Exact Inclusions of Gehring Classes in Muckenhoupt Classes
Copyright policy )Exact Inclusions of Gehring Classes in Muckenhoupt Classes
Let be \(|E|\) the Lebesgue measure of the set \(E\) and \[ -\hskip-.9em\int_E f(x) dx= {1\over|E|} \int_E f(x) dx. \] A measurable and nonnegative function \(f\) on the cube \(Q_0\subset\mathbb{R}^n\) belongs to the Muckenhoupt class \(A_q(B)\) on the cube \(Q_n\), if for any subcube \(Q\subset Q_0\) the following inequality holds \[ \Biggl(-\hskip-.9em\int_Q f(x) dx\Biggr) \Biggl(-\hskip-.9em\int_Q f^{-1/(q-1)}(x) dx\Biggr)^{q-1}\leq B. \] Such a function belongs to the Gehring class \(G_p(C)\), \(p> 1\), \(C> 1\), on the cube \(Q_0\subset\mathbb{R}^n\) if for any subcube \(Q\subset Q_0\) the function \(f\) satisfies the reverse Hölder inequality \[ \Biggl(-\hskip-.9em\int_Q f^p(x) dx\Biggr)^{1/p}\leq C\Biggl(-\hskip-.9em\int_Q f(x) dx\Biggr). \] The relation between the Gehring and Muckenhoupt classes was studied by Coifman and Fefferman, Bojarski, Wik, Sbordone and d'Apuzzo, Korenovskij and others. The aim of this paper is to find in the one-dimensional case exact bounds for exponents for which an inclusion of the following form exists \(G_{p_1}(C_1)\subset A_{q_1}(B_1)\).
Classification of real functions; Baire classification of sets and functions, Gehring class, Quasi-analytic and other classes of functions of one complex variable, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Muckenhoupt class, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Classification of real functions; Baire classification of sets and functions, Gehring class, Quasi-analytic and other classes of functions of one complex variable, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Muckenhoupt class, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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