English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This master thesis is based on the paper "Sharp commutator estimates viaharmonic extensions" by Lenzmann and Schikorra, in which they proposed a methodto prove estimates for commutators involving Riesz transforms, fractionalLaplacians and Riesz potentials, see arXiv:1609.08547. These proofs onlyinvolve harmonic extensions to the upper half-space and integration by partsnext to some elementary transfromations, since the deeper theory isconcentrated in a variety of trace characterization results which can be usedas a blackbox.  In the first half of this thesis, after collecting some elementary resultsfor the s-harmonic extension by Caffarelli and Silvestre, we use this method toprove a variety of commutator estimates, closely following Lenzmann andSchikorra except for shortening some proofs.  In the second half, we prove generalized versions of the blackbox estimateslisted by Lenzmann and Schikorra and discuss the different building blockswhich make up these blackbox estimates, including Triebel-Lizorkin andBesov-Lipschitz space characterizations as well as square function estimates.

Estimates for commutators of fractional differential operators via harmonic extension