By Makoto Namba (auth.)

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42). 14) can be reformulated as 1 1 (1,− p1 ) = 1. 28 we restrict ourselves to an example merely, since the main idea of envelopes just relies on those cases where we do not have the corresponding embeddings. We explain this in detail in the next chapters. However, this immediately proves the sharpness of the above assertions, too (and is also known already). 38), we present a family of functions hν that belong to Wn1 , but not to L∞ , as long as 0 < ν < 1 − n1 . For x ∈ Rn , consider the radial functions hν (x) = logν 1 + 1 |x| − logν 2, 0 < |x| < 1 0 , otherwise hν (x) , x ∈ Rn .

N p Rn Step 4. 38). 53) 32 Envelopes and sharp embeddings of function spaces and for p ≤ r < ∞, r u|Lr |u(x)|p |u(x)|r−p dx ≤ = u|C r−p u|Lp p Rn ≤ c u|Wpk r−p u|Wpk p = c u|Wpk r . In view of the above argument it is sufficient to deal with the case of u ∈ C0∞ ∩Wpk such that supp u ⊂ Ω with |Ω| < ∞, and to check the dependence of the constants upon Ω. For simplicity we may even assume from the beginning that Ω is the above cube with edges parallel to the axes of coordinates, and side-length b ≥ 1.

28 we restrict ourselves to an example merely, since the main idea of envelopes just relies on those cases where we do not have the corresponding embeddings. We explain this in detail in the next chapters. However, this immediately proves the sharpness of the above assertions, too (and is also known already). 38), we present a family of functions hν that belong to Wn1 , but not to L∞ , as long as 0 < ν < 1 − n1 . For x ∈ Rn , consider the radial functions hν (x) = logν 1 + 1 |x| − logν 2, 0 < |x| < 1 0 , otherwise hν (x) , x ∈ Rn .