Islam encourages us to give the family of the Prophet (peace and blessings be upon him) due esteem and there are some verses in the Qur’an that put them in a supreme position. An example is the saying of the Almighty:
(Allah only wants to remove from you all that might be loathsome, O you members of the [Prophet’s] household, and to purify you to utmost purity.)(Al-Ahzab 33: 33)
The reason behind the scarceness of traditions reported by them, is that memorizing and reporting Hadith was one of the main concerns of some specific companions such Abu Hurairah (may Allah be pleased with him). Due to the young age of both Al-Hasan and Al-Husayn (may Allah be pleased with them) they were not in the position to memorize and report Hadith. Fatimah (may Allah be pleased with her) passed away only six months after the demise of the Prophet (peace and blessings be upon him) and, at that time, the phase of collecting Hadith has not yet begun.
In response to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
Your love for the family of the Prophet (peace and blessings be upon him) is indeed extremely commendable. Love of the Prophet’s family is part of our love for him, who is the final Messenger of Allah, and our perfect role model and exemplar. Therefore, no Muslim can afford to cherish other than utmost love for his family including Fatimah as well his grandchildren Al-Hasan and Al-Husayn (may Allah be pleased with them).
Having said this, however, I must also point out that the Prophet’s family also includes his wives and his children. This is evident in the following reference in the Qur’an: (O wives of the Prophet! You are not like any of the [other] women, provided that you remain [truly] conscious of Allah. Hence, be not over-soft in your speech, lest any whose heart is diseased should be moved to desire [you]: but, withal, speak in a kindly way. And abide quietly in your homes, and do not flaunt your charms as they used to flaunt them in the old days of pagan ignorance; and be constant in prayer, and render the purifying dues, and pay heed unto Allah and His Messenger: for Allah only wants to remove from you all that might be loathsome, O you members of the [Prophet’s] household, and to purify you to utmost purity. And retain all that is recited in your homes of Allah’s messages and [His] wisdom: for Allah is unfathomable [in His wisdom], all-aware.)(Al-Ahzab 33: 32-34).
Therefore, it is unfair to say that the family of the Prophet (peace and blessings be upon him) is limited to his daughter Fatimah and his children from her; even as it is unfair to exclude them from his family on the ground that his wives are the main focus of the above verses.
Now coming to the question about the transmission of the teachings of Islam: As can very well be inferred from the above verses, this task was entrusted to his wives and the prominent companions from among the Muhajirun (the Prophet’s companions who migrated with him from Makkah to Madinah) and Ansar(the people of Madinah who hosted the Prophet and the Muhajirun) as mentioned elsewhere: (And the first to lead the way, of the Muha’jirin and the Ansar, and those who followed them in goodness Allah is well pleased with them and they are well pleased with Him and He hath made ready for them Gardens underneath which rivers flow, wherein they will abide for ever. That is the supreme triumph.)(Al-Tawbah 9: 100).
The pioneers referred to above include Fatimah, al-Hasan and al-Husayn, but in large measure they also refer to `Ali ibn Abi Talib and other companions. A greater share of the burden of transmission fell on the shoulders of `A’ishah, the beloved wife of the Prophet (peace and blessings be upon him) as well as his other wives, who dealt with him on a daily basis more intimately than anyone else. Thus, as it has been documented in the sources, `A’ishah transmissions from the Prophet, (peace and blessings be upon him), far exceed those of all the other members of Prophet’s family as well as those of other companions.
The reasons for this are clear: Fatimah, despite being considered as the beloved daughter of the Prophet (peace and blessings be upon him) she did not survive more than six months after his death; while al-Hasan and al-Husayn (may Allah be pleased with them) were still in their tender years when the Prophet (peace and blessings be upon him) was ali ve. Al-Hasan (may Allah be pleased with him) was born three years after the Hijrah ( migration from Makkah to Madina) of the Prophet (peace and blessings be upon him), while al-Husayn was born one year after. That is why their father ‘Ali retained and transmitted more hadiths than all of them. He is undoubtedly one of the greatest sources of Islamic knowledge along with other prominent companions.
These facts explain the reasons for the scarcity of reports from Fatimah, al-Hasan and al-Husayn, (may Allah be pleased with them all), but there is no shortage of reports about them in the sources transmitted by `A’ishah as well others, (may Allah be pleased with them all).
In conclusion, as Muslims we can be truly proud of the fact that the sources of our faith have been meticulously preserved for us, thanks to the efforts of men and women surrounding the Prophet (peace and blessings be upon him) whose earnestness, piety and commitment to the cause of Islam are unsurpassed. May Allah grant us all love for them all, and may He grant us the honor to emulate their beautiful examples.