New characterizations of first order Sobolev spaces

Abstract We provide new characterizations of Sobolev spaces that are true under some mild conditions. We study modified first order Sobolev spaces on metric measure spaces: $${{\,\mathrm{\textrm{TC}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>TC</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> -Newtonian space, $$\hat{{{\,\mathrm{\textrm{TC}}\,}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mspace/> <mml:mtext>TC</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>^</mml:mo> </mml:mover> </mml:math> -Newtonian space, and Ambrosio–Gigli–Savaré-like space. We prove that if the measure is Borel regular and $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -finite, then the modified $${{\,\mathrm{\textrm{TC}}\,}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>TC</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> -Newtonian space is equivalent to the Hajłasz–Sobolev space. Moreover, if additionally the measure is doubling then all modified spaces are equivalent to the Hajłasz–Sobolev space.