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

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By Rene Erlin Castillo, Humberto Rafeiro

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

This booklet is dedicated completely to Lebesgue areas and their direct derived areas. specific in its sole commitment, this e-book explores Lebesgue areas, distribution features and nonincreasing rearrangement. furthermore, it additionally bargains with susceptible, Lorentz and the more moderen variable exponent and grand Lebesgue areas with huge element to the proofs. The booklet additionally touches on uncomplicated 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.

Abstract Harmonic Analysis
Functional research

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Then f g ∈ L1 (X, A , μ ) and ˆ | f g| dμ ≤ f p g q. e. Proof. e. e. therefore ⎞ ⎛ ˆ ˆ ⎟ ⎜ | f g| dμ ≤ ⎝ | f | dμ ⎠ g ∞ , X X ˆ thus | f g| dμ ≤ f 1 g ∞. X Now, suppose that 1 < p < ∞, 1 < q < ∞ and f ≥ 0, g ≥ 0. Define h(x) = [g(x)]q/p , then g(x) = [h(x)] p/q = [h(x)] p−1 . 19 we have pt f (x)g(x) = pt f (x)[h(x)] p−1 ≤ (h(x) + t f (x)) p − [h(x)] p . 16) we have ˆ ( h p + t f p)p − h p f (x)g(x) dμ ≤ t p p X Taking f (t) = ( h p +t f p) p , we get f (0) = h pp . Then ˆ f g dμ ≤ lim p t→0 f (t) − f (0) = f (0) t X = p( h p ) p−1 f p.

Taking n → ∞, we obtain ∞ ∑ | f (ek )|q k=1 where { f (ek )}k∈N ∈ q. 1 q ≤ f 1 q 30 2 Lebesgue Sequence Spaces Now, we affirm that: (i) T is onto. In effect given b = (βk )k∈N ∈ q , we can associate a bounded linear functional g ∈ ( p )∗ , given by g(x) = ∑∞k=1 αk βk with x = (αk )k∈N ∈ p (the boundedness is deduced by H¨older’s inequality). Then g ∈ ( p )∗ . (ii) T is 1-1. This is almost straightforward to check. (iii) T is an isometry. We see that the norm of f is the | f (x)| = q norm of T f ∑ αk f (ek ) k∈N ≤ ∑ |αk | 1 p p k∈N = x ∑ | f (ek )| ∑ | f (ek )| 1 q q k∈N 1 q .

Let X = N, A = P(N), μ = # the counting measure and the function f : N → N given by n → n . We state that A = {M > 0 : #({x ∈ X : | f (x)| > M}) = 0} = 0. / In fact, let M > 0 be arbitrary, and choose k > M, k ∈ N then #({x ∈ X : | f (x)| > M}) ≥ #({k}) = 1, which implies that M ∈ / A and since M is arbitrary, we conclude that A = 0, / therefore f ∞ = ∞. 2 Lebesgue Spaces with p ≥ 1 We now study the set of p-th integrable functions. 6. Let (X, A , μ ) be a measure space and p a positive real number.

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