If Psalm 110 was NOT written by David, then did Jesus make an error in stating that "David himself calls him [the messiah] 'lord'" in the Psalm?

If Psalm 110 was NOT written by David, then did Jesus make an error in stating that "David himself calls him [the messiah] 'lord'" in the Psalm (Mark 12:36-37; Mat. 22:43-45; Luke 20:42-44), or is there a better way of looking at this? I've read a few commentaries (expositional and scholarly) on the New Testament passages and Psalm in question (Expositor's Bible Commentary: Carson (Matthew), Wessel and Strauss (Mark), Liefeld and Pao (Luke), VanGemeren (Psalms); Word Biblical Commentary: Evans (Mark), Hagner (Matthew), Nolland (Luke), Allen (Psalms 101-150); and the NET Bible's notes to name a few). At least one of the Expositor's commentators recommended Allen's commentary. I agree that his appears to provide the most thorough analysis of the Psalm's original context of the commentaries I've read and also best addresses the question at hand. While he concludes that the Psalm was most likely written *about* David rather than *by* David (as also the NET concludes), he also writes, > "An understanding of the heading of the psalm in terms of Davidic authorship features twice in argumentation, at Mark 12:35–37 (and parallels) and Acts 2:33. This understanding, already as old in principle as the redactional characterization of the block of Davidic psalms in Pss 3–71 as “the prayers of David son of Jesse” in Ps 72:20, accords with what R. N. Longenecker has called the “circumstantial” or “descriptive” type of interpretation, based on ancient cultural norms, to be found in the NT, as distinct from the normative kind of exegesis practiced today (TynBul 21 36–38; Biblical Exegesis, 193–98)." I've since read some of the recommended book by Longenecker, *Biblical Exegesis in the Apostolic Period*, learning much about Jewish exegetical practices around the 1st century. I also read Dr. Michael Brown's take on Jesus' use of Psalm 110 in his book, *Answering Jewish Objections to Jesus, Vol.* 3. While he prefers defending the Psalm originally being written about the Messiah, he also concedes, > "Even if the psalm was originally written by a court poet for his lord, King David, it would still point to David’s priestly calling (as a prototype of the Messiah) as well as to his worldwide reign, fulfilled only through David’s greater descendant, King Messiah. This would mean, then, that Jesus was pointing to Jewish interpretation of the day, interpretation that attributed the authorship of this psalm to David, thereby proving that Messiah had to be greater than David, but without making a definitive statement about the authorship of the psalm." This is the best and most direct answer I've found so far. The difficulty now is reconciling the assumption that Jesus was not "making a definitive statement about the authorship of the psalm" with the language that he uses, which appears to be definitive. Matthew's version seems to be most easily reconciled with this approach, since Davidic authorship is merely an assumed part of Jesus' questions to his opponents. Mark and Luke are more difficult, Mark's version reading, "David himself said by the Holy Spirit,... David himself calls him 'Lord'." (I also explored the possibility that *David* could merely be a colloquial designation for the Davidic Psalms, attempting to replace *David* with *the Davidic author* in Jesus' quote. This, however, doesn't seem to work well, since the argument in the gospels revolves around the question of how the messiah could be *David's son*, whenever *David*, the assumed author of the Psalm, calls him 'Lord'.) I also have some deeper questions, which I think are pertinent to how we answer the main question: - Does Jesus himself believe that David wrote the psalm? (It's easier to account for other New Testament writers' use of Jewish tradition, since they aren't themselves *divine*.) If so, how should that inform our Christology? (Which part of His argument is divine and which part is human?) - Is the point Jesus tries to make undermined if his argument is based on a false premise? - If Jesus said that David wrote this Psalm, but it actually wasn't written *by* David, how do we reconcile that with the doctrine of inerrancy?