Maximal function characterizations of Hardy spaces associated to homogeneous higher order elliptic operators

Abstract

Abstract Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mo>-</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>L</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>L</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(p_{-}(L),p_{+}(L))}$ be the maximal interval of exponents <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>q</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> ${q\in[1,\infty]}$ such that the semigroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo>{</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mi>L</m:mi> </m:mrow> </m:mrow> </m:msup> <m:mo>}</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> ${\{e^{-tL}\}_{t&gt;0}}$ is bounded on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${L^{q}(\mathbb{R}^{n})}$ . In this article, the authors establish the non-tangential maximal function characterizations of the associated Hardy spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${H_{L}^{p}(\mathbb{R}^{n})}$ for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>p</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>L</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${p\in(0,p_{+}(L))}$ , which when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> ${p=1}$ , answers a question asked by Deng, Ding and Yao in [21]. Moreover, the authors characterize <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${H_{L}^{p}(\mathbb{R}^{n})}$ via various versions of square functions and Lusin-area functions associated to the operator L .

Locations

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