Nonisotropic strongly singular integral operators

Shayya, Bassam

doi:10.1090/S0002-9947-02-03097-0

Nonisotropic strongly singular integral operators
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Trans. Amer. Math. Soc. 354 (2002), 4893-4907 Request permission

Abstract:

We consider a class of strongly singular integral operators which include those studied by Wainger, and Fefferman and Stein, and extend the results concerning the $L^p$ boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp.

References

Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Akihiko Miyachi, On some Fourier multipliers for $H^{p}(\textbf {R}^{n})$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 1, 157–179. MR 573335
Per Sjölin, $L^{p}$ estimates for strongly singular convolution operators in ${\rm {\bf }R}^{n}$, Ark. Mat. 14 (1976), no. 1, 59–64. MR 412722, DOI 10.1007/BF02385824
Per Sjölin, An $H^{p}$ inequality for strongly singular integrals, Math. Z. 165 (1979), no. 3, 231–238. MR 523124, DOI 10.1007/BF01437558
Per Sjölin, Convolution with oscillating kernels, Indiana Univ. Math. J. 30 (1981), no. 1, 47–55. MR 600031, DOI 10.1512/iumj.1981.30.30004
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Elias M. Stein and Stephen Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. MR 508453, DOI 10.1090/S0002-9904-1978-14554-6
Stephen Wainger, Special trigonometric series in $k$-dimensions, Mem. Amer. Math. Soc. 59 (1965), 102. MR 182838