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A characterization of potential spaces. (English) Zbl 0577.46035
Functions in the Bessel potential spaces \(L^ p_ a\) are characterized in terms of their approximability by polynomials. More precisely, if f is locally in \(L^ 1\), define for \(t>0\), \(x\in R^ n\) \[ E_ kf(x,t)=Ef(x,t)=\sup_{x\in Q,| Q| =t^ n}(\inf_{P\in {\mathbb{P}}^ k}\int_{Q}\quad | f-P| dz/| Q|) \] with \({\mathbb{P}}^ k\) the polynomials of degree \(\leq k\). Then we prove (theorem 2) that for \(a>0\), \(k=[a]\) and \(1<p<\infty\), \(f\in L^ p_ a\) iff \(f\in L^ p\) and \(G_ af\in L^ p\), where \[ G_ af(x)=(\int^{\infty}_{0}Ef(x,t)^ 2dt/t^{2a+1})^{1/2}. \] Also, if a is not an integer, the following variant of Ef, involving the Taylor polynomial Pf(y,x) of f at x can also be used instead of Ef \[ \Omega f(x,t)=\int_{| y| \leq t}| f(x+y)-Pf(y,x)| dy/t^ n. \] These results extend characterizations due to R. Strichartz in terms of vector valued means of differences, and of E. Stein in terms of Marcinkiewicz integrals.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26A16 Lipschitz (Hölder) classes
41A10 Approximation by polynomials
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