We describe convex sets on the (1 p , 1 q)-plane for which the well-known Bochner-Riesz operator with the symbol (1 − |ξ| 2) −α + (0 < Re α < n+1 2) is bounded from L p into L q .
We consider a class of multidimensional potential-type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)-plane for which these operators are bounded from Lp into Lq and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration.
Изучается один Фурье-мультипликатор, вырождающийся или имеющий особенности на единичной сфере в Rn. Получены необходимые и достаточные условия принадлежности этого мультипликатора классу Хермандера Mqp.
Within the framework of the method of approximative inverse operators we construct the inversion of potential-type operators in R n with kernels aðjtjÞe ijtj =jtj nÀa , 0 < Re a < n both in the elliptic and non-elliptic cases. The characteristic aðrÞ, being sufficiently smooth in some neighbourhood of the point r ¼ 1, locally belongs to L 1 . The non-elliptic case, in which the symbol of the corresponding potential simultaneously has zeroes and singularities, ''spread'' over some spheres in R n , is the most difficult.
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