Pablo Raúl Stinga

Extension Problem and Harnack's Inequality for Some Fractional Operators

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy–Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (− Δ + |x|2)σ to deduce a Harnacks inequality. A pointwise formula for H σ f(x) and some maximum and comparison principles are derived.

Transference of Fractional Laplacian Regularity

In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus \(\mathbb{T}^{n}\) from the fractional Laplacian on \(\mathbb{R}^{n}\). Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that L 2 functions on the torus cannot be identified with L 2 functions on \(\mathbb{R}^{n}\). The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise formulas and Holder regularity estimates.

Full length article: The fractional Bessel equation in Hölder spaces

Motivated by the Poisson equation for the fractional Laplacian on the whole space with radial right hand side, we study global Holder and Schauder estimates for a fractional Bessel equation. Our methods stand on the so-called semigroup language. Indeed, by using the solution to the Bessel heat equation we derive pointwise formulas for the fractional operators. Appropriate Holder spaces, which can be seen as Campanato-type spaces, are characterized through Bessel harmonic extensions and fractional Carleson measures. From here the regularity estimates for the fractional Bessel equations follow. In particular, we obtain regularity estimates for radial solutions to the fractional Laplacian.

The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity

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Harnack inequality for the fractional nonlocal linearized Monge–Ampère equation

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Fractional elliptic equations, Caccioppoli estimates and regularity

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Fractional Laplacian on the torus

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Extension problem and fractional operators: semigroups and wave equations

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Maximum principles, extension problem and inversion for nonlocal one-sided equations☆

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Fractional semilinear Neumann problems arising from a fractional Keller–Segel model

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