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In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x0 in D, then is D "almost'' a ball with center x0? This is the stability counterpart of the rigidity question (the statement above, without the two "almost'') for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x0.

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at “$t=- \infty$” for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.

We consider degenerate Kolmogorov–Fokker–Planck operators L u = ∑ i , j = 1 m 0 a ij (x , t) ∂ x i x j 2 u + ∑ k , j = 1 N b jk x k ∂ x j u - ∂ t u ≡ ∑ i , j = 1 m 0 a ij (x , t) ∂ x i x j 2 u + Y u (with (x , t) ∈ R N + 1 and 1 ≤ m 0 ≤ N ) such that the corresponding model operator having constant a ij is hypoelliptic, translation invariant w.r.t. a Lie group operation in R N + 1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix (a ij) i , j = 1 m 0 is symmetric and uniformly positive on R m 0 . The coefficients a ij are bounded and Dini continuous in space, and only bounded measurable in time. This means that, setting (i) S T = R N × - ∞ , T , (ii) ω f , S T (r) = sup (x , t) , (y , t) ∈ S T ‖ x - y ‖ ≤ r | f (x , t) - f (y , t) | (iii) ‖ f ‖ D (S T) = ∫ 0 1 ω f , S T (r) r d r + ‖ f ‖ L ∞ S T we require the finiteness of ‖ a ij ‖ D (S T) . We bound ω u x i x j , S T , ‖ u x i x j ‖ L ∞ (S T) ( i , j = 1 , 2 ,... , m 0 ), ω Y u , S T , ‖ Y u ‖ L ∞ (S T) in terms of ω L u , S T , ‖ L u ‖ L ∞ (S T) and ‖ u ‖ L ∞ S T , getting a control on the uniform continuity in space of u x i x j , Y u if L u is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients a ij and L u are log-Dini continuous, meaning the finiteness of the quantity ∫ 0 1 ω f , S T r r log r d r , we prove that u x i x j and Yu are Dini continuous; moreover, in this case, the derivatives u x i x j are locally uniformly continuous in space and time. [ABSTRACT FROM AUTHOR]

Let \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential \Upsilon belonging to the reverse Hölder class B_{Q/2}, where Q is the homogeneous dimension of the stratified Lie group \mathbb {G}. Inspired by Shen's pioneer work and Li's work, we study fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \mathcal {L}-harmonic functions on \mathbb {G}. [ABSTRACT FROM AUTHOR]

We prove the well-posedness of a Cauchy problem of the kind: { L u = f , in D ′ (R N × (0 , + ∞)) , u (x , 0) = g (x) , ∀ x ∈ R N , where f is Dini continuous in space and measurable in time and g satisfies suitable regularity properties. The operator L is the degenerate Kolmogorov-Fokker-Planck operator L = ∑ i , j = 1 q a i j (t) ∂ x i x j 2 + ∑ k , j = 1 N b k j x k ∂ x j − ∂ t where { a i j } i j = 1 q is measurable in time, uniformly positive definite and bounded while { b i j } i j = 1 N have the block structure: { b i j } i j = 1 N = ( O ... O O B 1 ... O O ⋮ ⋱ ⋮ ⋮ O ... B κ O ) which makes the operator with constant coefficients hypoelliptic, 2-homogeneous with respect to a family of dilations and traslation invariant with respect to a Lie group. We prove the well-posedness of a Cauchy problem of the kind: where is Dini continuous in space and measurable in time and satisfies suitable regularity properties. The operator is the degenerate Kolmogorov-Fokker-Planck operator where is measurable in time, uniformly positive definite and bounded while have the block structure: which makes the operator with constant coefficients hypoelliptic, 2-homogeneous with respect to a family of dilations and traslation invariant with respect to a Lie group. [ABSTRACT FROM AUTHOR]

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of asymptotic average solutions making pointwise solvable every Poisson equation ℒ u (x) = − f (x) with continuous data f, where ℒ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form. [ABSTRACT FROM AUTHOR]

The Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x0 be a point of D. If u(x0) = ∫Du(y)dy for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x0. On the other hand, on every sufficiently smooth domain D and for every point x0 in D there exist Radon measures μ such that u(x0) = ∫Du(y)dμ(y) for every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D. [ABSTRACT FROM AUTHOR]

In this paper we reconsider the original Kolmogorov normal form algorithm [26] with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the normalization procedure. Besides, we select the frequencies of the final invariant torus and determine a posteriori the corresponding starting ones. In particular, we replace the classical translation step with a change of the frequencies. The algorithm is based on the original scheme of Kolmogorov, thus exploiting the fast convergence of the Newton-Kantorovich method. [ABSTRACT FROM AUTHOR]

Let \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V belonging to the reverse Hölder class B_{Q}, where Q is the homogeneous dimension of the Heisenberg group \mathbb {H}^n. In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to \mathcal {L}. As an application, we characterize the space BMO_{\mathcal {L}}(\mathbb {H}^n), associated to the Schrödinger operator \mathcal {L}, in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup \{e^{-s\mathcal {L}}\}_{s>0} and the Poisson kernel of the semigroup \{e^{-s\sqrt {\mathcal {L}}}\}_{s>0}, respectively. At last, we pose a conjecture about the converse characterization of BMO_{\mathcal {L}}(\mathbb {H}^n). [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that, for general second order uniformly elliptic equations of non-divergence form in unbounded cylinders with zero boundary condition, the space of solutions with a given exponential growth is of finite dimension. In particular, we generalize a result of Hang and Lin (1999) [ 4 ] in the non-divergence's setting by using a different approach. This allows us to consider the general lower order terms. Furthermore, a sharp estimate of the dimension is established by using the mean value inequality. [ABSTRACT FROM AUTHOR]

Let L be a sub-Laplacian on LN and let G = (LN, ?, d?) be its related homogeneous Lie group. Let E be a Euclidean subgroup of LN such that the orthonormal projection p : G → E is a homomorphism of homogeneous groups, and let 〈, 〈 be an inner product in E. Given a ? E, a ≠ 0, define O(a) := {x ? G : 〈a, p(x)〈 > 0}. We prove the following Liouville-type theorem. If u is a nonnegative L-superharmonic function in O(a) such that u ? L¹(O(a)), then u = 0 in O(a). [ABSTRACT FROM AUTHOR]

By an easy trick taken from caloric polynomial theory, we construct a family B of almost regular domains for the caloric Dirichlet problem. B is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain. [ABSTRACT FROM AUTHOR]

Let G G be a semisimple linear algebraic group defined over rational numbers, K K be a maximal compact subgroup of its real points and Γ Γ be an arithmetic lattice. One can associate a probability measure μ H μ H on Γ ∖ G Γ ∖ G for each subgroup H H of G G defined over Q Q with no non-trivial rational characters. As G acts on Γ ∖ G Γ ∖ G from the right, we can push forward this measure by elements from G G. By pushing down these measures to Γ ∖ G / K Γ ∖ G / K , we call them homogeneous. It is a natural question to ask what are the possible weak- ∗ ∗ limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of Γ ∖ G / K Γ ∖ G / K for H H generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general H H with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on S L n SL n proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words. [ABSTRACT FROM AUTHOR]

In this note we point out and correct a mistake in our paper "GlobalLpestimates for degenerate Ornstein–Uhlenbeck operators with variable coefficients", published in Math. Nachr. 286 (2013), no. 11–12, 1087–1101. [ABSTRACT FROM AUTHOR]

Let D be an open subset of R n with finite measure, and let x 0 ∈ D . We introduce the p-Gauss gap of D w.r.t. x 0 to measure how far are the averages over D of the harmonic functions u ∈ L p (D) from u (x 0) . We estimate from below this gap in terms of the ball gap of D w.r.t. x 0 , i.e., the normalized Lebesgue measure of D \ B , being B the biggest ball centered at x 0 contained in D. From these stability estimates of the mean value formula for harmonic functions in L p -spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space W 1 , p ′ , where p ′ is the conjugate exponent of p. [ABSTRACT FROM AUTHOR]