Many apparently have believed even from Caesar’s household. These are people who would otherwise have walked proudly and thought of nothing but Caesar. The power of the gospel has been revealed to these people. Many others who have believed are humble people. He equally greets them all, humbly and affably, wherever they are. The word “especially” in relation to “those of Caesar’s household” makes it apparent that they are taking pains to be pleasing in service.
"All the saints salute you, especially they that are of Caesar's household. The grace of the Lord Jesus Christ be with your spirit."
He elevated them and strengthened them, by showing that his preaching had reached even to the king's household. For if those who were in the king's palace despised all things for the sake of the King of Heaven, far more ought they to do this. And this too was a proof of the love of Paul, and that he had told many things of them, and said great things of them, whence he had even led those who were in the palace to a longing for them, so that those who had never seen them saluted them. Especially because the faithful were then in affliction, his love was great. And those who were absent from each other were closely conjoined together as if real limbs. And the poor man was similarly disposed toward the rich, and the rich toward the poor, and there was no preeminence, in that they were all equally hated and cast out, and that for the same cause. For as, if captives taken from divers cities should arise and come to the same towns, they eagerly embrace each other, their common calamity binding them together; thus too at that time they had great love one toward another, the communion of their afflictions and persecutions uniting them.
[AD 202] Irenaeus on Philippians 4:22