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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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##### References:
  Richard J. Bagby, A characterization of Riesz potentials, and an inversion formula, Indiana Univ. Math. J. 29 (1980), no. 4, 581 – 595. · Zbl 0415.42011  Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071  S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137 – 160 (Italian). · Zbl 0133.06801  R. De Vore and R. Sharpley, Maximal operators and smoothness, Mem. Amer. Math. Soc. No. 293 (1984).  C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107 – 115. · Zbl 0222.26019  Svante Janson, Mitchell Taibleson, and Guido Weiss, Elementary characterizations of the Morrey-Campanato spaces, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 101 – 114. · Zbl 0521.46022  E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc. 67 (1961), 102 – 104. · Zbl 0127.32002  Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501  Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031 – 1060. · Zbl 0145.38301
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