## Aspects of the theory of bounded integral operators in Lp-spaces |

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### Contents

Preface | 1 |

The Lebesgue Integral | 69 |

Inequalities for Integral Operators | 190 |

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### Common terms and phrases

a-finite measure space Banach space bounded linear operator bounded operator change of variables Chapter Clearly complex numbers complex-valued concludes the proof considered convolution defined Definition denoted disjoint Euclidean space Exercise referring Exercise relating extends finite constant follows at once follows from Theorem for/in Fourier operator Fourier transform Fourier transform multiplier fractional integral Fubini's theorem function f given Hilbert space Holder's inequality 3.2.2 identity If(R implies inner product space integer integral operators inversion isometry Jx Jx kernel Lebesgue convergence theorem Lebesgue integrable Lebesgue measurable function Lemma let f LP(R mapping measurable function non-increasing normed linear space obtained Okikiolu outer measure Poisson operator product formula Prove real interval real number real-valued rectangle representation required conclusion follows results involving semi-group of operators simple functions step function Suppose Ty(f vector vector-valued Young's inequality