2. If f, g € LP(R), then |(Tƒ)(x) − (Tg)(x)| < x¯‡||ƒ − 9||p₁ \x>0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
part 2 measure and operator theory
![Problem 2. Let p be a real number such that 1 < p < ∞ and let q be its conjugate.
(i.e.+¹=1).
P 9
1. Let f LP(R). Consider the operator T defined by
(Tf)(x) = == f*f(t)dt
Show that the operator T is well defined for all x > 0, and the function Tf is
continuous on ]0,00[ and satisfies
|(Tƒ)(x) < x¯||ƒ||p,
2. If f, g € LP(R), then
|(Tf)(x) - (Tg)(x)| ≤x|fg|₁x>0.
3. Let g be a continuous function with compact support in 10, ∞o[. Set G(x)
Sg(t)dt.
(a) Show that G is of class C¹(R) and that 0 ≤ G(x) < ||9||.
+00
(b) Deduce that lim (G(x)) = 0 and that fo (G(x)) dx < +∞o.
(c) Show that
√xG¹(x) (G(x))³-¹ dx + √ (G(x))³ dx = = √ [9(2)| (G(x))³-¹ dx.
(d) Deduce that
P = ¹ [~ (G(x))" dx = * \9(2)| (G(x))³-¹ dr.
P
(e) Use Hölder inequality to deduce that
||G||||||, and Tg||p²|||||p
p-1
p-1
4. Recall that the space of continuous function with compact support is dense in
LP(R). Use the previous parts and apply Fatou's lemma to deduce the Hardy
inequality:
||Tf|p<_P_||f||p-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64778d0a-957e-44d1-a0d8-dc4ce9d2d848%2Fac440bee-91ab-4d1e-9631-fa965e89b886%2F8pgroch_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2. Let p be a real number such that 1 < p < ∞ and let q be its conjugate.
(i.e.+¹=1).
P 9
1. Let f LP(R). Consider the operator T defined by
(Tf)(x) = == f*f(t)dt
Show that the operator T is well defined for all x > 0, and the function Tf is
continuous on ]0,00[ and satisfies
|(Tƒ)(x) < x¯||ƒ||p,
2. If f, g € LP(R), then
|(Tf)(x) - (Tg)(x)| ≤x|fg|₁x>0.
3. Let g be a continuous function with compact support in 10, ∞o[. Set G(x)
Sg(t)dt.
(a) Show that G is of class C¹(R) and that 0 ≤ G(x) < ||9||.
+00
(b) Deduce that lim (G(x)) = 0 and that fo (G(x)) dx < +∞o.
(c) Show that
√xG¹(x) (G(x))³-¹ dx + √ (G(x))³ dx = = √ [9(2)| (G(x))³-¹ dx.
(d) Deduce that
P = ¹ [~ (G(x))" dx = * \9(2)| (G(x))³-¹ dr.
P
(e) Use Hölder inequality to deduce that
||G||||||, and Tg||p²|||||p
p-1
p-1
4. Recall that the space of continuous function with compact support is dense in
LP(R). Use the previous parts and apply Fatou's lemma to deduce the Hardy
inequality:
||Tf|p<_P_||f||p-
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