Weighted Calderón–Zygmund and Rellich inequalities in $$L^p$$ L p

Abstract

We find necessary and sufficient conditions for the validity of weighted Rellich and Calderón-Zygmund inequalities in L p , 1 ≤ p ≤ ∞, in the whole space and in the half-space with Dirichlet boundary conditions. General operators like L = ∆ + c x |x| 2 · ∇ − b |x| 2 are considered. We compute best constants in some situations.Mathematics subject classification (2010): 26D10, 35PXX, 47F05.

Rellich inequalities in bounded domains

“…If Ω does not contain the origin the constant C above is not optimal, in general, see again [24,Section 6] for the case of the half space.…”

“…In particular they proved that (3) is verified for p = 2 if and only if α = N/2 + n, α = −N/2 + 2 − n for every n ∈ N 0 . In [24] the results in [7] are extended to 1 ≤ p ≤ ∞, computing also best constants in some cases. It is shown that (3) holds if and only if α = N/p ′ + n, α = −N/p + 2 − n for every n ∈ N 0 .…”

“…This proof allows also to compute the best constant C := b + N p − 2 + α N p ′ − α + c . For the other values of α appearing in (4), the best constant is unknown unless p = 2, see [7], [24], or when p is generic but special subspaces of L p are considered, see [24].…”

“…In the range (6), Rellich inequalities have essentially a one dimensional structure, since the (approximate) extremants are radial functions and best constants can therefore be computed. Outside of this range, however, the problem loses its rotational symmetry and the extremants, in special subspaces, involve spherical harmonics, see [24], again. This explains also why symmetrization arguments based on spherical rearrangements do not work and a spectral analysis appears.…”

Kernel estimates for Schrödinger type operators with unbounded coefficients and critical exponents

“…The paper [4] deals with the case β > α − 2 obtaining similar results on the generated semigroup. The critical homogeneous case + c x |x| 2 .∇ − b |x| 2 has been investigated in [16]. In [5,6] heat kernel estimates have been given in the case α ≥ 2 and β > α − 2.…”