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• W1966115728 abstract "We give sufficient conditions for estimates of the form<disp-formula content-type=math/mathml>[<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript q Baseline d mu left-parenthesis x right-parenthesis less-than-or-slanted-equals upper C double-vertical-bar u double-vertical-bar Subscript s comma p Superscript 1 Baseline comma u element-of upper H Superscript s comma p Baseline comma><mml:semantics><mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mo>∫<!-- ∫ --></mml:mo><mml:msup><mml:mrow class=MJX-TeXAtom-ORD><mml:mrow><mml:mo>|</mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>u</mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>q</mml:mi></mml:msup></mml:mrow><mml:mi>d</mml:mi><mml:mi>μ<!-- μ --></mml:mi><mml:mo stretchy=false>(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=false>)</mml:mo><mml:mo>⩽<!-- ⩽ --></mml:mo><mml:mi>C</mml:mi><mml:msubsup><mml:mrow><mml:mo symmetric=true>‖</mml:mo><mml:mi>u</mml:mi><mml:mo symmetric=true>‖</mml:mo></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mo>,</mml:mo><mml:mspace width=2em /><mml:mi>u</mml:mi><mml:mo>∈<!-- ∈ --></mml:mo><mml:mrow class=MJX-TeXAtom-ORD><mml:msup><mml:mi>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mrow><mml:annotation encoding=application/x-tex>{int {left | {u(x)} right |} ^q}dmu (x) leqslant Cleft | u right |_{s,p}^1,qquad u in {H^{s,p}},</mml:annotation></mml:semantics></mml:math>]</disp-formula>to hold, where<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=mu left-parenthesis x right-parenthesis><mml:semantics><mml:mrow><mml:mi>μ<!-- μ --></mml:mi><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>mu (x)</mml:annotation></mml:semantics></mml:math></inline-formula>is a measure and<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar u double-vertical-bar Subscript s comma p><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:msub><mml:mrow><mml:mo symmetric=true>‖</mml:mo><mml:mi>u</mml:mi><mml:mo symmetric=true>‖</mml:mo></mml:mrow><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:annotation encoding=application/x-tex>{left | u right |_{s,p}}</mml:annotation></mml:semantics></mml:math></inline-formula>is the norm of the Sobolev space<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript s comma p><mml:semantics><mml:mrow class=MJX-TeXAtom-ORD><mml:msup><mml:mi>H</mml:mi><mml:mrow class=MJX-TeXAtom-ORD><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:annotation encoding=application/x-tex>{H^{s,p}}</mml:annotation></mml:semantics></mml:math></inline-formula>. If<inline-formula content-type=math/mathml><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d mu equals d x><mml:semantics><mml:mrow><mml:mi>d</mml:mi><mml:mi>μ<!-- μ --></mml:mi><mml:mo>=</mml:mo><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow><mml:annotation encoding=application/x-tex>dmu = dx</mml:annotation></mml:semantics></mml:math></inline-formula>, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section." @default.
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• W1966115728 date "1987-01-01" @default.
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• W1966115728 title "Weighted norm estimates for Sobolev spaces" @default.
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• W1966115728 doi "https://doi.org/10.1090/s0002-9947-1987-0911089-3" @default.
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