Two classical properties of the Bessel
                    quotient 𝐼_{𝜈+1}/𝐼_{𝜈} and their implications in
                    pde’s

Garofalo, Nicola

doi:10.1090/conm/748/15056

Cited by 9 publications

(2 citation statements)

References 42 publications

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“…The final statement of the theorem follows from the observation that when z = 0, then in [22,Proposition 4.1,(4.3)] it is shown that for any a > −1, one has…”

Section: Functional Inequalities Etcmentioning

confidence: 95%

Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type

Garofalo

Tralli

2020

Nonlinear Analysis

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We consider a class of second-order partial differential operators A of Hörmander type, which contain as a prototypical example a well-studied operator introduced by Kolmogorov in the '30s. We analyze some properties of the nonlocal operators driven by the fractional powers of A , and we introduce some interpolation spaces related to them. We also establish sharp pointwise estimates of Harnack type for the semigroup associated with the extension operator. Moreover, we prove both global and localised versions of Poincaré inequalities adapted to the underlying geometry.= tr(Q∇ 2 u)+ < BX, ∇u > . 1 2 FUNCTIONAL INEQUALITIES ETC.Here, we have denoted by X the variable in R N (N ≥ 2), whereas Q and B indicate two given N × N matrices with real constant coefficients. For a N × N matrix A the notation tr A indicates the trace of A, A * the transpose of A, ∇ 2 u the Hessian matrix of u.The aim of the present note is to complement the above cited works, as well as our work in preparation [26], and also establish two results of independent interest. We remark that when Q = I N and B = O N in (1.2), then A = ∆ and (1.1) gives that K = ∆ − ∂ t is the standard heat operator in R N +1 . Although we will at times refer to this classical non-degenerate case for comparison or illustrative purposes, our primary focus is the genuinely degenerate setting in which Q = Q * ≥ 0, and B = O N . In such framework, the class (1.1) encompasses various evolution equations of interest in mathematics and physics.Perhaps the best known example dates back to Kolmogorov's 1934 note [33] on Brownian motion and the theory of gases, and it is given byOther examples of degenerate equations in the form (1.1) of interest in physics were studied in [15]. We emphasise that the operator K 0 badly fails to be parabolic since it is missing the diffusive term ∆ x u. Nonetheless, it is hypoelliptic. This remarkable fact was proved by Kolmogorov himself, who found the following explicit fundamental solution p 0 (X, Y, t) = c n t 2n exp −

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