domain name system - DNS for a subdomain

I have: - two DNS servers: 1.1.1.1 and 2.2.2.2 - domain name: domain.com pointed to 1.1.1.1

I want to point sub domain: sub.domain.com to 2.2.2.2 (obviously www.sub.domain.com too)

on 1.1.1.1 I've create a DNS A record pointing sub.domain.com to 2.2.2.2

accessing sub.domain.com now lead to 2.2.2.2 successfully

the problem is the DNS records on 2.2.2.2 such as CNAME www do not work means accessing www.sub.domain.com DNS lookup results to: domain does not exist.

what should I do in order to be sure that all sub.domain.com sub-domains and alias are managed by 2.2.2.2?

Your help is appreciated, thank you.

---

Update 1

@Tonin: yes, I'm looking to delegate the zone.

look to what I did:

on 1.1.1.1 I've set the zone:

sub.domain.com  A  2.2.2.2
                NS 2.2.2.2

on 2.2.2.2 I've set the zone:

www  CNAME  base (sub.domain.com)
mail A      2.2.2.2

DNS lookup of sub.domain.com is correct
DNS to www.sub.domain.com or other records do not lead to the intended results (Non-Existent Domain!)