## Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |

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Page 17

0l%IWI2, where is any positive constant dependent on K. Hence L maps

a e (0,l) and each compact set K of the finite plane. * From the Ascoli theorem it

follows ...

0l%IWI2, where is any positive constant dependent on K. Hence L maps

**bounded sets**of L (D) into equi -bounded, equi-continuous sets of Ca(K) for eacha e (0,l) and each compact set K of the finite plane. * From the Ascoli theorem it

follows ...

Page 20

Hef)(z)| <^||f||2 / |h (2,C)|2dTc (1.28) < - J^||f ||2 for z e K , where is a constant for

each compact set K. Similarly for z,zq e ... x D. Thus H£ maps

D) into locally equibounded, locally equicontinuous sets of functions on D. Hence

, ...

Hef)(z)| <^||f||2 / |h (2,C)|2dTc (1.28) < - J^||f ||2 for z e K , where is a constant for

each compact set K. Similarly for z,zq e ... x D. Thus H£ maps

**bounded sets**of L (D) into locally equibounded, locally equicontinuous sets of functions on D. Hence

, ...

Page 21

compact in the sense of almost uniform convergence. 2 Moreover, since h £ L (

D X D), it follows that and we may apply the Lebesgue dominated convergence

theorem ...

**bounded sets**of L (D) into**bounded sets**of L (D) which are harmonic andcompact in the sense of almost uniform convergence. 2 Moreover, since h £ L (

D X D), it follows that and we may apply the Lebesgue dominated convergence

theorem ...

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