The phrase itself originates in the Zohar, where Zohar III:92b teaches that an act performed below (עובדא דלתתא) arouses a corresponding act above, so that a person's deed of chesed or rachamim awakens that same quality in the heavenly realm, and Zohar I:164a similarly states that the daily offering exists because 'באתערותא דלתתא אתער לעילא' — through arousal from below, arousal is drawn from above, sustaining both worlds.
Chassidic sources build directly on this Zoharic formula: Likkutei Torah (Vayikra, Acharei Mot 2) speaks of the elevation of מ״ן through אתעדל״ת bringing down אתעדל״ע, while also noting that an awakening from above can itself stir the arousal below, and Shem MiShmuel (Mishpatim 14:5) applies this pattern to the Song at the Sea and the mitzvah of shekalim as instances of אתערותא דלתתא following אתערותא דלעילא.
Reb Nachman's Likutei Halakhot repeatedly grounds the created order itself in this principle: Likutei Halakhot, Choshen Mishpat, Laws of Artisans 3:2:1 holds that God deliberately left the world incomplete so that it would be sustained and perfected only through human אתערותא דלתתא, Likutei Halakhot, Yoreh Deah, Laws of Honouring One's Father and Mother 2:2:1 identifies honoring God through mitzvah-observance as the essential אתערותא דלתתא that gives God's honor its completeness, and Likutei Halakhot, Yoreh Deah, Laws of Charity 2:3:2 ties the filling-in of the moon's deficiency, accomplished specifically through tzedakah, to this same arousal from below.
Tanya frames the ultimate redemption in parallel terms, teaching in Tanya, Part I; Likkutei Amarim 37:1 that the final revelation of divine light in this world depends on human deeds throughout galut, since performing a mitzvah draws the light of the Ein Sof downward into the physical world.
Modern mussar and halachic writers apply the same language to practical religious life: Michtav MeEliyahu (Rav Dessler) teaches that an אתערותא דלתתא has real, lasting substance for a person, unlike an influx from above without a preceding arousal below, which remains fleeting; Shiurei Daas 2 p. 10 explains the verse 'תנו עוז לאלקים' as meaning that human action literally grants strength to the divine attribute, so that giving עוז לאלקים through אתערותא דלתתא causes the subsequent flow from above to below; and Peninei Halakhah, Women's Prayer 20:11 applies the concept to Birkat Kohanim, explaining that the blessing bestowed by God is linked to our own effort, in kabbalistic language 'באתערותא דלתתא תליא אתערותא דלעילא.'
Two further sources use the phrase in related contexts: Chochma uMusar 1 p. 193 invokes אתערותא דלתתא as bringing vitality to the world, and Valle on Bereishis 49 explains that even the upper worlds require assistance from below, since an אתערותא דלתתא arouses an awakening above, applying this to the renewal of the moon's diminished power in Birkat HaLevanah.
Related material that touches the same theme without using the precise phrase includes Etz Chaim, Shaar 39:1 on the need for human deeds to complete the refinement that the upper worlds alone could not achieve, Sha'arei Orah, Shaar 10 on the illumination effected through the kindling of the Menorah's lights, Pesikta DeRav Kahana 5:6 on the idea that a small human opening prompts a much greater divine opening, and Maamarim Melukatim ש"פ תולדות, מבה"ח וער"ח כסלו ה'תשמ"ה on how human self-nullification draws down new divine light, paralleling the moon's disappearance before its renewal.