By Vladimir Maz'ya, Tatyana O. Shaposhnikova
The objective of this publication is to provide a complete exposition of the idea of pointwise multipliers performing in pairs of areas of differentiable features. the speculation used to be primarily built by way of the authors over the last thirty years and the current quantity is principally in accordance with their effects.
Part I is dedicated to the speculation of multipliers and encloses the subsequent issues: hint inequalities, analytic characterization of multipliers, relatives among areas of Sobolev multipliers and different functionality areas, maximal subalgebras of multiplier areas, strains and extensions of multipliers, crucial norm and compactness of multipliers, and miscellaneous houses of multipliers.
Part II issues a number of functions of this concept: continuity and compactness of differential operators in pairs of Sobolev areas, multipliers as recommendations to linear and quasilinear elliptic equations, better regularity within the unmarried and double layer capability conception for Lipschitz domain names, regularity of the boundary in $L_p$-theory of elliptic boundary price difficulties, and singular critical operators in Sobolev spaces.
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Extra resources for Theory of Sobolev Multipliers: With Applications to Differential and Integral Operators
Example text
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.