1. Let fE LP(R). Consider the operator T defined by 1 (Tf)(x) = So x f(t)dt Show that the operator T is well defined for all x > 0, and the function Tf is continuous on 10, [ and satisfies |(Tf)(x)| ≤ x ||f||p> > 2. If f, g € LP(R), then |(Tf)(x) − (Tg)(x)| 0.
1. Let fE LP(R). Consider the operator T defined by 1 (Tf)(x) = So x f(t)dt Show that the operator T is well defined for all x > 0, and the function Tf is continuous on 10, [ and satisfies |(Tf)(x)| ≤ x ||f||p> > 2. If f, g € LP(R), then |(Tf)(x) − (Tg)(x)| 0.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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Transcribed Image Text:1. Let f € LP(R). Consider the operator T defined by
1
(Tf)(x) = -√√f(t)dt
Show that the operator T is well defined for all x > 0, and the function Tf is
continuous on 10, [ and satisfies
|(Tf)(x) < x ||f||
2. If f,gE LP (R+), then
|(Tf)(x) — (Tg)(x)| ≤ x¯||f9|p₁ x > 0.
3. Let g be a continuous function with compact support in 10, co[. Set G(x)
//Sog(t) dt.
(a) Show that G is of class C¹(R) and that 0 < G(x) <||||.
(b) Deduce that lim (G(x)) = 0 and that fo (G(x)) dx < +∞o.
8418
(c) Show that
=
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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=1).
1. Let ƒ € L¹(R). Consider the operator T defined by
1
(Tf)(x) = = √ √ ² f(t)dt
Show that the operator T is well defined for all x > 0, and the function Tf is
continuous on ]0, ∞[ and satisfies
|(Tƒ)(x)| ≤ x¯ ³ ||ƒ||px>
2. If f, g € LP(R), then
|(Tƒ)(x) − (Tg)(x)| < x¯ ||ƒ-9||p₂\x>0.
3. Let g be a continuous function with compact support in 10, ∞[. Set G(x)
So g(t)dt.
(a) Show that G is of class C¹(R) and that 0 < G(x) < ||g||.
(b) Deduce that lim (G(x)) = 0 and that f (G(x)) dx < +∞.
(c) Show that
8+←H
[*_xG'(x) (G(x))²-¹ dx + √ (G(x))³ dx = · g(x)| (G(x))³-¹ dr.
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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=1).
1. Let ƒ € L¹(R). Consider the operator T defined by
1
(Tf)(x) = = √ √ ² f(t)dt
Show that the operator T is well defined for all x > 0, and the function Tf is
continuous on ]0, ∞[ and satisfies
|(Tƒ)(x)| ≤ x¯ ³ ||ƒ||px>
2. If f, g € LP(R), then
|(Tƒ)(x) − (Tg)(x)| < x¯ ||ƒ-9||p₂\x>0.
3. Let g be a continuous function with compact support in 10, ∞[. Set G(x)
So g(t)dt.
(a) Show that G is of class C¹(R) and that 0 < G(x) < ||g||.
(b) Deduce that lim (G(x)) = 0 and that f (G(x)) dx < +∞.
(c) Show that
8+←H
[*_xG'(x) (G(x))²-¹ dx + √ (G(x))³ dx = · g(x)| (G(x))³-¹ dr.
=
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