Hi,

https://msdn.microsoft.com/en-us/library/office/dn567668(v=exchg.150).aspx

I need to search for a specific subject email for current date in currently logged in On premise SharePoint 2010 environment. My code is inside a SP Web Part. I also tried to access exchange web service in JQuery but same issue.

It is getting access denied due to authentication.

It does work in a console app on laptop. However it does not work in a WCF service on App server or in a SharePoint 2010 Web part. SharePoint is configrued as claims authentication. I do convert Claims to Windows authentication. 

How to resolve the authentication issue?

I'd start with http://blogs.msdn.com/b/emeamsgdev/archive/2012/07/26/exchange-web-services-and-sharepoint-without-applicationimpersonation.aspx which details how your configuration should be to do this.

Cheers
Glen