GENARAL ANALYSIS (PhD Program Qualify Exam: Feb. 23, 2001 ) I. (15%) Given a

GENARAL ANALYSIS

(PhD Program Qualify Exam: Feb. 23, 2001 )

I. (15%) Given a_n∈ {0, 1, 2, 3, 4} and bn∈ {0, 1, 2, 3, 4, 5, 6} and define the set P by

P = n

(x, y)∈ [0, 1] × [0, 1]¯¯x = X∞ n=1

an

5ⁿ, y = X∞ n=1

bn

7ⁿ an∈ {0, 2} , bn∈ {1, 3, 5}o

What does P look like? Compute the Lebesgue measure of P . Is P open, closed, compact, perfect?

Can you evaluate the Hausdorff dimension of P ?

II. (15%) Given x = (x₁, x₂,· · · , xn)∈ Rⁿ with|x| = (x²1+ x²₂+· · · + x²n)^1/2. Show that the function ρ(x) defined in Rⁿ by ρ(x) = exp(_|x|2¹₋₁) if|x| < 1 and ρ(x) = 0 if |x| > 1 belongs to Cc(Rⁿ), the space of infinitely differentiable with compact support. Compute the integralR

Rⁿρ(x)dx.

III. (15%) If the function f (x) is absolutely continuous on [a, b], then the length s of the curve y = f (x) can be computed according to the formula

s = Z b

a

p1 +|f⁰|²(x)dx

(You need to start from the definition of arc-length!)

IV. (20%) Let Ω be a bounded domain in Rⁿ, If u is a measurable function on Ω such that|u|^p∈ L¹(Ω) for some p∈ R, we define

Φp(u)≡

· 1

|Ω|

Z

Ω

|u|^pdx

¸1/p

where|Ω| denotes the measure of Ω. Show that (a) lim

p→∞Φ_p(u) = sup

Ω

|u|;

(b) lim

p→−∞Φp(u) = inf

Ω |u|;

(c) lim

p→0Φp(u) = exp

· 1

|Ω|

Z

Ω

log|u|dx

¸ .

(d) Φ is logarithmically convex in 1/p, i.e. if p≤ q ≤ r and

1/q = λ/p + (1− λ)/r then log Φq ≤ λ log Φp+ (1− λ) log Φr

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V. (15%) Give 1≤ p < ∞ and a sequence {fn}^∞n=1⊂ L^p(Ω) and f∈ L^p(Ω).

(a) Can you define the weak convergence of f_n to f in L^p(Ω) (denoted by f_n* f ).^w (b) For 1 < p <∞ show that fn

* f in Lw ^p(Ω) if and only if sup_nkfnkL^p(Ω)<∞ and R

Efndx→R

Ef dx

for all bounded measurable set E ⊂ Ω. Is it true for p = 1?

VI. (20%) Let (X, µ) be a measure space, and let 1 ≤ p ≤ ∞ and C > 0. Suppose K is a measurable function on X× X such thatR

X|K(x, y)|dµ(y) ≤ C for all x ∈ X and R

X|K(x, y)|dµ(x) ≤ C for all y∈ X. Define the function T f by

T f (x)≡ Z

X

K(x, y)f (y)dµ(y) .

(a) Show that T f is well defined almost everywhere and is in L^p(X), andkT fkL^p(X) ≤ CkfkL^p(X). (Hint: H¨older inequality)

(b) Use (a) to show that if f ∈ L¹(Rⁿ) and g∈ L^p(Rⁿ), 1≤ p ≤ ∞, then the convolution f ∗g ∈ L^p(Rⁿ) andkf ∗ gkL^p(Rⁿ)≤ kfkL¹(Rⁿ)kgkL^p(Rⁿ)

(c) Let ρε(x)≡ ε⁻ⁿρ(x/ε), ρ being the same as problem II, show that if f ∈ L^γ(Rⁿ) (1≤ γ < ∞) then f∗ ρε→ f in L^γ(Rⁿ) strongly.

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