Gemara

(Rava): If one said 'this Esrog is yours on condition that you return it to me', the recipient fulfilled the Mitzvah only if he returned it.

Sukah 41b (Beraisa): "Lachem" teaches that on the first day one is not Yotzei with another's Lulav, unless he gave it for a gift.

Once, Chachamim were on a ship and only R. Gamliel had a Lulav. He was Yotzei with it, then gave it for a gift to R. Yehoshua, who was Yotzei... and gave it to R. Akiva. R. Akiva was Yotzei with it and returned it to R. Gamliel.

Nedarim 48a (Mishnah): A case occurred in Beis Choron in which Yakov was Mudar Hana'ah from his son Reuven. Reuven was marrying off his son. He told a friend David 'the courtyard and the banquet are yours, only in order that my father can eat with us.' David said 'if they are mine, I say that they are Hekdesh!'

Reuven: I did not give them to you to make them Hekdesh!

David: You gave them to me in order that you and your father will eat together, and I will bear the sin!

Chachamim: If the recipient of a gift cannot make it Hekdesh, it is not a gift.

Rishonim

The Rif and Rosh (Sukah 3:30) bring the Gemara.

Question: 'On condition that you return it to me' connotes 'in a way that I can use.' Returning a Hekdesh Esrog is no good. A gift in which the recipient cannot be Makdish is not a gift (Nedarim 48a)!

Answer #1 (Rashbam Bava Basra 137b): The text should say 'on condition that you return it', without the words 'to me'.

Answer #2 (Ra'avad, brought in Ran DH v'Yesh): The recipient can be Makdish his Tovas Hana'ah (the value of the right to use it).

Answer #3 (Ran, ibid. and Nedarim 48a DH Kol): 'Any gift that is not (a gift, rather, a mere ruse)*, in which the recipient cannot be Makdish it, e.g. the gift in Beis Choron, is not a gift. Here, the Lulav was a true gift while he had it.

Ran (ibid.): We learn that the recipient can give it to another, and he to another, as long as the giver gets it back in the end.

Rambam (Hilchos Lulav 8:10): If one received a Lulav for a gift on condition to return it he is Yotzei and returns it, for this is a valid gift. If he did not return it he was not Yotzei, for it is as if he stole it.

Rosh (3:30): If the giver said 'this is a gift until you are Yotzei, and then it reverts to me', the recipients would not be Yotzei, for this is like a loan. One should not be Makneh his Lulav to a child on the first day, because a child acquires, but he cannot be Makneh (46b). If a limited time gift was valid, one could give to a child and get it back, even though the child cannot give!

Ba'al ha'Itur (brought in Rosh): Since the recipient knows that the giver needs is Lulav back, there is no need to specify. Rather, he gives it Stam (without specifying) and the recipient returns it Stam, just like Chachamim on the boat.

Rosh (ibid.): Surely, a gift on condition to return is better (there is no doubt that he intends for a loan). The Ba'al ha'Itur teaches that even if it was given Stam, it was on condition to return it.

Ba'al ha'Itur (ibid.): Normally, returning the value of an item is as good as returning the item (Gitin 74a). However, regarding Lulav it is not; the giver wanted back his Lulav to be Yotzei with it. Returning money does not suffice.

Poskim

Shulchan Aruch (OC 658:4): If one received a Lulav for a gift on condition to return it, he is Yotzei and returns it, for this is a valid gift. If he did not return it he was not Yotzei, even if he paid its value or it was lost through Ones.

Question (Beis Yosef DH v'Ika and DH v'Im Tomar): It is not his until he returns it, and then he cannot be Yotzei! A Tanai pertaining to (contradicting) the Ma'aseh is an invalid Tanai, so he should be Yotzei in any case!

Answer (Beis Yosef, ibid.): Rava holds that a Tanai pertaining to the Ma'aseh is a valid Tanai. Alternatively, he holds that 'Al Manas' connotes 'from now', so the gift works retroactively, so there is no contradiction.

Beis Yosef (DH Kosav ha'Roke'ach): The Roke'ach requires saying only 'on condition to return it'. If he adds 'to me', the gift is invalid, for the recipient cannot make it Hekdesh. This is very difficult. Was Rava imprecise?!

Note: Presumably, the Roke'ach relies on the text of the Rashbam.

Mishnah Berurah (15): Most Poskim say that even if it was lost through Ones, paying its value does not fulfill the condition. The Me'iri says that it helps for the giver to say 'it is as if I received it back.' The Chayei Adam is unsure.

Darchei Moshe (4): One with Yir'as Shomayim who cherishes Mitzvos will get his own four Minim to shake them properly (during Hallel). Also, most people think that one may bless on a jointly owned Esrog. The Maharil says that even when the first day is Shabbos one should strive to have his own Minim.

Kaf ha'Chayim (21): If he said 'it is a gift and you will return it to me', the recipient is not Yotzei.

Question: One may not give gifts on Yom Tov!

Answer (Kaf ha'Chayim 22): The Magid Mishnah permits a Kinyan Arai needed just then. The Mordechai permits a Kinyan needed for a Mitzvah.

Shulchan Aruch (ibid.): If he returned it after the time of the Mitzvah, he was not Yotzei.

Mishnah Berurah (16) and Bi'ur Halachah (DH l'Acher): If the giver (and his children) already blessed on it, returning it before the giver needs it tomorrow fulfills 'on condition that you return it to me.' If he did not say 'to me', even if he returns it after Sukos, when it is worthless, he was Yotzei. Some disagree.

Rema: One may give a gift on condition not to make it Hekdesh, for this is no worse than a gift on condition to return it.

Shulchan Aruch (5): If Reuven gave it Stam, it is as if he said on condition that you return it to me. We assume so because Reuven has no other Lulav and needs it back.

Rema: Shimon must return it for a gift, so Reuven can be Yotzei with it.

Source (Gra DH v'Tzarich): This is why one should not give his Lulav to a child.

Mishnah Berurah (20): The Rema means that if it is not returned like a gift, the condition was not fulfilled and Shimon never owned it, and he was not Yotzei. In any case Reuven is Yotzei with it.

Shulchan Aruch (ibid.): Even if Shimon gave it to Levi, and Levi to Yehudah, as long as Reuven gets it back in the end Shimon was Yotzei.

Mishnah Berurah (21,22): This teaches that one who receives a gift on condition to return it may give it to others, as long as it will be returned to the giver, unless the giver specified not to do so. If the recipient disobeyed, it was not a gift. If it seems that the giver would not want it passed around lest it get ruined, it is forbidden. Some require permission to give it even to one other.