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A rough hypersingular integral operator with an oscillating factor |
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| Authors: | Daning Chen Hung Viet Le |
| Affiliation: | a Department of Mathematics, Jackson State University, Jackson, MS 39217, USA b Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA c Department of Mathematics, Central China (Huazhong Normal University), Wuhan 430074, PR China d Department of Mathematics, Southwestern Oklahoma State University, Weatherford, OK 73096, USA |
| Abstract: | We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|−n−α(α>0) at the origin, an oscillating factor ei|y|−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space  to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0. |
| Keywords: | Singular integrals Hardy spaces on spheres Maximal operators Sobolev spaces |
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