How one configure his web site to let users search only on his domain using omnibox?
Example:
- type youtube.com then press tab

 - or type yahoo.com then press

 - trying with vimeo wouldn't work
 
I didn't find anyhing in the source code.
How one configure his web site to let users search only on his domain using omnibox?
Example:


I didn't find anyhing in the source code.
http://dev.chromium.org/tab-to-search has the information on how to do it, potentially you could use a chrome addon to force the page to have the needed code for this to function, but I'm not sure how well such would work.
Yes it's possible. I did this for my blog.
 <link rel="search" type="application/opensearchdescription+xml" title="The Sheng Blog" href="/resources/opensearch.php"/>
opensearch.php looks likes this:
 <?xml version="1.0"?>
 <OpenSearchDescription xmlns="http://a9.com/-/spec/opensearch/1.1/">
     <ShortName>The Sheng Blog (Beta)</ShortName>
     <Description>The Sheng Blog Search</Description>
     <Developer>Sheng Slogar</Developer>
     <LongName>Search the entire Sheng Blog</LongName>
     <InputEncoding>UTF-8</InputEncoding>
     <OutputEncoding>UTF-8</OutputEncoding>
     <Query role="example" searchTerms="code"/>
     <SyndicationRight>open</SyndicationRight>
     <AdultContent>false</AdultContent>
     <Language>en-us</Language>
     <Contact>contact@theshengblogg.comule.com</Contact>
     <Tags>code posts tutorials ideas playground</Tags>
     <Image width="16" height="16" type="image/x-icon">data:/ico;base64,AAABA...(Icon in base64)</Image>
     <Url type="text/html" template="http://theshengblogg.comule.com/search.php?s={searchTerms}"></Url>
     <Url type="application/x-suggestions+json" method="GET" template="http://theshengblogg.comule.com/autocomplete.php?search={searchTerms}&json=true"/>
 </OpenSearchDescription>
I learned this from http://www.opensearch.org/Specifications/OpenSearch/1.1. The autocomplete part is optional. Return it in JSON format.
For example, if you search "a", return ["a","about","across","all","and"]. (Notice I put your query in the array as item 0.)
My autocomplete only seems to work in Firefox. This might be something with my sub domain. I also couldn't get it to work in IE or Chrome.