By Luc Tartar

After publishing an creation to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with one other set of lecture notes in keeping with a graduate direction in elements, as indicated through the identify. A draft has been to be had on the web for many years. the writer has now revised and polished it right into a textual content obtainable to a bigger audience.

**Read or Download An Introduction to Sobolev Spaces and Interpolation Spaces (Lecture Notes of the Unione Matematica Italiana) PDF**

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After publishing an creation to the Navier–Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with one other set of lecture notes in keeping with a graduate direction in elements, as indicated by means of the name. A draft has been to be had on the net for many years. the writer has now revised and polished it right into a textual content obtainable to a bigger viewers.

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**Extra info for An Introduction to Sobolev Spaces and Interpolation Spaces (Lecture Notes of the Unione Matematica Italiana)**

**Example text**

For a vector a ∈ RN , and f ∈ L1loc (RN ), τa f denotes the function deﬁned by (τa f )(x) = f (x − a) for almost every x ∈ RN . This means that the graph of τa f is obtained from that of f by a translation of vector (a, 0). Of course, one has τb (τa f ) = τa+b f for all a, b ∈ RN , f ∈ L1loc (RN ). 5) An important property of convolution is that it commutes with translation; this is of course related to the fact that the Lebesgue measure is invariant by translation. 18 19 Oliver HEAVISIDE, English engineer, 1850–1925.

N . dx = R ϕ(x) dx = As x pv x1 , ϕ = pv x1 , x ϕ = limn→∞ |x|≥ 1 x ϕ(x) x n 1, ϕ for all ϕ ∈ Cc∞ (R), one has x pv x1 = 1. Notice that x pv x1 + C δ0 = 1 for all C, but pv x1 can be shown to be the only solution T of x T = 1 which is odd. ] 8 9 Ren´e-Louis BAIRE, French mathematician, 1874–1932. He worked in Montpellier and in Dijon, France. Hugo Dyonizy STEINHAUS, Polish mathematician, 1887–1972. He worked in Lw´ ow (then in Poland, now Lvov, Ukraine) until 1941, and after 1945 in Wroclaw, Poland.

1. For a nonnegative integer m, for 1 ≤ p ≤ ∞ and for an open set Ω ⊂ RN , the Sobolev space W m,p (Ω) is the space of (equivalence classes of ) functions u ∈ Lp (Ω) such that Dα u ∈ Lp (Ω) for all derivations Dα of length |α| ≤ m. It is a normed space equipped with the norm ||u|| = α |α|≤m ||D u||p , or the equivalent norm 1/p α p dx if 1 ≤ p < ∞ ||u||m,p = Ω |α|≤m |D u| α ||u||m,∞ = max|α|≤m ||D u||∞ if p = ∞. 2. (i) For 1 ≤ p ≤ ∞ and m ≥ 0 the Sobolev space W m,p (Ω) is a Banach space. (ii) For p = 2, W m,2 (Ω) is denoted1 by H m (Ω) and is a Hilbert space, for the scalar product ⎛ ⎞ ⎝ ((u, v)) = Ω Dα u Dα v ⎠ dx.