A HOLDER-YOUNG-LIEB INEQUALITY FOR NORMS OF GAUSSIAN WICK PRODUCTS

作者：Da Pelo Paolo^*; Lanconelli Alberto; Stan Aurel I

来源：Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011, 14(3): 375-407.

DOI：10.1142/S0219025711004456

• Wick product
• second quantization operator
• convolution product
• Holder inequality
• Young inequality
• Lieb inequality
• exponential functions

An important connection between the finite-dimensional Gaussian Wick products and Lebesgue convolution products will be proven first. Then this connection will be used to prove an important Holder inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the Holder and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian Holder inequality and classic Holder inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite-dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite-dimensional framework.

• 出版日期2011-9