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We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.
The paper considers the representation of non-degenerate bilinear forms on the non-Archimedean Hilbert space by linear operators. More precisely, upon making some suitable assumptions we prove that if is a non-degenerate bilinear form on , then is representable by a unique linear operator whose adjoint operator exists.
The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then is representable by a unique self-adjoint (possibly unbounded) operator .
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