  The recent analysis of low-redshift supernovae (SN) has increased the
apparent tension between the value of $H_0$ estimated from low and high
redshift observations such as the cosmic microwave background (CMB) radiation.
At the same time other observations have provided evidence of the existence of
local radial inhomogeneities extending in different directions up to a redshift
of about $0.07$. About $40\%$ of the Cepheids used for SN calibration are
directly affected because are located along the directions of these
inhomogeneities. We derive a new simple formula relating directly the
luminosity distance to the monopole of the density contrast, which does not
involve any metric perturbation. We then use it to develop a new inversion
method to reconstruct the monopole of the density field from the deviations of
the redshift uncorrected observed luminosity distance respect to the $\Lambda
CDM$ prediction based on cosmological parameters obtained from large scale
observations.
  The inversion method confirms the existence of inhomogeneities whose effects
were not previously taken into account because the $2M++$ density field maps
used to obtain the peculiar velocity for redshift correction were for $z\leq
0.06$, which is not a sufficiently large scale to detect the presence of
inhomogeneities extending up to $z=0.07$. The inhomogeneity does not affect the
high redshift luminosity distance because the volume averaged density contrast
tends to zero asymptotically, making the value of $H_0^{CMB}$ obtained from CMB
observations insensitive to any local structure. The inversion method can
provide a unique tool to reconstruct the density field at high redshift where
only SN data is available, and in particular to normalize correctly the density
field respect to the average large scale density of the Universe.
