In general, the people that are often referred to as white slaves were indentured servants. So what is the difference between slaves and indentured servants?
They are collectively often called unfree labor. Servants, like slaves, were not able to simply leave and work for someone else. They owed work to a specific person. The difference comes in the terms that they are bound by. African slaves were owned as chattel. This means that not only their labor but their persons were owned by their master. Legally, they were property (though their legal existence is more complex than that). This means that they may be bought and sold and they are owned for life; there is not a point where they must be freed.
Indentured servants, however, were bound for a specific term and existed as a legal person with rights. In theory, indentured servants freely signed a contract (indenture) that bound them to labor for a specific person for a specified period of time. Seven years was a common term. After this period, they were free to move wherever they wanted and did not owe continued labor to their former master. In fact, part of the indenture was often that the master must provide them some amount of funds or food once their indenture was complete. They owed their labor to someone, but they were not owned by that person in the same way as a black slave. Another difference is that indentured servants had substantially stronger legal protections. They could use the courts to enforce the contract, argue for their rights, and improve treatment in a way that the overwhelming majority of black slaves were never able. In fact, as the United States continued, they tended to gain rights and improve their treatment. This was part of the very complex process that led to the dramatic increase of black slavery and the decrease of indentured servitude in the seventeenth and eighteenth centuries.
This leads me to more directly answer your question. Indentured servitude declined drastically during the eighteenth century. Indentured servants were still used in the nineteenth century (Antebellum period), but their use was significantly smaller than in the seventeenth or eighteenth centuries. Black slaves were the overwhelming source of unfree labor and black slavery dominated the culture and concerns of the Antebellum South. As for black ownership of slaves, Anthony Johnson is a relatively famous example of a black man who owned slaves himself. However, he lived during the seventeenth century rather than the Antebellum. The rights of blacks decreased significantly after he died. I do not know of any free blacks who owned slaves in the Antebellum period. Someone else might be able to provide an example, but there was certainly not a large black slaveholding class.
Note: this is not to say that white indentured servants were treated particularly well or that there were not severe moral issues with the methods of recruiting them and binding them in service. Additionally, I have made the categories seem far more different than they might have been in practice because I used the legal definitions. In reality, slavery and servitude were closer than the law would lead one to believe. And the distance between servant and slave changed dramatically throughout time. However, they were drastically different than slaves and were not the important source of labor by the time of the Antebellum South.
I would be happy to expand on the treatment of indentured servants compared to slaves if you wish. It is fairly complicated and changed over time so I don't want to include it if you aren't interested.
Sources:
Jordan, Winthrop D. White Over Black: American Attitudes Toward the Negro, 1550-1812. 2nd ed. Chapel Hill, NC: University of North Carolina Press, 2012.
Berlin, Ira. Many Thousands Gone: The First Two Centuries of Slavery in North America (Cambridge, MA: Harvard University Press, 1998.
Edit: Added explanation.
It's important to keep in mind that terms like "white" and "black" are not scientific terms, but cultural terms that have a multiplicity of meanings--and during the antebellum south, they were legal distinctions that might have nothing to do with the way a 21st century person would define race. So it all depends upon how you define "white" and "black". In the antebellum era, any child born to an enslaved woman was also a slave, regardless of paternity. Many slaves had a lot of white ancestry and may have looked "white", depending upon people's definition of "white". But they were still slaves and still considered legally to be black. People who looked white and who lacked any known black ancestry were not legally enslaved in the US. Mark Twain draws attention to and satirizes the idea of people who were in appearance indistinguishable from slave owners, and who were the children of slave owners, being enslaved and considered to be black, in Pudd'nhead Wilson. In the novel, an enslaved woman who is 15/16ths white has a blond haired, blue-eyed son who is 31/32nds white, according to Twain, who adds that "by a fiction of law and custom [he was] a Negro." She then switches her child with the child of the slave master. No one notices the switch, highlighting how indistinguishable a slave may be from a slave owner, and her son is raised to be a slave master and the (legitimate) child of the slave master is raised to be a slave.
In the book 12 Years a Slave, Solomon Northup comments upon seeing many slaves who were white in appearance, but, just as Twain said, were "by a fiction of law and custom," black.
So no, there were no people who were legally white and legally enslaved. But there were plenty who "looked white" who were legally black and therefore legally enslaved.
Each state would have their own Slave Codes defining who can be enslaved. In general whites were not considered slaves even if they were held in indentured servitude. Some examples:
Virginia, 1705: "All servants imported and brought into the Country...who were not Christians in their native Country...shall be accounted and be slaves. All Negro, mulatto and Indian slaves within this dominion...shall be held to be real estate."
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South Carolina, 1712: "Be it therefore enacted, by his Excellency, William, Lord Craven, Palatine.... and the rest of the members of the General Assembly, now met at Charles Town, for the South-west part of this Province, and by the authority of the same, That all negroes, mulattoes, mestizo's or Indians, which at any time heretofore have been sold, or now are held or taken to be, or hereafter shall be bought and sold for slaves, are hereby declared slaves; and they, and their children, are hereby made and declared slaves..."