×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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 · doi:10.1512/iumj.1980.29.29044
[2] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[3] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137 – 160 (Italian). · Zbl 0133.06801
[4] R. De Vore and R. Sharpley, Maximal operators and smoothness, Mem. Amer. Math. Soc. No. 293 (1984).
[5] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107 – 115. · Zbl 0222.26019 · doi:10.2307/2373450
[6] 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 · doi:10.1007/BFb0069154
[7] E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc. 67 (1961), 102 – 104. · Zbl 0127.32002
[8] 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
[9] Robert S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031 – 1060. · Zbl 0145.38301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.