## Averages of functions over hypersurfaces in $$\mathbb R^n$$.(English)Zbl 0626.42009

Given a $$C^\infty$$ hypersurface $$S$$ in $$\mathbb R^n$$, $$n\geq 3$$, the ‘mean-value’ operators $$M_t$$ are defined by $M_t(f)(x)=\int_{S}f(x-ty)\psi (y)\,d\sigma (y),\quad t>0,$ where $$\psi$$ is a fixed function in $$C_0^\infty(S)$$ and $$d\sigma$$ is the induced Lebesgue measure on $$S$$. The authors study $$L^p$$ inequalities for the maximal operator $$\mathcal M$$ defined by $$(\mathcal Mf)(x)=\sup_{t>0}| M_t(f)(x)|.$$ It is shown that if the Gaussian curvature of $$S$$ does not vanish of infinite order at each point of $$S$$, then $$\mathcal M$$ is bounded in $$L^p$$ in a certain range $$p>p_0(S)$$.
Reviewer: B.P.Duggal

### MSC:

 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B25 Maximal functions, Littlewood-Paley theory

### Keywords:

smooth functions; Lp inequalities; maximal operator
Full Text:

### References:

 [1] Greenleaf, A.: Principal curvature and harmonic analysis. Indiana Math. J.30, 519-537 (1982) · Zbl 0517.42029 [2] Hörmander, L.: The Analysis of Linear Partial Differential Operators, I. New York-Berlin: Springer 1983 · Zbl 0521.35001 [3] Milnor, J.: Morse Theory. Ann. Math. Stud. Princeton, New Jersey: Princeton Univ. Press 1973 [4] Randol, B.: On the asymptotic behavior of the Fourier transform of the indicator function of a convex set. Trans. Am. Math. Soc.139, 279-285 (1969) · Zbl 0183.26905 [5] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton, New Jersey: Princeton Univ. Press 1970 · Zbl 0207.13501 [6] Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Ann. Math. Stud. Princeton, New Jersey: Princeton Univ. Press 1970 · Zbl 0193.10502 [7] Stein, E.M.: Maximal functions: spherical means. Proc. Natl. Acad. Sci. USA73, 2174-2175 (1976) · Zbl 0332.42018 [8] Stein, E.M., Wainger, S.: Problems in harmonic analysis related to curvatures. Bull. Am. Math. Soc.84, 1239-1295 (1978) · Zbl 0393.42010 [9] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton, New Jersey: Princeton Univ. Press 1971 · Zbl 0232.42007 [10] Svenson, I.: Estimates for the Fourier transform of the characteristic function of a convex set. Arkiv För Mathematik9, 11-22 (1971) · Zbl 0221.52001 [11] Zygmund, A.: Trigonometric Series. Cambridge-New York: Cambridge Univ. Press 1979 · JFM 58.0296.09
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.