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Sci. Math.</a> <a href="/?q=in%3A304441" title="Articles in this Issue">136, No. 5, 521-573 (2012)</a>. </div> <div class="abstract">Summary: This paper deals with the fractional Sobolev spaces \(W^{s,p}\). We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results.<br class="zbmathjax-paragraph"> Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6064171">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6064171">Cited in <strong>2769</strong> Documents</a></div></div> <div class="mathoverflow"> <h3>MathOverflow Questions:</h3> <a href="https://mathoverflow.net/a/457063">What is the relationship between Hölder spaces and differentiability?</a><br> </div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E35" title="MSC2020">46E35</a> </td> <td class="space"> Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35S30" title="MSC2020">35S30</a> </td> <td class="space"> Fourier integral operators applied to PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35S05" title="MSC2020">35S05</a> </td> <td class="space"> Pseudodifferential operators as generalizations of partial differential operators </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+Sobolev+spaces">fractional Sobolev spaces</a>; <a href="/?q=ut%3AGagliardo+norm">Gagliardo norm</a>; <a href="/?q=ut%3Afractional+Laplacian">fractional Laplacian</a>; <a href="/?q=ut%3Anonlocal+energy">nonlocal energy</a>; <a href="/?q=ut%3ASobolev+embeddings">Sobolev embeddings</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1252.46023" data-ciurl="/ci/06064171" data-biburl="/bibtex/06064171.bib" data-amsurl="/amsrefs/06064171.bib" data-xmlurl="/xml/06064171.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06064171.pdf" title="Zbl 1252.46023 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.bulsci.2011.12.004" aria-label="DOI for “Hitchhiker's guide to the fractional Sobolev spaces”" title="10.1016/j.bulsci.2011.12.004">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1104.4345"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Adams, R. 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