Aspects of the Theory of Bounded Integral Operators in Lp-spaces |
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a₁ Banach space bounded linear operator bounded operator c₁ Clearly complex numbers considered convolution d/dx Deduce defined Definition denoted dm(x dmx(x Do(A estimate Exercise referring Exercise relating extends fe L'(R finite constant follows at once follows from Theorem Fourier operator Fourier transform Fourier transform multiplier fractional integral Fubini's theorem function f ƒ and g given Hilbert transform Hölder's inequality Hölder's inequality 3.2.2 identity integral operators isometry k₁ kernel L¹(R L²(R Lebesgue integrable Lebesgue measurable function Lemma let f let ƒ LP(R math measurable function non-increasing non-negative normed linear space o-finite measure space obtained Okikiolu one-dimensional P₁ P₂ Poisson operator product formula Prove real number representation results involving semi-group of operators sequence simple functions step function subspace Suppose vector vector-valued X-measurable Young's inequality