Schur Multipliers of Divided Differences and Multilinear Harmonic Analysis
J. Reimann (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Martijn Caspers (TU Delft - Analysis)
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Abstract
It was first shown by D. Potapov and F. Sukochev in 2009 that Lipschitz functions are also operator-Lipschitz on Schatten class operators Sp, 1<p<∞, which is related to a conjecture by M. Krein. Their proof combined Schur multiplication, a generalisation of component-wise matrix multiplication, with the so-called first order divided difference of a function, an approximation of its derivative. Showing that the Schur multipier associated with a divided difference function is bounded relies on a so-called transference technique, the boundedness of certain Schur multipliers can be inferred from the boundedness of associated Fourier multipliers. Soon after, this boundedness result was extended by D. Potapov, A. Skripka, and F. Sukochev to multilinear Schur multipliers of divided differences of arbitrary order, i.e. approximations of higher derivatives.
In this thesis, we offer an alternative boundedness proof for bilinear Schur multipliers of second order divided differences, in which we use recent results of multilinear harmonic analysis towards a multilinear transference proof, as well as recently found sufficient conditions for the boundedness of linear Schur multipliers which cannot be studied by transference. These methods were not known at the time Potapov, Skripka, and Sukochev proved their result.
Moreover, we show that this new proof improves the growth of the bound on the norm of the considered Schur multiplier for p→∞ significantly. Finally, we give an outlook on further steps towards an alternative boundedness proof of multilinear Schur multipliers of divided differences of arbitrary order.