Lecture Notes 7 for 247b

Once one has these two properties, one obtains several additional properties for free, such as boundedness on L(R) for all 1 < p < ∞; also T and T ∗ both map L to BMO. However, it is not always easy to verify these properties in practice. The singular kernel properties are usually not too difficult, as the kernel can often be expressed directly, and then estimated pointwise in a fairly straightforward fashion. The L boundedness however can often be tricky we are asking to bound Tf for all functions f ∈ L(R), which is a large class to exhaust over. With assumptions such as translation invariance one can diagonalise T , which makes verification of L boundedness much easier (it is equivalent to the Fourier multiplier symbol being bounded), but in general such invariance is not available. (In the case of pseudodifferential operators there is an almost orthogonality property which is sort of a weak version of translation invariance and serves to almost block-diagonalise T ; again, this is not always available in general.)