Gemara

5b (R. Avin citing R. Yochanan): If Reuven was laden with food and drink, and went back and forth between Reshus ha'Rabim and Reshus ha'Yachid all day, he is exempt until he stops.

(Rav Safra citing R. Yochanan): If one moved objects from one corner to another, and reconsidered to take them out, he is exempt, since his initial intention was not to take them out.

(Beraisa): If one was Motzi from a store (Reshus ha'Yachid) to the market (a Reshus ha'Rabim) via a Stav (platforms where merchants sit), he is liable;

Ben Azai exempts.

Question: Granted, Ben Azai considers one who walks like one who stops. (Each step is Hanachah. It is as if he was Motzi to Karmelis, and then from Karmelis to Reshus ha'Rabim);

However, what is Chachamim's reason?

Answer #2: Rather, this is like Hotza'ah from a Reshus ha'Yachid to Reshus ha'Rabim via Tzidei Reshus ha'Rabim (e.g. a breach in a wall between a Chatzer and Reshus ha'Rabim). Even though he would be exempt if he was Mani'ach in Tzidei Reshus ha'Rabim, when he is Mani'ach at the end, he is liable. The same applies here.

(R. Yochanan): Ben Azai agrees that one who throws [from Reshus ha'Yachid to Reshus ha'Rabim via Karmelis] is liable. (It did not rest in the middle.)

Support (Beraisa): If one was Motzi from a store to the market via a Stav, he is liable;

The same applies if he brought in (from the market to the store), whether he carried, threw, or passed the item.

Ben Azai exempts one who carried, and obligates one who threw or passed.

8b (Beraisa): Someone standing on a threshold [between two Reshuyos] may take from or give to a Ba'al ha'Bayis [in Reshus ha'Yachid], or take from or give to an Oni [in Reshus ha'Rabim].

The threshold is Mekom Petur, e.g. it is less than four by four;

(Rav Dimi): If a place is less than four by four, both those in Reshus ha'Yachid and Reshus ha'Rabim may adjust loads on it, as long as they [people in different Reshuyos] do not switch loads.

(Beraisa): He may not take from the Ba'al ha'Bayis and give the item to the Oni, nor vice-versa. If he did so, all three are exempt.

Suggestion: This refutes Rava!

(Rava): One who transfers an item four Amos in Reshus ha'Rabim is liable, even if he carried it above 10 Tefachim (i.e. through Mekom Petur)!

Rejection: No. in Rava's case the item did not rest [in the middle]. Here, it was Nach [in the middleman's hands].

91b (Mishnah): If one was Motzi food and put it on a ledge, and later he was Motzi it, he is exempt, for the Melachah was not done at once.

The ledge is a Karmelis. Had the food not rested in the middle, he would be liable. This is unlike Ben Azai.

Rishonim

Rif (2a): R. Yochanan taught that Ben Azai agrees that one who throws is liable. A Beraisa supports R. Yochanan.

Rosh (1:7): The Rif (brought R. Yochanan's teaching. This) connotes that he rules like Ben Azai. This is astounding. Why does he rule like an individual? Even though R. Yochanan taught about Ben Azai's opinion, Amora'im often explain Tana'im even though the Halachah does not follow them! It seems that the Halachah follows Rabanan. The Rambam rules like Rabanan. Also R. Yonah questioned the Rif's ruling. We find that R. Yochanan himself holds that one who walks is unlike one who stops. He exempts one who was laden and went back and forth between Reshus ha'Rabim and Reshus ha'Yachid all day, until he stops. He also exempts one who moved objects from one corner to another, and reconsidered to take them out, since his initial intention was not to take them out. If one who walks is like one who stops, when he decided to take them out, he should be liable, just like if he stopped (and reconsidered)! In Kesuvos (31a) we challenged R. Yochanan from a Beraisa that obligates one who reconsidered, and answered that the Beraisa is like Ben Azai.

Beis Yosef (OC 346 DH v'Chasvu): Perhaps also the Rif rules unlike Ben Azai. He brought R. Yochanan's teaching, lest one think that the first Tana obligates only for throwing, but he exempts walking, like Ben Azai. The Rif taught that even Ben Azai obligates one who throws, so we must say that (they argue about walking, and) the first Tana obligates even one who walks.

Rambam (Hilchos Shabbos 14:15): If one was Motzi from Reshus ha'Yachid to Reshus ha'Rabim and while walking he passed through a Mekom Petur in between, he is liable. One who walks is not considered to be at rest. We need not say that one who threw and it passed through Mekom Petur is liable, for it is not considered as if it rested in the middle. If one was standing in Mekom Petur and took an item from Reshus ha'Yachid or from someone standing there, and deposited it in Reshus ha'Rabim or in the hand of someone standing there, he is exempt. Likewise, if he entered something from Reshus ha'Rabim to Reshus ha'Yachid while standing in Mekom Petur, he is exempt.

Rosh (1:17): The Gemara said that the threshold is a Mekom Petur when it is in Reshus ha'Rabim, not between the doorposts, and it has no roof. It is [at least] three tall, and less than four by four wide. This is like R. Yochanan taught, that any place less than four by four, people of Reshus ha'Yachid and people of Reshus ha'Rabim may adjust loads on it, as long as they do not switch.

Ran (3a DH Adam): The threshold is less than four by four. Mekom [Petur] is a specific place, a little wide and three tall, next to Reshus ha'Rabim and Reshus ha'Yachid. In Eruvin we say that it is three tall.

Poskim

Shulchan Aruch (OC 346:1): One who stands in Reshus ha'Yachid, and is Motzi, Machnis, extends or throws to or from Reshus ha'Rabim through Mekom Petur, or vice-versa, is liable.

Taz (2): Why does the Shulchan Aruch say Motzi or Moshit? Also Motzi is Moshit, for he remained standing in Reshus ha'Yachid! Rather, we must say that Motzi discusses one who stood in Reshus ha'Yachid and walked and was Motzi or Machnis, or stood in place and extended or threw.

Beis Yosef (DH u'Mah she'Chasav u'Vilvad): Our texts of the Tur say "as long as he does not take from Reshus ha'Yachid to Mekom Petur through Reshus ha'Rabim." The Tur rules that one who walks is unlike one who stops. Therefore if he did not stop to rest in the middle, he is like one who takes from Reshus ha'Yachid to Mekom Petur, which is permitted. Even so, since if he would stop in Reshus ha'Rabim [to rest] he would be liable, we decree lest he stand. An old text of the Tur, and a corrected version, say 'as long as he does not take from Reshus ha'Yachid to Mekom Petur, and transfer from Mekom Petur to Reshus ha'Rabim, or vice-versa.' This is like what he wrote above (that one in Mekom Petur may not transfer from Reshus ha'Yachid to Reshus ha'Rabim).

Rebuttal (Taz 2): After this, the Tur wrote "similarly, one who stands in Reshus ha'Yachid..." I.e. he walks from Reshus ha'Yachid to Mekom Petur, and then from Mekom Petur to Reshus ha'Rabim. According to the "old text", the Tur already taught this! This is good according to the first text the Beis Yosef brought. The Bach challenged this text, for we do not find such a decree lest one stop to rest. Letter of the law, l'Chatchilah one may take from Reshus ha'Yachid to Mekom Petur through Reshus ha'Rabim. The Bach is astounding. This is included in the Gemara's decree that one in Mekom Petur may not take from the Ba'al ha'Bayis and give to the Oni, lest one overtly transfer from Reshus ha'Yachid to Reshus ha'Rabim. The same applies here! Also, if we would permit, because one who walks is unlike one who stops, we should permit also from Reshus ha'Yachid to Reshus ha'Yachid through Reshus ha'Rabim. The Tur (348) forbids. Rather, one who walks is unlike one who stops regarding Chiyuv, but we always forbid due to disgrace of Shabbos. However, according to the Beis Yosef's text, why did the Tur teach first what is not explicit in the Gemara? The Beis Yosef's original text is primary. "Or Moshit" means that Motzi or Machnis is forbidden, whether it is through extending or throwing.

Kaf ha'Chayim (8): Tosefes Shabbos says that if one intended to take from Reshus ha'Yachid to Reshus ha'Rabim through Mekom Petur, and he stopped in the middle, he is exempt if he did not initially intend to stop in the middle.

Gra (DH Aval): The Stam Mishnah (91b) is unlike Ben Azai. In Eruvin (98a), the Gemara toiled to give difficult answers before establishing the Mishnah like Beitzim.

Mishnah Berurah (5): He is liable only if he did not stop in Mekom Petur. However, even if he stopped there, it is an Isur mid'Rabanan, lest he overtly take from Reshus ha'Yachid to Reshus ha'Rabim. Likewise, one may not take from Reshus ha'Yachid to Mekom Petur through Reshus ha'Rabim, even if he is careful not to stop in Reshus ha'Rabim.

Shulchan Aruch (ibid.): Similarly, one may not stand in Mekom Petur and take an item from someone in Reshus ha'Rabim and give it to someone in Reshus ha'Yachid, or vice-versa.

Beis Yosef (DH Aval): Rashi says that one may not take from the Ba'al ha'Bayis and give the item to the Oni, for this disgraces the Isur Shabbos, to l'Chatchilah cause a Hotza'ah from Reshus ha'Yachid to Reshus ha'Rabim. We decree lest he overtly transfer from Reshus ha'Yachid to Reshus ha'Rabim. . vice-versa. If he did so, all three are exempt.

Mishnah Berurah (6): "And similarly" is not precise, for in this case he is exempt. Since he himself is stationary in Mekom Petur, once the item is by him it is as if it rested a little. The Shulchan Aruch said "similarly" because also this is forbidden, since through him there is Hotza'ah from Reshus ha'Yachid to Reshus ha'Rabim, and people will come to take straight from Reshus ha'Yachid to Reshus ha'Rabim.

Mishnah Berurah (7): the same applies to taking from the ground of Reshus ha'Yachid and putting it on the ground of Reshus ha'Rabim.

Kaf ha'Chayim (12): Rabbeinu Zalman says that if he did not rest his hand in Mekom Petur at all, he is liable.

Kaf ha'Chayim (14): Rabbeinu Zalman forbids even though through two people, e.g. Ploni was Motzi from Reshus ha'Yachid to Mekom Petur, and Almoni took it from Mekom Petur to Reshus ha'Rabim, lest one person take it from Reshus ha'Yachid to Reshus ha'Rabim.