your written solutions to the graded problems.

Problem 1Fix s • 0 and a multi-index ↵ of order m :“ |↵| P N.

(a) Use the Fourier transform and Fourier inverse operators F ,F ˚ : L2pRnq Ñ L2pRnq

to write down an explicit formula for the unique extension of B↵ : S pRnq Ñ S pRnqto a bounded linear operator B↵ : Hs`mpRnq Ñ H

spRnq.(b) p˚q Show that a sequence fj P H

mpRnq converges in the Hm-norm to f P H

mpRnqif and only if B�

fj Ñ B�f in the L

2-norm for all multi-indices � of order |�| § m.[3pts]

(c) p˚q Show that for any scalar-valued function f P L1pRnq, the convolution with f

defines a bounded linear operator HspRnq Ñ H

spRnq : g fiÑ f ˚ g, and if g PH

s`mpRnq, then B↵pf ˚ gq “ f ˚ B↵g. [4pts]

Hint: Problem Set 7 #6 proves the formula zf ˚ g “ pfpg for f, g P L1pRnq. For the

present problem, you may assume this formula also holds when f P L1pRnq and

g P L2pRnq; this case was omitted from the lecture due to lack of time, but is proved

via an easy density argument as Theorem 8.18 in the lecture notes.

(d) Show that for any f P HmpRnq and any approximate identity ⇢j : Rn Ñ r0,8q with

shrinking support, the functions fj :“ ⇢j ˚ f are in HmpRnq XC

8pRnq and convergein the H

m-norm to f as j Ñ 8.

Problem 2Suppose ⇢ P S pRnq satisfies ⇢ • 0 and

≥Rn ⇢pxq dx “ 1, and define ⇢jpxq :“ j

n⇢pjxq for

j P N.

(a) p˚q Show that for any s • 0 and f P HspRnq, the sequence ⇢j ˚ f P C

8pRnq satisfies

}⇢j ˚ f}Hs § }f}Hs and ⇢j ˚ fH

s›Ñ f as j Ñ 8.

Note that the convergence does not follow from Problem 1(d) since we are notassuming s P N. [4pts]

(b) Show that the same result holds if ⇢j P S pRnq is instead defined as F ˚ j for a

sequence of smooth functions j : Rn Ñ r0, 1s with compact support in the increa-singly large ball Bj`1p0q Ä Rn and j |Bjp0q ” 1.

Problem 3Prove that for every m P N, functions f P C

mpTnq are also in HmpTnq and the inclusion

CmpTnq ãÑ H

mpTnq is continuous.

1

density

of mi

Espaces

Pj fHepE hfHµ Gm If gHepE H

ftp.HGKP

2

Problem Set 8

Problem 4Fix constants a, b ° 1 with b P N and consider the periodic function fpxq :“ ∞8

k“01ake2⇡ibkx,

which is continuous since the series converges absolutely and uniformly. Prove:

(a) p˚q f P HspS1q if and only if s † logb a. [4pts]

(b) f P C0,↵pS1q for every ↵ P p0, 1q with ↵ § logb a.

Hint: Use Lemma 9.27 from the lecture notes. Note that the partial sums are conti-

nuously di↵erentiable: estimate their C0,1

-norms.

Remark: f is a variant of the function famously introduced by Weierstrass in 1872, which

is of class C1if b † a, but nowhere di↵erentiable if b • a. Part (a) establishes a weak

version of the latter statement by proving f R H1pS1q, which implies f R C

1pS1q via

Problem 3. Notice that while the Sobolev embedding theorem provides a continuous in-

clusion HspS1q ãÑ C

0,↵pS1q whenever ↵ § s ´ 1{2, f turns out to be in a wider range of

Holder spaces than is guaranteed by that theorem. (That is just a coincidence—there is

no interesting phenomenon behind it that I am aware of.)

Problem 5Assume ⌦ is a compact subset of either Rn or Tn, and 0 † ↵ † � § 1.

(a) Prove via the Arzela-Ascoli theorem that the inclusion C0,�p⌦q ãÑ C

0p⌦q is compact.

(b) Show that if fk P C0,�p⌦q is a uniformly C

0,�-bounded sequence that is C0-convergentto f P C

0p⌦q, then f is also in C0,�p⌦q.

Caution: Do not try to prove that fk is also C0,�

-convergent to f—that is not gene-

rally true.

(c) Show that the inclusion C0,�p⌦q ãÑ C

0,↵p⌦q is also compact.Hint: Given fk Ñ f as in part (b), use the relation

|gpxq ´ gpyq||x ´ y|↵ “

ˆ |gpxq ´ gpyq||x ´ y|�

˙↵{�

¨ |gpxq ´ gpyq|1´↵� .

for the functions g :“ f ´ fk.

Problem 6The linear inhomogeneous Cauchy-Riemann equation for functions f : C Ñ C of a complexvariable z “ x ` iy is a first-order PDE taking the form

Bf :“ Bxf ` iByf “ g.

Use the coordinates px, yq to identify C with R2 and consider functions that are fullyperiodic on R2; these are equivalent to complex-valued functions on the torus T2. Prove:

(a) p˚q If f P H1pT2q and g “ Bf P H

mpT2q for some m P N, then f P Hm`1pT2q. [6pts]

(b) If f P C1pT2q and g “ Bf is smooth, then f is smooth.

2

Problem Session 9

f c Cm Tn f is m times continuouslydifferentiable on Tn compact D If12km is bounded D Zdf c L2 Fn

for Klem Fn is of finite measure

f E CmcTn f e HMCFn

want HfHµm E Chthon

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a HHuses 241 1542 tax

E 22 b atc 2 ba cc Ea

awant forf c Hscs

lose aTaking logy we get

f c Hscs 0 0 Ss logiathe

Lemma9o2F_ Suppose f k is a sequence of continuous

functions on Sc IR converging uniformly to f

and there are constants a L b Id C O

and BE Oil s t

Hf fall E Ea and lfklco.ptCbk

Then f e 42 for a BItlogab

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If tiako Ine

b

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B L

that I tan e Neo

find bounds on the derivative of fincxwhich are finitely many terms

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