2022/01/26 by Óscar Domínguez, Sergey Tikhonov
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Mathematical Analysis and Transform Methods
paper · doi:10.1090/tran/8632
In this paper we study sharp pointwise inequalities for maximal operators. In particular, we strengthen DeVore’s inequality for the moduli of smoothness and a logarithmic variant of Bennett–DeVore–Sharpley’s inequality for rearrangements. As a consequence, we improve the classical Stein–Zygmund embedding deriving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper B With dot Subscript normal infinity Superscript d slash p Baseline upper L Subscript p comma normal infinity Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis right-arrow with hook BMO left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>B</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msubsup> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> ↪ </mml:mo> <mml:mtext>BMO</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex"> Bd/p_∞ Lp,∞ (\mathbb Rd) \hookrightarrow \text BMO(\mathbb Rd)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p greater-than normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p > ∞</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Moreover, these results are also applied to establish new Fefferman–Stein inequalities, Calderón–Scott type inequalities, and extrapolation estimates. Our approach is based on the limiting interpolation techniques.