Photosynthesis is the process by which plants capture sunlight energy in order to convert carbon dioxide and water into oxygen gas and glucose (a sugar). The chlorophyll that is contained within the thylakoids of chloroplasts is the green pigment that captures the sunlight energy.
Photosynthesis is less in deserts because there are fewer plants in deserts. The plants that do exist have adapted in order to survive the hot, dry climates of deserts. One of these adaptations is the reduction of leaves. Leaves are broad and flat, providing much surface area from which transpiration can occur. Instead, desert plants have needles in order to prevent this water loss. With fewer leaves, there is less chlorophyll. With less chlorophyll to capture the sunlight energy, the rate of photosynthesis decreases.
There is also less water in the desert. Water is one of the products of photosynthesis. With less water, the rate of the reaction decreases (it serves as a limiting factor).