Mean oscillation bounds on rearrangements
نویسندگان
چکیده
We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements R n {\mathbb {R}^n} . For decreasing rearrangement alttext="f asterisk"> f ∗ encoding="application/x-tex">f^* of a rearrangeable function alttext="f"> encoding="application/x-tex">f bounded (BMO) cubes, we improve classical inequality Bennett–DeVore–Sharpley, alttext="double-vertical-bar f asterisk Baseline double-vertical-bar Subscript B M O left-parenthesis double-struck Sub plus right-parenthesis less-than-or-equal-to C n right-parenthesis"> ‖<!-- ‖ <mml:msub> BMO stretchy="false">( + stretchy="false">) ≤<!-- ≤ <mml:mi>C encoding="application/x-tex">\|f^*\|_{{\operatorname {BMO}}(\mathbb {R}_+)}\leq C_n \|f\|_{{\operatorname {R}^n)} , by showing growth alttext="upper encoding="application/x-tex">C_n in dimension alttext="n"> encoding="application/x-tex">n is not exponential but at most order alttext="StartRoot EndRoot"> encoding="application/x-tex">\sqrt {n} This achieved comparing cubes family rectangles which one can dimension-free Calderón–Zygmund decomposition. By polar rectangles, provide first proof that an analogous holds symmetric rearrangement, S f"> S encoding="application/x-tex">Sf
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2022
ISSN: ['2330-0000']
DOI: https://doi.org/10.1090/tran/8629