Problem 2. (i.e.+1=1). 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) = 1/²*1 (1) dt Show that the operator T is well defined for all x > 0, and the function Tf is continuous on 10, ∞[ and satisfies |(Tƒ)(x)| ≤ x¯ ||f||p
Problem 2. (i.e.+1=1). 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) = 1/²*1 (1) dt Show that the operator T is well defined for all x > 0, and the function Tf is continuous on 10, ∞[ and satisfies |(Tƒ)(x)| ≤ x¯ ||f||p
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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part 3 measure theory a
![**Problem 2:**
Let \( p \) be a real number such that \( 1 < p < \infty \) and let \( q \) be its conjugate (i.e., \( \frac{1}{p} + \frac{1}{q} = 1 \)).
1. Let \( f \in L^p(\mathbb{R}_+) \). Consider the operator \( T \) defined by
\[
(Tf)(x) = \frac{1}{x} \int_0^x f(t)dt
\]
Show that the operator \( T \) is well defined for all \( x > 0 \), and the function \( Tf \) is continuous on \( ]0, \infty[ \) and satisfies
\[
|(Tf)(x)| \leq x^{-\frac{1}{p}} \|f\|_p, \quad \forall x > 0.
\]
2. If \( f, g \in L^p(\mathbb{R}_+) \), then
\[
|(Tf)(x) - (Tg)(x)| \leq x^{-\frac{1}{p}} \|f - g\|_p, \quad \forall x > 0.
\]
3. Let \( g \) be a continuous function with compact support in \( ]0, \infty[ \). Set \( G(x) = \frac{1}{x} \int_0^x g(t)dt \).
(a) Show that \( G \) is of class \( C^1(\mathbb{R}_+) \) and that \( 0 \leq G(x) \leq \frac{1}{x} \|g\|_1 \).
(b) Deduce that \( \lim_{x \to +\infty} (G(x))^p = 0 \) and that \( \int_0^\infty (G(x))^p dx < +\infty \).
(c) Show that
\[
\int_0^\infty xG'(x) (G(x))^{p-1} dx + \int_0^\infty (G(x))^p dx = \int_0^\infty](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9b724e5-7939-4f6f-9751-b1616d7d7fa6%2F9e13a318-7efc-4a22-b559-b580423045dc%2F3vpv93d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2:**
Let \( p \) be a real number such that \( 1 < p < \infty \) and let \( q \) be its conjugate (i.e., \( \frac{1}{p} + \frac{1}{q} = 1 \)).
1. Let \( f \in L^p(\mathbb{R}_+) \). Consider the operator \( T \) defined by
\[
(Tf)(x) = \frac{1}{x} \int_0^x f(t)dt
\]
Show that the operator \( T \) is well defined for all \( x > 0 \), and the function \( Tf \) is continuous on \( ]0, \infty[ \) and satisfies
\[
|(Tf)(x)| \leq x^{-\frac{1}{p}} \|f\|_p, \quad \forall x > 0.
\]
2. If \( f, g \in L^p(\mathbb{R}_+) \), then
\[
|(Tf)(x) - (Tg)(x)| \leq x^{-\frac{1}{p}} \|f - g\|_p, \quad \forall x > 0.
\]
3. Let \( g \) be a continuous function with compact support in \( ]0, \infty[ \). Set \( G(x) = \frac{1}{x} \int_0^x g(t)dt \).
(a) Show that \( G \) is of class \( C^1(\mathbb{R}_+) \) and that \( 0 \leq G(x) \leq \frac{1}{x} \|g\|_1 \).
(b) Deduce that \( \lim_{x \to +\infty} (G(x))^p = 0 \) and that \( \int_0^\infty (G(x))^p dx < +\infty \).
(c) Show that
\[
\int_0^\infty xG'(x) (G(x))^{p-1} dx + \int_0^\infty (G(x))^p dx = \int_0^\infty
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