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• W1592814570 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X 1 comma upper X 2 comma ellipsis comma upper X Subscript q Baseline> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>X_1,X_2,ldots ,X_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a system of real smooth vector fields, satisfying Hörmander’s condition in some bounded domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega subset-of double-struck upper R Superscript n> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega subset mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n greater-than q> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>n>q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). We consider the differential operator <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L equals sigma-summation Underscript i equals 1 Overscript q Endscripts a Subscript i j Baseline left-parenthesis x right-parenthesis upper X Subscript i Baseline upper X Subscript j Baseline comma> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>q</mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} mathcal {L}=sum _{i=1}^qa_{ij}(x)X_iX_j, end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where the coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript i j Baseline left-parenthesis x right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>a_{ij}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu StartAbsoluteValue xi EndAbsoluteValue squared less-than-or-equal-to sigma-summation Underscript i comma j equals 1 Overscript q Endscripts a Subscript i j Baseline left-parenthesis x right-parenthesis xi Subscript i Baseline xi Subscript j Baseline less-than-or-equal-to mu Superscript negative 1 Baseline StartAbsoluteValue xi EndAbsoluteValue squared> <mml:semantics> <mml:mrow> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>q</mml:mi> </mml:munderover> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:msup> <mml:mi>μ<!-- μ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} mu |xi |^2leq sum _{i,j=1}^qa_{ij}(x)xi _ixi _jleq mu ^{-1}|xi |^2 end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for a.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>xin Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, every <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi element-of double-struck upper R Superscript q> <mml:semantics> <mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>xi in mathbb {R}^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, some constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu> <mml:semantics> <mml:mi>μ<!-- μ --></mml:mi> <mml:annotation encoding=application/x-tex>mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, we assume that the coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript i j> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>a_{ij}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belong to the space <italic>VMO</italic> (“Vanishing Mean Oscillation”), defined with respect to the subelliptic metric induced by the vector fields <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X 1 comma upper X 2 comma ellipsis comma upper X Subscript q Baseline> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>X_1,X_2,ldots ,X_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove the following local <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L Superscript p> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathcal {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-estimate: <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar upper X Subscript i Baseline upper X Subscript j Baseline f double-vertical-bar Subscript script upper L Sub Superscript p Subscript left-parenthesis normal upper Omega prime right-parenthesis Baseline less-than-or-equal-to c left-brace double-vertical-bar script upper L f double-vertical-bar Subscript script upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline plus double-vertical-bar f double-vertical-bar Subscript script upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline right-brace> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow> <mml:mo symmetric=true>‖</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>f</mml:mi> <mml:mo symmetric=true>‖</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mrow> <mml:mo symmetric=true>‖</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo symmetric=true>‖</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mo symmetric=true>‖</mml:mo> <mml:mi>f</mml:mi> <mml:mo symmetric=true>‖</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} left |X_iX_jfright |_{mathcal {L}^p(Omega ’)}leq cleft {left |mathcal {L}fright |_{mathcal {L}^p(Omega )}+left |fright |_{mathcal {L}^p(Omega )}right } end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> for every <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega prime subset-of subset-of normal upper Omega> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>⊂<!-- ⊂ -->⊂<!-- ⊂ --></mml:mo> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega ’subset subset Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <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>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also prove the local Hölder continuity for solutions to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L f equals g> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {L}f=g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g element-of script upper L Superscript p> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>gin mathcal {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> large enough. Finally, we prove <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper L Superscript p> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>L</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathcal {L}^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-estimates for higher order derivatives of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whenever <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the coefficients <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript i j> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>a_{ij}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are more regular." @default.
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• W1592814570 date "1999-09-21" @default.
• W1592814570 modified "2023-09-25" @default.
• W1592814570 title "𝐿^{𝑝} estimates for nonvariational hypoelliptic operators with 𝑉𝑀𝑂 coefficients" @default.
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