Inclusions Among Mixed-norm Lebesgue Spaces

Download or read book Inclusions Among Mixed-norm Lebesgue Spaces written by Wayne Robert Grey and published by . This book was released on 2015 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: A mixed Lp norm of a function on a product space is the result of successive classical Lp norms in each variable, potentially with a different exponent for each. Conditions to determine when one mixed norm space is contained in another are produced, generalizing the known conditions for embeddings of Lp spaces. The two-variable problem (with four Lp exponents, two for each mixed norm) is studied extensively. The problem's unpermuted case simply reduces to a question of Lp embeddings. The other, permuted case further divides, depending on the values of the Lp exponents. Often, they fit the Minkowski case, when Minkowski's integral inequality provides an easy, complete solution. In the non-Minkowski case, the solution is determined by the structure of the measures in the component Lp spaces. When no measure is purely atomic, there can be no mixed-norm embedding in the non-Minkowski case, so for such measures the problem is solved. With at least one purely atomic measure, the non-Minkowski case divides further based on the structure of the measures and the values of the exponents. Various necessary conditions and sufficient conditions are found, solving a number of subcases. Other subcases are shown to be genuinely complicated, with their solutions expressed in terms of an optimization problem known to be computationally difficult. With some di cult cases already present in the two-variable problem, it is impractical to cover every case of the multivariable problem, but results are presented which fully solve some cases. When no measure is purely atomic, the multivariable problem is solved by a reduction to the Minkowski case of certain two-variable subproblems. The multivariable problem with un-weighted ̀p spaces has a similar reduction to easy two-variable subproblems. It is conjectured that this applies more generally; that, regardless of the structures of the involved measures, when every permuted two-variable subproblem fits the Minkowski case, the full multivariable mixed norm inclusion must hold.