Mail form (PHP, AJAX)

Question

I am trying to make a simple website for a hotel, but I cannot recieve any e-mail from the website to my inbox. I bought a bootstrap theme, i inserted the e-mail to which i want to recieve the e-mail and i can send it as well but when i look in the inbox, i do not see any e-mail. The file "mail.php" has the following code:

<?php
if($_POST)
{    
    ##################################################################################################################
    
    // Email Settings
    $site_title     = "Hotel Centro Cavour";
    $to_email       = "info@hotelcentrocavour.it"; 
    $subject        = "Hotel Centro Cavour - New Message"; 
    
    // Output Messages
    $success_mssg   = "Your message has been sent successfully. Thank you.";               
    $error_mssg     = "An error has occurred. Please check your PHP email configuration."; 
    $short_mssg     = "Too short message! Please enter something.";                      
    $empty_fields   = "Input fields are empty! Please enter something.";              
    $name_mssg      = "Name is too short or empty! Please enter something.";             
    $email_mssg     = "Please enter a valid email!";                                  
    
    //Email Text
    $tr_name    = "Name";
    $tr_email   = "Email";
    $tr_message = "Message";
    $tr_phone   = "Phone Number";
    
    ##################################################################################################################

    //Check if its an ajax request, exit if not
    if(!isset($_SERVER['HTTP_X_REQUESTED_WITH']) AND strtolower($_SERVER['HTTP_X_REQUESTED_WITH']) != 'xmlhttprequest') {
    
        //Exit script outputting json data 
        $output = json_encode(
        array(
            'type'=>'error', 
            'text' => 'Request must come from Ajax'
        ));
        
        die($output);
    } 
    
    //Check $_POST vars are set, exit if any missing
    if( !isset($_POST["user_name"]) || !isset($_POST["user_email"]) || !isset($_POST["user_message"]) )
    {
        $output = json_encode(array('type'=>'error', 'text' => $empty_fields));
        die($output);
    }

    //Sanitize input data using PHP filter_var(). *PHP 5.2.0+
    $user_name        = filter_var($_POST["user_name"], FILTER_SANITIZE_STRING);
    $user_email       = filter_var($_POST["user_email"], FILTER_SANITIZE_EMAIL);
    $user_message     = filter_var($_POST["user_message"], FILTER_SANITIZE_STRING);
    $user_phone       = filter_var($_POST["user_phone"], FILTER_SANITIZE_STRING);
    $user_subject     = filter_var($_POST["user_subject"], FILTER_SANITIZE_STRING);

    if(!empty($user_subject)){
        $subject = $user_subject;
    }
    
    //Additional php validation
    if(strlen($user_name)<4) // If length is less than  it will throw an HTTP error.
    {
        $output = json_encode(array('type'=>'error', 'text' => $name_mssg));
        die($output);
    }
    //Check Email
    if(!filter_var($user_email, FILTER_VALIDATE_EMAIL)) //email validation
    {
        $output = json_encode(array('type'=>'error', 'text' => $email_mssg));
        die($output);
    }

    //Check Message
    if(strlen($user_message)<10) 
    {
        $output = json_encode(array('type'=>'error', 'text' => $short_mssg));
        die($output);
    }

    //Headers
    $headers = 'From: '.$site_title.' <'.$user_email.'>' . PHP_EOL .
    'Reply-To: '.$user_name.' <'.$user_email.'>' . PHP_EOL .
    'MIME-Version: 1.0' . PHP_EOL .
    'Content-type:text/html;charset=utf-8' . PHP_EOL .
    'X-Mailer: PHP/' . phpversion();
    
    $message = "Name: $user_name <br> -- Email: $user_email <br> -- Phone: $user_phone <br> -- Message: $user_message";
    
    $sendemail = @mail($to_email, $subject, $message, $headers);
    
    if(!$sendemail)
    {
        $output = json_encode(array('type'=>'error', 'text' => $error_mssg));
        die($output);
    }else{
        $output = json_encode(array('type'=>'message', 'text' => $success_mssg));
        die($output);
    }
}else{

    header('Location: ../404.html');

}
?>