By J.F. Ritt
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Clearly, |∇l γ(x)| ∼ c |x|−l (log |x|−1 )l(µ−1)−ν . 2 we obtain λ ∈ Wpl (Rn ) ⇐⇒ l(µ − 1) < ν − 1/p, λ ∈ M Wpl (Rn ) ⇐⇒ l(µ − 1) ≤ ν − 1 for lp = n. 1 Extension from a Half-Space Let Rn+ = {z = (x, xn ) : x ∈ Rn−1 , xn > 0}. The classical extension operator π is defined for functions given on Rn+ by ⎧ ⎪ for xn > 0, ⎪ ⎨v(z) l π(v)(z) = ⎪ αj v(x, −jxn ) for xn < 0, ⎪ ⎩ j=1 where αj satisfy the conditions l (−1)k j k αj = 1, 0 ≤ k ≤ l − 1. 1. Suppose that γ ∈ M (Wpm (Rn+ ) → Wpl (Rn+ )), where 0 ≤ l ≤ m and p ∈ [1, ∞).
And thus γρ ∈ M (Wpm−l → Lp ). 11) γρ M (Wpm−l →Lp ) ≤ c γρ M (Wpm →Wpl ) . 11) for all γ ∈ M (Wpm → Wpl ). 7) for j = l. 5) is contained in the following lemma. 5. Let γ ∈ M (Wpm−l → Lp ) and let ∇l γ ∈ M (Wpm → Lp ). Then γ ∈ M (Wpm → Wpl ) and the estimate γ holds. 12) 42 2 Multipliers in Pairs of Sobolev Spaces Proof. 13) where j = 1, . . , l − 1. For any u ∈ C0∞ , l ∇l (γu) Lp ≤c |∇j γ| |∇l−j u| ≤c Lp ∇l γ M (Wpm →Lp ) j=0 l−1 + γ M (Wpm−l →Lp ) ∇j γ + M (Wpm−l+j →Lp ) u Wpm . 13) that ∇l (γu) Lp ≤c ∇l γ M (Wpm →Lp ) + γ M (Wpm−l →Lp ) u Wpm .
1 Introduction In the present chapter we study multipliers acting in pairs of spaces Wpk and wpk , where k is a nonnegative integer. The concepts of this chapter prove to be prototypes for the subsequent study of multipliers in other pairs of spaces. Using the result of Sects. 1, we derive necessary and sufficient conditions for a function to belong to the space of multipliers M (Wpm → Wpl ) and M (wpm → wpl ), where m ≥ l ≥ 0 and p ∈ [1, ∞) (Sects. 8). The case of the half-space Rn+ is treated in Sect.