Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Problem 2. (i.e.+¹=1). Let p be a real number such that 1

Problem 2. (i.e.+¹=1). Let p be a real number such that 1

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
See similar textbooks
icon
Related questions
Question
part3 d measure and operator theory ( using previous part as result)
Problem 2. (i.e.+¹=1). Pq hat do you Let p be a real number such that 1 < p <∞ and let q be its conjugate 1. Let f LP(R). Consider the operator T defined by (Tf)(x) = f(1) di Show that the operator T is well defined for all a > 0, and the function Tf is. continuous on 10,00[ and satisfies |(Tf)(x)| ≤x|f||p> 2. If f,g LP(R), then |(Tf)(x) - (Tg)(x)| < xƒ-9|p₁\x>0. 3. Let g be a continuous function with compact support in 10,00[. Set G(x) = fog(t)dt. (a) Show that G is of class C¹ (R) and that 0≤G(z) <|9|| (b) Deduce that lim (G(x)) = 0 and that (G(x)) dx < +∞o. (c) Show that ∞0++-3 SzG'(x) (G(x))³-¹ dx + + [** (G(x)) dx = : - * \9 (2)| (G(x)) ³-¹ dr. (d) Deduce that p-1 P=¹ (G(x)) dx = - * \9(2)| (G(x))"~¹ dx. (e) Use Hölder inequality to deduce that ||G||<₁||||, and ||7g||p²|||||p 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: ||TS||||||
Transcribed Image Text:Problem 2. (i.e.+¹=1). Pq hat do you Let p be a real number such that 1 < p <∞ and let q be its conjugate 1. Let f LP(R). Consider the operator T defined by (Tf)(x) = f(1) di Show that the operator T is well defined for all a > 0, and the function Tf is. continuous on 10,00[ and satisfies |(Tf)(x)| ≤x|f||p> 2. If f,g LP(R), then |(Tf)(x) - (Tg)(x)| < xƒ-9|p₁\x>0. 3. Let g be a continuous function with compact support in 10,00[. Set G(x) = fog(t)dt. (a) Show that G is of class C¹ (R) and that 0≤G(z) <|9|| (b) Deduce that lim (G(x)) = 0 and that (G(x)) dx < +∞o. (c) Show that ∞0++-3 SzG'(x) (G(x))³-¹ dx + + [** (G(x)) dx = : - * \9 (2)| (G(x)) ³-¹ dr. (d) Deduce that p-1 P=¹ (G(x)) dx = - * \9(2)| (G(x))"~¹ dx. (e) Use Hölder inequality to deduce that ||G||<₁||||, and ||7g||p²|||||p 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: ||TS||||||
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS