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• W2068366883 abstract "The potential theoretic idea of the thinness of a set at a given point is extended to the weighted nonlinear potential theoretic setting—the weights representing in general singularities/degeneracies—and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis integral left-parenthesis upper I Subscript alpha Baseline asterisk f right-parenthesis Superscript q Baseline v d x right-parenthesis Superscript 1 slash q Baseline less-than-or-slanted-equals left-parenthesis integral f Superscript p Baseline w d x right-parenthesis Superscript 1 slash p Baseline comma> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msub> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> <mml:mi>v</mml:mi> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mi>w</mml:mi> <mml:mspace width=thinmathspace /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{left ( {int {{{({I_alpha }{ast }f)}^q}v,dx} } right )^{1/q}} leqslant {left ( {int {{f^p}w,dx} } right )^{1/p}},</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than p less-than-or-slanted-equals q 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>q</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>1 > p leqslant q > infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which at least in the first-order case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis alpha equals 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(alpha = 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have found some use in a number of places in analysis." @default.
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• W2068366883 date "1986-01-01" @default.
• W2068366883 modified "2023-10-05" @default.
• W2068366883 title "Weighted nonlinear potential theory" @default.
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