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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-

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-

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part 3 a b 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-
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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