Derivative estimates of averaging operators and extension

Abstract

We study the derivative operator of the generalized spherical mean {{\displaystyle S}}_t^{\gamma }. By considering a more general multiplier {{\displaystyle m}}_{\gamma ,b}^{\mathrm{\Omega }}={{\displaystyle V}}_{\frac{n-\mathrm{2}}{\mathrm{2}}+\gamma }(\left|\xi \right|){\left|\xi \right|}^b\mathrm{\Omega }(\xi \text{'}) and finding the smallest \gamma such that {{\displaystyle m}}_{\gamma ,b}^{\Omega } is an H^p multiplier, we obtain the optimal range of exponents (\gamma ,\beta ,p) to ensure the H^p({ℝ}^n) boundedness of {\partial }^{\beta }{{\displaystyle S}}_{\mathrm{1}}^{\gamma }f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p({ℝ}^n) spaces.

Keywords

Generalized spherical mean / Bessel function / H^p multiplier / wave equation / oscillatory integrals

References

[1] Bennett C, Sharpley R. Interpolation of Operators. Pure Appl Math, Vol 129. Boston: Academic Press, 1988

[2] Bourgain J. Averages in the plane over convex curves and maximal operators. J Anal Math, 1986, 47: 69–85

[3] Carbery A, Rubio de Francia J L, Vega L. Almost everywhere summability of Fourier integrals. J Lond Math Soc (2), 1988, 38(3): 513–524

[4] Chen J C, Fan D S, Sun L J. Hardy space estimates for the wave equation on compact Lie groups. J Funct Anal, 2010, 259(12): 3230–3264

[5] Fan D S, Zhao F Y. Approximation properties of combination of multivariate averages on Hardy spaces. J Approx Theory, 2017, 223: 77–95

[6] Gelfand I M, Shilov G E. Generalized Functions. Vol I: Properties and Operations. New York-London: Academic Press, 1964

[7] Grafakos L. Classical Fourier Analysis. 3rd ed. Grad Texts in Math, Vol 249. New York: Springer, 2014

[8] Lu S Z. Four Lectures on Real H^p Spaces. River Edge: World Scientific Publishing Co, Inc, 1995

[9] Miyachi A. On some singular Fourier multipliers. J Fac Sci Univ Tokyo Sect IA Math, 1981, 28(2): 267–315

[10] Peral J C. L^pestimates for the wave equation. J Funct Anal, 1980, 36(1): 114–145

[11] Stein E M. Maximal functions I: Spherical means. Proc Natl Acad Sci USA, 1976, 73(7): 2174–2175

[12] Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math Ser, Vol 43. Princeton: Princeton Univ Press, 1993

[13] Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ Press, 1971

[14] Zhao J Y, Fan D S. Certain averaging operators on Lebesgue spaces. Banach J Math Anal (to appear)

Rights & permissions

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature