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Traces of multipliers in the space of Bessel potentials. (English) Zbl 0694.46028

The space of the multipliers on a Banach function space S is denoted by MS. Let H p ( n ), 0, 1<p<, be the space of Bessel potentials on n , defined as the completion of C 0 ( n ) by the norm (1-Δ) /2 u; n L p . According to E. M. Stein, let us define the extension Tγ (ξ,y) on n+m of a function γ (x) on n by Tγ(x,y)= n ξ(t)γ(x+|y|t)dt, where the kernel ζC ( n )L( n ) satisfies

(*) n ζ(x)dx=1, n x α ζ(x)dx=0, 0<|α|[r] for some r>0 and

(**) K n (1+|x|) j=0 [r] sup B |x| | j,x ζ|(1+|x|) j dx<, B |x| ={t n : |t|<|x|}·

In this paper it is proved that

(i) if ΓMH p ( n+m ) and γ(x)=Γ(x,0), it follows that γMW p -m/p ( n ) with γ; n MW p -m/p cΓ; n+m MH p ,, where W p -m/p ( n ), p>m, is the trace of H p ( n+m ) on n ,

(ii) if ζ satisfies (*) and (**) with r=1 and if γMW p -m/p ( n ) (resp. γL ( n ) and p<m), then it follows that TγMH p ( n+m ) with Tγ; n+m MH p cKγ; n MW p -m/p (resp. cKγ; n L ) (Theorem 1, 2).

The proofs are considerably hard partly because some of the notations adopted there are not familiar to the present reviewer.

Reviewer: K.Yoshinaga
MSC:
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B15Multipliers, several variables
References:
[1]S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).
[2]V. G. Maz’ya and T. O. Shaposhnikova, Multipliers in Spaces of Differentiable Functions [in Russian], Leningrad State Univ. (1986).
[3]R. S. Strichartz, ?Multipliers on fractional Sobolev spaces,? J. Math. Mech.,16, No. 9, 1031-1060 (1967).
[4]E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970).