According to the Catholic Church (and other Nicene Churches), does the divine name YHWH (יַהְוֶה) apply equally to all Persons of the Sacred Trinity?
For the Catholic Church and other Nicene churches (the Eastern Orthodox, the Oriental Orthodox, the Armenian Orthodox, the Assyrians; as well as the majority of Protestants), the divine name YHWH (which is closely linked to the expression “I am” or “I Am Who I Am” (see Ex. 3:14), applies to God in His divine nature—hence to all three Persons of the Trinity.
The attribution of YHWH begins with Jesus himself, who multiple times attributes to himself the divine name “I am.” The most straightforward example is John 8:58:
Jesus said to them, “Truly, truly, I say to you, before Abraham was, I am.”
Hence, Jesus, the Son, attributes the divine name YHWH to himself. It is less immediately clear from the Scriptures that the Holy Spirit can also be called YHWH, but it follows from our knowledge of Trinitiarian theology.
Each of the Persons of the Trinity is perfectly identical with the Divine Nature or Essence—in orthodox Trinitarian doctrine, the Persons are not “parts” or “manifestations” of the Trinity, but strictly relations of opposition. (See for instance, St. Augustine, De trinitate I, 6, and V, 5.) In the words of the Nicene-Constantinopolitan Creed, the Son and Spirit are both consubstantial (homoousios) with the Father. (The Creed does not explicitly say so with respect to the Holy Spirit, but that is the fully intentional implication of saying that the Holy Spirit is “worshiped and glorified” together with the Father and Son.)
The name YHWH, however, refers to the very essence of God: clearly, it pertains to all three Persons to “be” in the absolute sense. Hence, YHWH can be applied equally well to all three Persons, although it is frequently attributed to the Father, because He is the ultimate origin or monarch of the whole Trinity.
(The Creed actually implies this: in liturgical celebrations the divine name YHWH is traditionally not used—a holdover from the ancient Jewish reluctance to utter that name. That is why in most translations of the Old Testament, the name YHWH is generally translated as “the LORD”—translating the Hebrew “Adonai.” The Creed, however, calls the Holy Spirit the “Lord and giver of life,” hence the liturgical substitute for the divine name YHWH is used for the Holy Spirit, just as it is for the Son and the Father.)
The divine name (tetragrammaton: JHVH) applies also to the incarnate God, Jesus Christ.
The prophets clearly state that the Messias is God. Isaias says: "God Himself will come and will save you" (35:4); "Make ready the way of Jahweh" (40:3); "Lo Adonai Jahweh will come with strength" (40:10). That Jahweh here is Jesus Christ is clear from the use of the passage by St. Mark (1:3). (Catholic source: https://www.newadvent.org/cathen/07706b.htm)
The divine name also applies to the Holy Spirit:
At the beginning of time, we were slowly made aware of the reality of God the Father. The Great I AM. It was revealed to us through the prophets and patriarchs that there was one God and only one. But as time passed, we were also made aware of the Messiah who was the Son of God. As we came to know this Son, in the person of Jesus, we came to realize that He also was I AM. He also was God. Then the Son began to reveal to us that He would send His Advocate, the Holy Spirit. And we came to realize that this Holy Spirit is also God, also I AM. (Catholic source: https://mycatholic.life/the-my-catholic-life-series/my-catholic-faith/the-holy-spirit/)