My initial impression after reviewing the flight track last night was that the aircraft was experiencing roll control problems and finally rolled left at high speed into the water.
Although the maintenance history of Captain side pitot problems may be relevant, and there is always the possibility of maintenance action causing new problems, an airspeed indication problem of itself should not cause loss of the aircraft.
Those of you speculating on the possibility of stall warnings being triggered should remember that these warnings normally are generated from the angle of attack data which is not derived from airspeed but instead from the vanes on the sides of the cockpit.
The impact of defective air data on the aircraft feel systems needs to be discussed by someone with more 737 aircraft familiarity than myself but could potentially be a significant factor. Aircraft control appeared to be degraded early in the flight.
Speculation: Could both of the cockpit crew have been applying force to the control column and not have had a free hand for setting the power? The continued acceleration of the aircraft doesn't make good sense from a piloting perspective. They needed to set a reasonable power setting and investigate if they could not continue the climb or did not wish to continue the climb.