Resumen de Sobolev inequalities: isoperimetry and symmetrization

Walter Andrés Ortiz Vargas

• The first part of the thesis is devoted to obtain a Sobolev type embedding result for Besov spaces defined on a doubling metric space. This will be done by obtaining pointwise estimates between the special difference (called oscillation of and the X−modulus of smoothness defined by EX(f,r) ∶= ∥∫−B(x,r) ∣f(x) − f(y)∣dµ(y)∥X .

  is the decreasing rearrangement of for all t > 0 and X a rearrangement invariant space on Ω.

  In the second part of the thesis, to obtain symmetrization inequalities on probability metric spaces that admit a convex isoperimetric estimator which incorporate in their formulation the isoperimetric estimator and that can be applied to provide a unified treatment of sharp Sobolev-Poincar´e and Nash type inequalities.