In the painting "The Duchess," the figure has one hand rested and the other is holding something, near her chest. What is the thing in her second hand?
If this is something precious, maybe this explains why it is so high up her chest, close to her heart (my interpretation). It looks almost like a jewel embedded bottle, and I thought that maybe it would consist of poison, however, as far as I know, the Duchess was executed.
If you are referring to Thomas Gainsborough's painting "The Duchess of Devonshire," the woman in the painting is holding a rose in her hand.
If the painting in question is De LAszlo's "Portrait of the Duchess of York," the duchess is depicted holding a portion of her pearls in her hand.
Both being said, one can justifiable assume that the nearer something is to one's heart the more the object may have meaning or be dear to them. Given that the first painting depicts a rose, which is a historically recognized symbol of love, one could easily assume that the rose represents the love she has for another. In the case of the pearls, pearls symbolize honesty, wisdom and integrity. Therefore, the clutching of the pearls to the heart may show the importance of honesty, wisdom and integrity to the duchess.