Almost all the designer sunglasses have UV protection if the one you bought is real.
They are branded, and they are super expensive. It really can't make sense if they don't provide UV protection.
However, I think there is no need to buy designer sunglasses if you have any hesitation about the price. Most of the middle price sunglasses provide sunglasses with good quality as well. They also have UV protection.
Indeed, designer sunglasses both have excellent quality and supreme UV protections. Generally, the designer sunglasses are expensive because they have durable and good UV protection, high quality, unique design, superior hand craftsmanship. They lead the way in the eyewear market because they can the ability to make their products perfect. Customers always like to buy designer sunglasses to match with their designer clothes. Wonderful combination will make them confident and attractive among a group of people. So to protect your eye from the harm of sunshine, a pair of designer sunglasses is necessary.