Seven times in Ezekiel 6-7 we read the phrase "then they will know that I am the LORD" (and this "recognition formula" is found more than 60 times across the whole book!). Clearly, this is a great burden of Ezekiel: that those who read and hear the message would know that God is the Sovereign Lord.

If you were to write a list of what you know of the Lord God, what would start your list? How many words do you think you would write down before you wrote "righteous judge" or something equivalent? Yet, here in the context of Ezekiel, God this is what is front-loaded.

When God acts decisively in judgement (as he does in these early chapters of Ezekiel), the result is that the people "will know that I am the LORD". God's Word spoken through the prophet are written to make the people aware of both their sin and God's righteousness. His words, whether those brought by Ezekiel or found further back in the covenant curses of Leviticus 26 are not empty threats (Ezekiel 6:10). 

But this raises the question: what kind of knowledge of God do they receive in Exile? Is it the imposed knowledge of God's power like that received by the Egyptians when after being battered by plagues they saw horse and rider thrown into the sea? The purpose of that power encounter was so that they would know that God was LORD (Exodus 14:4, 18), but this was a knowledge of despair, not hope. Or is it the knowledge of God that comes to the repentant, whose love of sin is broken through discipline? Both possibilities are present in these chapters.

 The following paragraph by Ian Duguid is helpful:

To "remember" in the Old Testament is never simply the recalling to mind of the past, but includes the idea of a present action that flows from that recollection. Remembering the Lord seems to be universally positive, and can clearly be used to describe repentance (see Jonah 2:7). The content of what the people will remember about the Lord is His grief at their adulterous actions, and the result flowing from that remembrance will be self-loathing at their evil ways (Ezekiel 6:9). This seems to imply the possibility on the part of at least some of the exiles of a repentant return to the Lord. Even now they can act the part of the righteous remnant by remembering the God who acts, and repenting of their sin and the sins of their fellow countrymen in going after dead idols who cannot save (Isaiah 44:6-20). As much as their sin delighted them in the past, now it will become an object of horror in their eyes.

But Ezekiel's focus on this possibility seems to be only fleeting as he continues with the threat of total devastation (see Ezekiel 6:11-14). In either event, the Lord's justice will be seen and known. The audience of Ezekiel can remember the Lord and find hope, or they can continue on their present godless trajectory and be utterly destroyed. By including both paths, Ezekiel encourages us to consider our own condition.