Gemara

(Abaye): R. Yochanan permits under the Korah, but he forbids Bein ha'Lechayayim (the area parallel to the middle wall and even with the thickness of the Lechi).

(Rava): Also Bein ha'Lechayayim is permitted.

Support (Abaye, for himself - Rav): Bein ha'Lechayayim needs another Lechi [outside it] to permit it.

Suggestion: It is four by four. (It itself is a Karmelis!)

Rejection: Rav requires another Lechi to permit even if it is not four by four.

Rejection (Rava): That is when it is open to a Karmelis.

Inference: If it is open to Reshus ha'Rabim, it is permitted.

Objection: This is absurd! (Karmelis cannot be more stringent than Reshus ha'Rabim.)

Answer: It is reasonable. (Bein ha'Lechayayim would be a Karmelis if it were big enough.) When it opens to a Reshus (Karmelis) like itself, Matza Min Es Mino v'Ni'ur. (It joins with it to complete the Shi'ur.)

Rishonim

Rif (2b): We permit under the Korah or Bein ha'Lechayayim when it is open to Reshus ha'Rabim, but not if it is open to a Karmelis, even if it is not four by four, for Matza Min Es Mino v'Ni'ur.

R. Yehonason (DH Mishum): Karmelis is light in the eyes of those who use the Mavoy, for it is only an Isur mid'Rabanan. Karmelis is like inside the opening, for it has three walls. If we let people carry in the opening, they will comes to carry in the Karmelis next to it.

Rambam (Hilchos Shabbos 17:11): One may carry in a Mavoy under the Korah, or Bein ha'Lechayayim. This is if it was next to Reshus ha'Rabim. If it was next to Karmelis, one must make another Lechi to permit the opening, for Matza Min Es Mino v'Ni'ur.

Rebuttal (Ra'avad): Under the Korah is always permitted. Rava distinguishes between next to Reshus ha'Rabim and next to Karmelis only for Bein ha'Lechayayim. The Rambam rules like Rava. Some forbid Bein ha'Lechayayim, like Abaye.

Migdal Oz: The Sugya supports the Rambam. Rava said 'also Bein ha'Lechayayim is permitted.' This connotes that there is one reason for it and under the Korah. Also, the Gemara was astonished that Karmelis could be more stringent than Reshus ha'Rabim, and Rava answered that it is, due to Matza Min Es Mino v'Ni'ur. This applies also to under the Korah. My Rebbi, the Rashba, supported the Rif from this reason.

Rosh (1:10): One may carry in a Mavoy under the Korah, or Bein ha'Lechayayim, like Rava. This is if it was next to Reshus ha'Rabim, but not if it was next to Karmelis, for Matza Min Es Mino v'Ni'ur.

Shibolei ha'Leket (103, brought in Beis Yosef OC 346 DH Kasuv): R. Chananel says that if a house opens to Reshus ha'Rabim, one may not take anything from the house to outside or vice-versa, unless the threshold is higher than Reshus ha'Rabim, to distance Bnei Reshus ha'Rabim from the doorway. A roof does not permit carrying. Walls are required. A roof over Reshus ha'Rabim does not help. An Oni who takes from or brings into a house is liable [even if there is an awning above]. Proper walls are needed, in which there is more standing wall than gaps. We are lenient to rely on Lavud to permit as long as the gaps are less than three Tefachim only for a caravan that camped in a valley that is a Karmelis (Eruvin 16a). One may not take from a house to Reshus ha'Rabim even through a porch, all the more so if there are more than three Tefachim between the beams in a place where it is level with Reshus ha'Rabim. R. Avigdor said that if the porch is less than 10 tall and less than four by four Tefachim wide, one may hide the key between the door and the porch, for it is Mekom Petur. One may not hide it inside the door.

Beis Yosef (ibid.): I must investigate these matters.

Poskim

Rema (OC 346:3): Our awnings which jut out in front of the house, the law [of underneath them] is like the Reshus in front of them.

Taz (7): If a house's roof juts out into the street, and pillars are built underneath, some asked whether one may carry to there from the house. Some apply Pi Tikra Yored v'Sosem on the edge jutting out into the street, so it is as if a wall descends to the ground [so one may carry]. The Rema refutes this. It is like the Reshus in front of it. Shibolei ha'Leket says that if there are pillars to the side [of the area under the awning] of a house open to Reshus ha'Rabim, they do not change the law. One may not take from a house to Reshus ha'Rabim even through a porch, especially if there are more than three Tefachim between the beams. Really, it is similarly forbidden to take from the house to Karmelis. The only difference is that this makes Karmelis. Shibolei ha'Leket says that a roof does not help. This shows that we do not say Pi Tikra Yored v'Sosem. This is even on the porch, and all the more so in front of the opening [of the house]. Since there are not walls on the side, we do not say Pi Tikra Yored v'Sosem. Rather, it is like the Reshus in front of it, like the Rema says.

Mishnah Berurah (31): All the Acharonim agree with the Taz. If the awning has pillars under it on the side of Reshus ha'Rabim, one makes pillars opposite them on the side of the house, so there is Tzuras ha'Pesach on all three sides [so one may carry underneath].

Gra (DH v'Chen b'Gagin): Even though there are walls, we do not say Pi Tikra Yored v'Sosem since [the roof] is inclined (361:2).

Be'er Heitev (7, citing Rema mi'Pi'ano 112): The custom is that women stand under the threshold and hide the key under the door, even though the door is locked and the threshold is like outside. We do not tell people to do so, but one need not protest.

Shulchan Aruch (365:4): If under the Korah or Bein ha'Lechayayim is open to Karmelis, it is forbidden, for it joins with the Karmelis.

Beis Yosef (DH Mutar): The Rashba says that if a Korah was four wide and strong enough to hold plaster, one may carry under it even if it is open to Karmelis, for Pi Tikra Yored v'Sosem on the outer edge. Semag and Semak permit under the Korah and Bein ha'Lechayayim if they are Reshus ha'Rabim, like Rava. They also wrote that a Korah is a Heker, like Rava. Pi Tikra Yored v'Sosem permits under the Korah only if it is four wide. This is unlike the Rashba, for the Magid Mishneh says that the Rashba forbids under the Korah if it is open to Karmelis. If he permits what opens to Reshus ha'Rabim only if it is four wide, he equates it to Karmelis! Do not say that a Korah open to Karmelis permits only if it is strong enough to hold plaster, for if so, the same applies to Reshus ha'Rabim! Rather, the Rashba permits what opens to Reshus ha'Rabim in every case.

Magen Avraham (10): Also Semak and Semag discuss Karmelis.

Taz (6): Even though Karmelis is more lenient, here we say Matza Min Es Mino v'Ni'ur. I.e. the Karmelis Bein ha'Lechayayim did not have a Shi'ur Karmelis. When it finds its type outside it, it awakens. It was "sleeping", i.e. did not get the law of Karmelis next to a different Reshus. Then, it was Batel. See what I wrote above (346:7) about pillars next to houses. Surely one may not carry there. Here the Beis Yosef brought that if the Korah is four wide and strong enough, it is permitted even if it opens to Karmelis, for the outer edge is Yored v'Sosem. This does not contradict what I wrote about pillars. There, the edge is not Yored v'Sosem, for there is no wall around the roof, like the Rema says (346:3). A Mavoy has three walls, so we say Pi Tikra Yored v'Sosem as long as the Korah is four wide. Teshuvas Maharil (115) says that it is good to be stringent even in this case.

Mishnah Berurah (26): When it opens to Reshus ha'Rabim, which is not its Min at all, it does not join with it at all. It is Batel to the inside.

Bi'ur Halachah (DH Mipnei): Perhaps one may not carry to the Karmelis outside, for mid'Oraisa it is Reshus ha'Yachid, but 346:3 connotes that it is permitted. This requires investigation.