The central image is that on Rosh Hashanah Zohar, Emor 38-40 describes Yitzchak going out alone, embodying the attribute of strict judgment (din) without the tempering of mercy, and calling to Esav — the accuser — to bring him the 'foods of the world,' meaning to press charges over the deeds of mankind, a reading spelled out by Sulam on Zohar, Emor 217.
That same text teaches that it is specifically the voice of Yaakov rising through the shofar that approaches Yitzchak and transforms the judgment, so that the sounding of the shofar is what draws judgment back toward its proper, merciful configuration, as Zohar, Emor 38-40 and Sulam on Zohar, Emor 217 both indicate.
This fits the broader Zoharic teaching that on Rosh Hashanah and Yom Kippur the accuser stands ready to prosecute, and Israel is required to arouse themselves through the shofar, sounding a voice that unites the attributes and silences the prosecution, as explained in Sulam on Zohar, Vayera 381:1, Sulam on Zohar, Vayera 384, and Sulam on Zohar, Pekudei 722.
Zohar, Emor 38:191 further teaches that Israel specifically needs a shofar and not a mere horn-instrument on the day of judgment, because the shofar carries hidden words and deeds meant to stir concealed matters, rather than drawing attention to a fixed, identifiable place that judgment could attach to.
Parallel Zoharic passages reinforce that the day of judgment requires an offering or mechanism to deflect the accuser, whether through the goat given over to Esav's side on Yom Kippur so that it becomes a defender rather than a prosecutor in Zohar, Vayishlach 18:188, or through the bull brought on Rosh Chodesh so the world can bear the judgment in Zohar, Addenda, Volume I 25:242.
The Rambam frames the shofar's function in experiential terms, teaching that although the mitzvah is a scriptural decree, it hints at a call to awaken the sleeping from their slumber and rouse them to examine their deeds and return in repentance, as stated in Mishneh Torah, Repentance 3:4, while he also establishes in Mishneh Torah, Shofar, Sukkah and Lulav 1:1 that the obligation is to hear the terua specifically from a bent ram's horn, learned by tradition from the yovel's shofar.
Tzitz Eliezer directly addresses the connection between Yitzchak and the day of judgment, holding that God judges Yitzchak's descendants on Rosh Hashanah, and that if they seek merit through remembrance of the Binding of Yitzchak they should sound a shofar made from the horn of that very ram, as Tzitz Eliezer 17:27 rules.
Later sources carry this theme into practical terms, teaching that the Day of Judgment is itself a festival for Israel, on which the shofar is blown to arouse heavenly mercy, remove accusations, and bring judgment to a favorable light, as found in Valle, Inyanei Elul veRosh Hashana p. 54.
A related teaching ties the loud voice of the shofar's awakening to the custom of raising one's voice in prayer specifically on Rosh Hashanah and Yom Kippur, since a holy and upright cry pierces the firmaments on its way before God, as discussed in Kuntres Umaayan, Maamar 11.