Local real analysis in locally homogeneous spaces

Abstract

We introduce the concept of locally homogeneous space, and prove in this context L p and C α estimates for singular and fractional integrals, as well as L p estimates on the commutator of a singular or fractional integral with a B M O or V M O function. These results are motivated by local a priori estimates for subelliptic equations.

Lpand Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift

“…However, to prove our L p estimates (1.5), we also need some commutator estimates, of the kind of the well-known result proved by [13], that, as far as we know, are not presently available in the framework of general nondoubling quasimetric (or metric) spaces. For this reason, we have recently developed in [8] a theory of locally homogeneous spaces which is a quite natural framework where all the results we need about singular integrals and their commutators with BMO functions can be proved. To give a unified treatment to both L p and C α estimates, here we have decided to prove both exploiting the results in [8].…”

Fundamental solutions and local solvability for nonsmooth Hörmander’s operators

“…Finally, the explicit expression of the derivatives X i P, X j X i P, X 0 P allows us to repeat the argument used to prove (4. 7), showing that also (4.8) holds. Under the above assumptions and with the above notation, we have:…”

“…This means that in our situation (U, d, dx) is not a space of homogeneous type in the sense of Coifman-Weiss. However, (U, d, dx) fits the assumptions of locally homogeneous spaces as defined in [7]. We will apply some results proved in [7] which assure the local C α continuity of singular and fractional integrals defined by a kernel of the kind a (x) k (x, y) b (y) (with a, b smooth cutoff functions) provided that the kernel k satisfies natural assumptions which never involve integration over domains of the kind B (x, r) ∩ U , but only over balls B (x, r) ⋐ U, which makes our local doubling condition usable.…”

“…However, (U, d, dx) fits the assumptions of locally homogeneous spaces as defined in [7]. We will apply some results proved in [7] which assure the local C α continuity of singular and fractional integrals defined by a kernel of the kind a (x) k (x, y) b (y) (with a, b smooth cutoff functions) provided that the kernel k satisfies natural assumptions which never involve integration over domains of the kind B (x, r) ∩ U , but only over balls B (x, r) ⋐ U, which makes our local doubling condition usable. Before starting the proof of the above theorem we need the following We say that k (x, y) satisfies the standard estimates of singular integrals if the previous estimates hold with ν = 0 and some positive β.…”

“…Proof. By Proposition 5.1 in [7], it is enough to prove the following cancellation property for k 2 : there exists C > 0 such that for a.e. x ∈ B (x, R 0 ) and every ε 1 , ε 2 such that 0 < ε 1 < ε 2 and B (x, ε 2 ) ⊂ U, ε1<d(x,y)<ε2 k 2 (x, y) dy C. To handle C ε1,ε2 (x) we start rewriting We know that the kernel k 2 (x, y) satisfies the standard estimates of singular integrals with exponent β = 1 (see the end of the first part of this proof) and the cancellation property (5.33).…”