1,721,003 research outputs found
On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb
Full text linkIn 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order Ws; p. The case p 1⁄4 2 of their result implies the nonlocal isoperimetric inequality PsEÞ jEj N2s N b PsB1Þ jB1j N_2s N ; 0 < s < 1=2; where Ps indicates the fractional s-perimeter, and B1 is the unit ball in RN. In this note we explicitly compute the best constant, and show that for any 0 < s < 1=2, one has PsB1Þ jB1j N2s N 1⁄4 NpN 2 þsG12sÞ sGN 2 þ 12s NG1sÞGNþ22s 2:
Unique Continuation for Nonnegative Solutions of Schrodinger Operators
No full textChiarenza, F.; Garofalo, N.. (1984). Unique Continuation for Nonnegative Solutions of Schrodinger Operators. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2580
Two classical properties of the bessel quotient iν+1 /iν and their implications in pde’s
No full textTwo elementary and classical results about the Bessel quotient yν =I ν+1 state that on the half-line (0, ∞) one has for ν ≥ −1/2: Iν (i) 0 < yν < 1; (ii) yν is strictly increasing. In this paper we show that (i) and (ii) have some nontrivial and interesting applications to pde’s. As a consequence of them, we establish some sharp new results for a class of degenerate partial differential equations of parabolic type in Rn+1 + × (0, ∞) which arise in connection with the analysis of the fractional heat operator (∂t − Δ)s in Rn × (0, ∞), see Theorems 1.2, 1.4, 1.5 and 1.7 below
Feeling the heat in a group of Heisenberg type
No full textIn this paper we use the heat equation in a group of Heisenberg type G to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators Ls and Ls,
A heat equation approach to intertwining
No full textIn this paper we present a new approach based on the heat equation and extension problems to some intertwining formulas arising in conformal CR geometry
Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators
No full textIn this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on Brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case
A Class of Nonlocal Hypoelliptic Operators and their Extensions
Full text linkIn this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form Ku = Au -partial_t u = tr(Q nabla^2 u) + - partial_t u, introduced by Hormander in his 1967 hypoellipticity paper. We show that the nonlocal operators (-K)^s , (-A)^s can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in L^infty, in L^p for 1 = 0. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities
Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups
Full text linkIn his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting
Functional inequalities for a class of nonlocal hypoelliptic equations of Hörmander type
No full textWe 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 analyse 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
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