In the power of the Spirit - By the "influence" or direction of the Spirit.
A fame - A report. See Matthew 4:24.

Returned in the power of the Spirit - Εν τῃ δυναμει του πνευματος, In the mighty power of the Spirit. Having now conquered the grand adversary, he comes in the miracle-working energy of the Spirit to show forth his power, godhead, and love to the people, that they might believe and be saved. He who, through the grace of God, resists and overcomes temptation, is always bettered by it. This is one of the wonders of God's grace, that those very things which are designed for our utter ruin he makes the instruments of our greatest good. Thus Satan is ever duped by his own proceedings, and caught in his own craft.

And Jesus returned in the power of the Spirit,.... Of which he was full, and by which he was led into the wilderness, and had combated with Satan, and had got the victory over him; and by virtue of which, he entered on his public ministry, wrought miracles, and taught with authority. A like way of speaking is used by the Targumist, on Micah 3:8. I am filled, , "with the power of the spirit of prophecy", from before the Lord. Moreover, this phrase is used, to show that his return
into Galilee, where he had been brought up, and from whence he came to John at Jordan, did not arise from a natural love to his country, and a fond desire of being there again; but was owing to the powerful impulse of the Holy Spirit, which was in him, and moved him to return thither; where he was to begin his ministry, and work his miracles, and so fulfil a prophecy of him, in Isaiah 9:1 see Matthew 4:12.
And there went out a fame of him through all the region round about: throughout all Galilee and Syria, Decapolis and Judea; see Matthew 4:23, the report of his doctrines and miracles, was spread far and near; and on account of them, he became the subject of the common talk of people every where, who highly applauded and commended him for them.

Christ taught in their synagogues, their places of public worship, where they met to read, expound, and apply the word, to pray and praise. All the gifts and graces of the Spirit were upon him and on him, without measure. By Christ, sinners may be loosed from the bonds of guilt, and by his Spirit and grace from the bondage of corruption. He came by the word of his gospel, to bring light to those that sat in the dark, and by the power of his grace, to give sight to those that were blind. And he preached the acceptable year of the Lord. Let sinners attend to the Saviour's invitation when liberty is thus proclaimed. Christ's name was Wonderful; in nothing was he more so than in the word of his grace, and the power that went along with it. We may well wonder that he should speak such words of grace to such graceless wretches as mankind. Some prejudice often furnishes an objection against the humbling doctrine of the cross; and while it is the word of God that stirs up men's enmity, they will blame the conduct or manner of the speaker. The doctrine of God's sovereignty, his right to do his will, provokes proud men. They will not seek his favour in his own way; and are angry when others have the favours they neglect. Still is Jesus rejected by multitudes who hear the same message from his words. While they crucify him afresh by their sins, may we honour him as the Son of God, the Saviour of men, and seek to show we do so by our obedience.

Returned. After the Temptation. For the events before the return, see Matthew 4:12.
In the power of the Spirit. Full of Divine strength since his victory over Satan. To conquer evil always lifts to a higher power.

Jesus returned in the power of the Spirit - Being more abundantly strengthened after his conflict.