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 x = Table[i, {i, 1, 10}];
 y = Table[i, {i, 1, 10, 1}]
  Datah7 = Transpose@{x, y};
  h7 = ListLogLogPlot[Datah7, Frame -> True, 
    FrameLabel -> {"\[Omega]", 
     "\!\(\*OverscriptBox[SubscriptBox[\(T\), \(\(0\)\(,\)\(c\)\(\\\ \
  \)\)], \(_\)]\) - \!\(\*OverscriptBox[SubscriptBox[\(T\), \(c\)], \(_\
   \)]\)"}, FrameStyle -> Thickness[.003], 
   FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, 
   LabelStyle -> Directive[Black, Bold, Small], PlotStyle -> Blue, 
    PlotMarkers -> "OpenMarkersThick"]

I simply want to reduce the space between the x-axis and x-axes labels similarly to reduce space between y-axis and y axes labels.

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One option is to put the labels as Text in an Epilog

x = Range[10];
y = Range[10];
Datah7 = Transpose@{x, y};
h7 = ListLogLogPlot[Datah7, Frame -> True, 
  FrameStyle -> Thickness[.003], 
  FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, 
  LabelStyle -> Directive[Black, Bold, Small], PlotStyle -> Blue, 
  PlotMarkers -> "OpenMarkersThick",
  Epilog -> {Text["\[Omega]", Scaled[{0.5, -0.05}]], 
    Text[Rotate[
      "\!\(\*OverscriptBox[SubscriptBox[\(T\), \(\(0\)\(,\)\(c\)\(\\\ \
  \)\)], \(_\)]\) - \!\(\*OverscriptBox[SubscriptBox[\(T\), \(c\)], \
\(_   \)]\)", 90 Degree], Scaled[{-0.05, 0.5}]]},
  PlotRangeClipping -> False,
  ImagePadding -> {{23, 5}, {15, 5}},
  ImageMargins -> 0]

enter image description here

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