By Rene Erlin Castillo, Humberto Rafeiro

Introduces reader to contemporary themes in areas of measurable functions

Includes component of difficulties on the finish of every bankruptcy

Content permits 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. 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 newer variable exponent and grand Lebesgue areas with huge element to the proofs. The e-book additionally touches on easy harmonic research within the aforementioned areas. An appendix is given on the finish of the ebook giving it a self-contained personality. This paintings is perfect for lecturers, graduate scholars and researchers.

Topics

Abstract Harmonic Analysis

Functional research

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**Extra info for An Introductory Course in Lebesgue Spaces**

**Sample text**

17. 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).

Iii) T is an isometry. 14) 34 2 Lebesgue Sequence Spaces and | f (x)| = ∑ αk f (ek ) k∈N ≤ sup | f (ek )| ∑ |αk | = x j∈N k∈N 1 sup | f (ek )|. 15) we get that T f the spaces ( 1 )∗ and ∞ are isometric. ∞ ∞. 15) = f . We thus showed that One of the main difference between p and ∞ spaces is the separability issue. 8. 14. The space ∞ is not separable. Proof. Let us take any enumerable sequence of elements of where we take the sequences in the form (1) (1) (1) (1) (2) (2) (2) (2) (3) (3) (3) (3) ∞, namely {xn }n∈N , x1 = x1 , x2 , x3 , .

We will give another proof of the H¨older inequality using Minkowski’s inequality, but first an auxiliary lemma. 21. Let a, b, and θ be real numbers such that 0 < θ < 1 and a, b ≥ 0. Then n lim θ a1/n + (1 − θ )b1/n = aθ b(1−θ ) . n→+∞ n Proof. Let a, b > 0. Taking I(n) := θ a1/n + (1 − θ )b1/n , we have I(n) = exp n log θ a1/n + (1 − θ )b1/n ⎧ ⎫ ⎪ ⎨ ϕ 1n − ϕ (0) ⎪ ⎬ = exp 1 ⎪ ⎪ ⎩ ⎭ n where ϕ (t) = log θ at + (1 − θ )bt . Passing now to the limit, we get lim I(n) = exp ϕ (0) n→+∞ = exp θ log(a) + (1 − θ ) log(b) = aθ b1−θ , which ends the proof.