Geodesic
Estimates for some convolution operators with singularities of their kernels on spheres
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2011), pp. 17-23.

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In the framework of Hardy spaces $H^p$, we study multidimensional convolution operators whose kernels have power-type singularities on a finite union of spheres in $\mathbb R^n$. Necessary and sufficient conditions are obtained for such operators to be bounded from $H^p$ to $H^q$, $0$, from $H^p$ to BMO, and from BMO to BMO.
Mots-clés : convolution, multiplier, distribution.
Keywords: sphere, oscillating symbol, BMO, $(H^p{-}H^{q})$-estimates 
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A. V. Gil; A. I. Zadorozhnyi; V. A. Nogin. Estimates for some convolution operators with singularities of their kernels on spheres. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2011), pp. 17-23. http://geodesic.mathdoc.fr/item/VSGTU_2011_2_a1/

[1] Strichartz R. S, “Convolutions with kernels having singularities on a sphere”, Trans. Amer. Math. Soc., 146:2 (1970), 461–471 | DOI | MR

[2] Sjöstrand S., “On the Riesz means of the solutions of the Schrödinger equation”, Ann. Scuola Norm. Sup. Pisa., 24:3 (1970), 331–348 | MR | Zbl

[3] Peral J. C., “$L^p$ estimates for the wave equation”, J. Funct. Anal., 36:1 (1980), 114–115 | DOI | MR

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

[5] Karasev D. N., Nogin V. A., “On the $\mathcal L$-characteristic of some potential-type operators with radial kernels, having singularities on a sphere”, Fract. Calc. Appl. Anal., 4:3 (2001), 343–366 | MR | Zbl

[6] Karapetyants A. N., Karasev D. N., Nogin V. A., “Estimates for some potential-type operators with oscillating kernels”, J. Contemp. Math. Anal., 38:2 (2003) | MR | Zbl

[7] Karasev D. N., Nogin V. A., “On the boundedness of some potential-type operators with oscillating kernels”, Math. Nachr., 278:5 (2005), 554–574 | DOI | MR | Zbl

[8] Gil A. V., Nogin V. A., “Estimates for some potential-type operators with oscillating symbols”, Vladikavkaz. Matem. Zhurn., 12:3 (2010), 21–29 | MR

[9] Gil A. V., Nogin V. A., “$H^p{-}H^q$ estimates for some potential-type operators with oscillating symbols”, Izv. vuzov. Sev.-Kav. Region, 2010, no. 5, 8–13

[10] Samko S. G., Kilbas A. A., Marichev O. I., Integrals and derivatives of fractional order and some of their applications, Nauka i Tekhnika, Minsk, 1987, 688 pp. | MR | Zbl

[11] Watson G., A treatise on the theory of Bessel functions, Cambridge University Press, London, 1922, 804 pp. ; Vatson G. N., Teoriya besselevykh funktsii, Inostr. lit., M., 1949, 798 pp. | Zbl | Zbl

[12] Fedoryuk M. V., The saddle-point method, Nauka, Moscow, 1977, 368 pp. | MR