The Laws of the Lulav on the First Day of Yom Tov, 9 Seifim: 1. By Torah law, the mitzvah of lulav only applies on the first day outside of the Temple, but the Sages decreed that one should do it everywhere for all seven days.
2. It is not taken on Shabbat even if it is the first day. RAMA: It is forbidden to touch the lulav on Shabbat as it is likened to a stone (Ra"n the end of ch. "Lulav haGazul", Maggid ch. 4, Kol Bo). However, it is permitted to touch the etrog because it can be smelled. It is forbidden to put it on clothes even on Yom Tov because it imparts a scent (Mahari"l).
3. A person does not fulfill the obligation on the first day with a lulav loaned by somebody else because we require it to be "yours," of your possessions. Even if they said "This is yours until you fulfill your obligation, after which point it will be mine as it was initially," the obligation is not fulfilled because it is basically borrowed. However, if it is given as a gift, it is permitted.
4. If it was given on the condition that it be returned, the obligation is fulfilled and it should be returned, as a gift on the condition that it be returned is considered a gift. If it is not returned, the obligation is not fulfilled, even if the value is paid, and even if this is due to duress. Similarly, if it is returned after the time for the mitzvah is over, the obligation is not fulfilled. Rem"a: It is permitted to give an etrog on the condition that it will not be sanctified because this is no worse than a gift on the condition that it be returned (Ra"n ch. 5 of Nedarim).
5. If it was given with no specification, this is considered as if they said "on the condition that you return it to me," because we assume that with this intention it was given because it is needed for the obligation and they have no other. If it is not returned, the obligation is not fulfilled. Rem"a: One must return and give it back to its owner as a gift so that the owners will have it to use for the obligation (Ros"h and Rabbeinu Yerucham 8:4). That said, even if one did not give it back directly but rather to another and another to another and the last person gives it back to the owners, the obligation is fulfilled.
6. One should not give it to a child on the first day before fulfilling the obligation because a child can acquire but not bequeath by Torah law, and it will turn out that when it is returned it is not actually returned. There is one who says that if they have reached the age of understanding commerce, it is permitted. If one holds it together with the child, the obligation is fulfilled; because it has not left his hands, it is fine.
7. If two purchase a lulav or etrog together, neither can fulfill the obligation on the first day until one gives their share as a gift. Rem"a: This is specifically when they did not buy it for the mitzvah. However, if they did buy it for the mitzvah, they can fulfill their obligation with it because we assume that this was their intention when they bought it (Maggid).
8. If brothers buy etrogs from their joint inheritance and one takes the etrog, he fulfills the obligation if he could eat it and they wouldn't be angry. If they would be angry, he does not fulfill his obligation until they give their portions as a gift. If one bought an etrog and another bought a quince, or if they together bought an etrog, a pomegranate, and a quince from their inheritance, he cannot fulfill his obligation with the etrog until they give up their portions as a gift. Even were he to eat it, they would not be angry; because there is no other fruit of the same species, their presumed permission is not sufficient. However, if there is something else of the same species, even if one is better than the others, their presumed permission is sufficient because they will not be angry.
9. The practice in places where there are not etrogs wherein the community buys an etrog together [works] because since they bought it to fulfill the obligation, we assume that they specified that all of the community relinquishes their shares to whoever is using it to fulfill the obligation, on the condition that it be returned. Rem"a: We collect etrog money according to wealth, because beautification of the mitzvah applies more to the rich than to the poor. A woman is exempt from giving etrog money because she is not obligated in it (responsa of Mahari"l 114). Every man should take efforts and be hasty to perform the mitzvah of buying an etrog and lulav so as to fulfill the mitzvah fittingly (Hagahot Maimoni end of Laws of Lulav).