Synopsis
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the T(1) theorem, first proved by David & Journé (1984), describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn). The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1. Suppose that T is a continuous operator from Schwartz functions on Rn to tempered distributions, so that T is given by a kernel K which is a distribution. Assume that the kernel is standard, which means that off the diagonal it is given by a function satisfying certain conditions. Then the T(1) theorem states that T can be extended to a bounded operator on the Hilbert space L2(Rn) if and only if the following conditions are satisfied: * T(1) is of bounded mean oscillation (where T is extended to an operator on bounded smooth functions, such as 1). * T*(1) is of bounded mean oscillation, where T* is the adjoint of T. * T is weakly bounded, a weak condition that is easy to verify in practice.
Reseña del editor
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the T(1) theorem, first proved by David & Journé (1984), describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn). The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1. Suppose that T is a continuous operator from Schwartz functions on Rn to tempered distributions, so that T is given by a kernel K which is a distribution. Assume that the kernel is standard, which means that off the diagonal it is given by a function satisfying certain conditions. Then the T(1) theorem states that T can be extended to a bounded operator on the Hilbert space L2(Rn) if and only if the following conditions are satisfied: * T(1) is of bounded mean oscillation (where T is extended to an operator on bounded smooth functions, such as 1). * T*(1) is of bounded mean oscillation, where T* is the adjoint of T. * T is weakly bounded, a weak condition that is easy to verify in practice.
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