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• W2962960686 abstract "Abstract We investigate weighted Sobolev spaces on metric measure spaces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> {(X,mathrm{d},mathfrak{m})} . Denoting by ρ the weight function, we compare the space <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mi>𝔪</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {W^{1,p}(X,mathrm{d},rhomathfrak{m})} (which always coincides with the closure <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mi>𝔪</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1,p}(X,mathrm{d},rhomathfrak{m})} of Lipschitz functions) with the weighted Sobolev spaces <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>W</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {W^{1,p}_{rho}(X,mathrm{d},mathfrak{m})} and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{1,p}_{rho}(X,mathrm{d},mathfrak{m})} defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>⁢</m:mo> <m:mi>𝔪</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {W^{1,p}(X,mathrm{d},rhomathfrak{m})=H^{1,p}_{rho}(X,mathrm{d},mathfrak{% m})} . We also adapt the results in [23] and in the recent paper [27] to the metric measure setting, considering appropriate conditions on ρ that ensure the equality <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msubsup> <m:mi>W</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:mi>ρ</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>𝔪</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {W^{1,p}_{rho}(X,mathrm{d},mathfrak{m})=H^{1,p}_{rho}(X,mathrm{d},% mathfrak{m})} ." @default.
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• W2962960686 date "2016-05-18" @default.
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• W2962960686 title "Weighted Sobolev spaces on metric measure spaces" @default.
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