When the British arrived on Crete in October 1940, there was only one airfield on the island, the airstrip at Heraklion. Heraklion was a relatively rudimentary airfield at the time, so could only be used to refuel aircraft in transit between Egypt and Greece. As the Greek campaign progressed, the RAF expanded the Heraklion field and began to prepare airfields elsewhere on the island. A second strip was constructed near Heraklion to take bombers. Five more RAF airfields were planned, though only one was constructed, at Rethymno. In addition, the Fleet Air Arm operated a base at Maleme to cover the naval base at Souda Bay. Most of these airfields were relatively rudimentary, and lacked dedicated maintenance and repair facilities.

However, there were few aircraft available to fly from these bases. The RAF had committed some nine squadrons to defend Greece. Three of these were full fighter squadrons, while the remaining six had a mix of fighters and light bombers. These came to a total of 152 aircraft, mostly older aircraft that were badly outclassed by their Axis opposition. While the RAF were largely successful at holding off Italian attacks, the Germans were a different story. Their crushing numerical superiority (1,200 aircraft) over the RAF allowed them to quickly gain air superiority. British squadrons faced overwhelming odds in the air, leading to heavy losses. British airfields were often attacked to devastating effect; the RAF's official history describes an attack on No. 113 Squadron's airbase at Niamata where every aircraft in the squadron was damaged or destroyed. The rapid pace of the German advance worsened the situation. There was not enough time to repair damaged aircraft. Instead, any spare parts were stripped off them, to keep the squadron's other aircraft running to cover the retreat. Others had to be abandoned, as they could not be flown to Crete or loaded onto ships. This led to heavy aircraft losses by the time of the withdrawal to Crete. The RAF lost a total of 151 aircraft during the Greek Campaign, nearly their entire strength at the start of the battle. Of these, 87 had been abandoned due to damage. The withdrawal left less than 40 fighters on Crete. Many of the RAF's squadrons had left their maintenance equipment behind them during the evacuation. This combined with the lack of facilities at the airfields to mean that servicability rates for the fighters were low.

Crete could not easily be reinforced by shifting aircraft from elsewhere in the Mediterranean. This was largely because these aircraft did not exist. The RAF had stripped its forces in North Africa to reinforce Greece. There were just two fighter squadrons protecting the Allied troops in Libya; this cost the Allies greatly when the German Afrika Korps began attacking eastwards in April, causing heavy losses in troops and aircraft. The RAF also needed to protect Malta, Gibraltar and the Suez Canal, but there were dire shortages of fighters. A convoy with 50 crated Hurricanes arrived in Alexandria on the 12th May, but this was barely sufficient to make up the losses suffered in North Africa, let alone to reinforce Crete. RAF leaders were especially loathe to send more aircraft to Crete given the vulnerability of the airfields.

During the lead-up to the invasion of Crete, the Germans began a campaign of bombing and strafing the island's airfields. This would clear the RAF out of the way, making the aerial assault easier. The attack on the airfields was carried out by Fliegerkorps VIII, which had some 716 aircraft (outnumbering the RAF 18:1). The Germans made heavy strafing and bombing attacks on the Cretan airfields, with daily attacks being carried out from the 14th. On the 19th, the attacks were being carried out at a half-hourly frequency. By this time, the RAF's fighter strength on the island had been reduced to just seven aircraft. It was clear that they had no hope of holding out against the Germans, and General Freyberg, commanding the island's garrison, ordered them to withdraw to Egypt. The next day, the German airborne assault began.

I'm not sure I'd use 'dominant' to describe RAF efforts in Greece, the few squadrons of mostly obsolete aircraft that fought against similarly undistinguished Italian types performed well enough but were a pretty minor part of the campaign. The German invasion of 1941 saw them greatly outnumbered by significantly superior Luftwaffe aircraft, who were dominant - the RAF suffered heavily both on the ground and in the air, within a week it was down from around 200 aircraft (of which only 80 were serviceable) to 46. After two weeks the bombers had been withdrawn, the last few fighters were positioned to cover the British evacuation to Crete but most were destroyed on the ground. Following the evacuation the Luftwaffe focused their attentions on Crete and its two airfields; with aircraft in desperately short supply across the Mediterranean a few RAF reinforcements from Egypt dripped in over the weeks but by 18th May there were only five operational fighters compared to almost 200 German fighters and more than 400 bombers. Two Gladiators and two Hurricanes evacuated to Egypt that day, the final Hurricane left Crete on 19th May.