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A note on singular integral

  • Arup Maity EMAIL logo and Shyam Swarup Mondal
Published/Copyright: June 1, 2023

Abstract

In this paper, we prove that for n 2 - 1 2 < α < n 2 , α , the convolution operator

S α f ( x ) = | y | 1 f ( x - y ) ( | y | 2 - 1 ) - α 𝑑 y

is bounded from L p to L q 1 + L q 2 for certain values of p and q 1 , q 2 .

Acknowledgements

The first author gratefully acknowledges the support provided by Harish-Chandra Research Institute, Government of India. Second author thanks IIT Guwahati, India, for the support provided during the period of this work. The authors are deeply indebted to Professor P. K. Ratnakumar for several fruitful discussions and generous comments.

References

[1] B. Amri and M. Gaidi, L p - L q estimates for the solution of the Dunkl wave equation, Manuscripta Math. 159 (2019), no. 3–4, 379–396. 10.1007/s00229-019-01109-wSearch in Google Scholar

[2] I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. I: Properties and Operations, Academic Press, New York, 1964. Search in Google Scholar

[3] A. V. Gil and V. A. Nogin, L 1 - H 1 bounds for the generalized Strichartz potential, Izv. Vyssh. Uchebn. Zaved. Mat. (2011), no. 9, 10–18. 10.3103/S1066369X11090027Search in Google Scholar

[4] L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 104 (1960), 93–140. 10.1007/BF02547187Search in Google Scholar

[5] A. N. Karapetyants, On L p L q boundedness for convolutions with kernels having singularities on a sphere, Studia Math. 144 (2001), no. 2, 121–134. 10.4064/sm144-2-2Search in Google Scholar

[6] A. N. Karapetyants and V. A. Nogin, On characteristic of some potential type operators with oscillating symbols and singularities of the kernels on a sphere, Acta Math. Hungar. 92 (2001), no. 1–2, 1–9. Search in Google Scholar

[7] D. N. Karasev and V. A. Nogin, On the boundedness of some potential-type operators with oscillating kernels, Math. Nachr. 278 (2005), no. 5, 554–574. 10.1002/mana.200310258Search in Google Scholar

[8] A. Miyachi, On some estimates for the wave equation in L p and H p , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 331–354. Search in Google Scholar

[9] A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 2, 267–315. Search in Google Scholar

[10] A. Miyachi, Notes on Fourier multipliers for H p , BMO and the Lipschitz spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), no. 2, 221–242. Search in Google Scholar

[11] V. A. Nogin and D. N. Karasev, On the -characteristic of some potential-type operators with radial kernels, having singularities on a sphere, Fract. Calc. Appl. Anal. 4 (2001), no. 3, 343–366. Search in Google Scholar

[12] F. Ricci and G. Travaglini, Convex curves, Radon transforms and convolution operators defined by singular measures, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1739–1744. 10.1090/S0002-9939-00-05751-8Search in Google Scholar

[13] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492. 10.1090/S0002-9947-1956-0082586-0Search in Google Scholar

[14] E. M. Stein, Localization and summability of multiple Fourier series, Acta Math. 100 (1958), 93–147. 10.1007/BF02559603Search in Google Scholar

[15] R. S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471. 10.1090/S0002-9947-1970-0256219-1Search in Google Scholar

[16] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University, New York, 1944. Search in Google Scholar

[17] A. Zygmund, Trigonometric Series, Cambridge University, New York, 1959. Search in Google Scholar

Received: 2022-09-28
Accepted: 2023-05-18
Published Online: 2023-06-01
Published in Print: 2023-08-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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