Download An Introductory Course in Lebesgue Spaces by Rene Erlin Castillo, Humberto Rafeiro PDF

By Rene Erlin Castillo, Humberto Rafeiro

Introduces reader to fresh subject matters in areas of measurable functions
Includes portion of difficulties on the finish of every bankruptcy ​
Content makes it possible for use with mixed-level classes
Includes non-standard functionality areas, viz. variable exponent Lebesgue areas and grand Lebesgue spaces

This e-book is dedicated completely to Lebesgue areas and their direct derived areas. detailed in its sole commitment, this publication explores Lebesgue areas, distribution services and nonincreasing rearrangement. furthermore, it additionally bargains with susceptible, Lorentz and the more moderen variable exponent and grand Lebesgue areas with substantial element to the proofs. The publication additionally touches on easy harmonic research within the aforementioned areas. An appendix is given on the finish of the publication giving it a self-contained personality. This paintings is perfect for lecturers, graduate scholars and researchers.

Topics
Abstract Harmonic Analysis
Functional research

Show description

Read or Download An Introductory Course in Lebesgue Spaces PDF

Best functional analysis books

Elliptic theory and noncommutative geometry

The ebook offers with nonlocal elliptic differential operators. those are operators whose coefficients contain shifts generated by means of diffeomorphisms of the manifold on which the operators are outlined. the most objective of the research is to narrate analytical invariants (in specific, the index) of such operators to topological invariants of the manifold itself.

Measure Theory and Integration

Ways integration through degree, instead of degree through integration.

Weighted inequalities in Lorentz and Orlicz spaces

This set of chosen papers of Klingenberg covers the various vital mathematical elements of Riemannian geometry, closed geodesics, geometric algebra, classical differential geometry and foundations of geometry of Klingenberg. His contributions to Riemannian geometry have been major within the huge, in addition to beginning a brand new period in international Riemannian geometry.

Additional resources for An Introductory Course in Lebesgue Spaces

Example text

5) f p = f L p := ⎝ | f | p dμ ⎠ , X whenever 1 ≤ p < +∞. 5) does not define a norm when p < 1, we can take f = χ[0,1/2] , 1 g = χ[1/2,1] and we see that we have a reverse triangle inequality in L 2 ([0, 1], L , m). 79. We now want to see if the product of two functions in some L p is still in L p . The following example shows us that this is not always true. 18. Consider the function f (x) = |x|−1/2 if |x| < 1, 0 if |x| ≥ 1. 52 note that 3 Lebesgue Spaces ˆ ˆ dx = 4, |x| f (x) dx = R therefore f ∈ L1 (m), but [−1,1] ˆ ˆ f 2 (x) dx = R [−1,1] dx |x| is a divergent integral, therefore f ∈ / L1 (m).

Let {an }n∈Z and {bn }n∈Z be sequences of real numbers such that k= ∞ ∑ ∞ ∑ |an | < ∞ and n=−∞ |bm | p < ∞ m=−∞ where p > 1. Let Cn = ∑∞m=−∞ an−m bm . Prove that (a) |Cn | ≤ k1/q ∑∞m=−∞ |an−m ||bm | p (b) ∑∞n=−∞ |Cn | p 1/p 1/p ≤ k ∑∞n=−∞ |bn | p where 1/p 1 p + 1q = 1. 27. If an > 0 for n = 1, 2, 3, . . show that ∞ ∑ n=1 √ n ∞ a1 a2 · · · an ≤ e ∑ an . n=1 42 2 Lebesgue Sequence Spaces If a1 ≥ a2 ≥ · · · ≥ ak ≥ · · · ≥ an ≥ 0 and α ≥ β > 0. Demonstrate that n 1/α ∑ aαk ≤ k=1 1/β n β ∑ ak . 28.

Castillo, H. 1007/978-3-319-30034-4 3 43 44 3 Lebesgue Spaces f Note that if A = 0, / then 0 is a lower bound on A, and thus inf(A) ∈ R. Let α = ∞ < ∞, we state that α ∈ A. Notice that Eα = {x ∈ X : | f (x)| > α } = ∞ {x ∈ X : | f (x)| > α + 1/n} n=1 moreover, for each n the set {x ∈ X : | f (x)| > α + 1/n} ∈ A. e. e. it follows f ∞ = f∗ ∞ = sup | f ∗ (x)| = sup | f (x)|. 2. We define L∞ (X, A , μ ), called the set of essentially bounded functions, by L∞ (X, A , μ ) = f : X → R is an A -measurable function and f ∞ <∞ .

Download PDF sample

Rated 4.06 of 5 – based on 21 votes