Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all ${p \in (1,\infty)}$ , but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space ${\mathrm{C}(\overline{\Omega})}$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in ${\mathrm{C}(\overline{\Omega})}$ is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on ${\mathrm{C}(\overline{\Omega})}$ .     

Partial regularity for nonlinear elliptic systems with continuous growth exponent

For weak solutions of nonlinear elliptic systems of the type ${- {\rm div}a(x, u(x), Du(x)) = 0,}$ with nonstandard p(x) growth, we show interior partial Hölder continuity for any Hölder exponent ${\alpha \in (0,1)}$ , provided that the exponent function is ‘logarithmic Hölder continuous’. The result also covers the up to now open partial regularity for systems with constant growth with exponent p less than two in the case of merely continuous dependence on the spacial variable x.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

The Hölder continuity of the solution map to the b-family equation in weak topology

We prove that the solution map of the $b$ -family equation is Hölder continuous as a map from a bounded set of $H^s(\mathbb{R }), s>\frac{3}{2}$ with $H^r(\mathbb{R })$ ( $0\le r<s$ ) topology, to $C([0, T], H^r(\mathbb{R }))$ for some $T>0$ . Moreover, we show that the obtained exponent of the Hölder continuity is optimal when $s-1<r<s$ .     

A Note on Variable Exponent Hörmander Spaces

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that ${{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}$ is isomorphic to ${{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}$ (Ω open set in ${{\mathbb{R}^n, p? > 1}}$ and the Hardy–Littlewood maximal operator M is bounded in ${L_p(\cdot))}$ extending a Hörmander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.     

Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces

In this paper, we give new results about existence, uniqueness and regularity properties for solutions of Laplace equation $$\Delta u = h \quad {\rm in} \, \Omega$$ where Ω is a cusp domain. We impose nonhomogeneous Dirichlet conditions on some part of ?Ω. The second member h will be taken in the little Hölder space ${h^{2 \sigma}(\bar{\Omega})}$ with ${\sigma \, \in \, ]0, \, 1/2[}$ . Our approach is based essentially on the study of an abstract elliptic differential equation set in an unbounded domain. We will use the continuous interpolation spaces and the generalized analytic semigroup theory.     

Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ .     

Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Let $\mathcal{X}$ be a metric space with doubling measure and L a nonnegative self-adjoint operator in $L^{2}(\mathcal{X})$ satisfying the Davies–Gaffney estimates. Let $\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such that φ(x,?) is an Orlicz function, $\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2?I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space $H_{\varphi,L}(\mathcal{X})$ , by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space $\mathrm{BMO}_{\varphi,L}(\mathcal{X})$ , which is further proved to be the dual space of $H_{\varphi,L}(\mathcal{X})$ and hence whose φ-Carleson measure characterization is deduced. Characterizations of $H_{\varphi,L}(\mathcal{X})$ , including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize $H_{\varphi,L}(\mathcal{X})$ in terms of the Littlewood–Paley $g^{\ast}_{\lambda}$ -function $g^{\ast}_{\lambda,L}$ and establish a Hörmander-type spectral multiplier theorem for L on $H_{\varphi,L}(\mathcal{X})$ . Moreover, for the Musielak–Orlicz–Hardy space H _φ,L (? ⁿ ) associated with the Schrödinger operator L:=?Δ+V, where $0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ , the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ?L ^?1/2 is bounded from H _φ,L (? ⁿ ) to the Musielak–Orlicz space L ^φ (? ⁿ ) when i(φ)∈(0,1], and from H _φ,L (? ⁿ ) to the Musielak–Orlicz–Hardy space H _φ (? ⁿ ) when $i(\varphi)\in(\frac{n}{n+1},1]$ , where i(φ) denotes the uniformly critical lower type index of φ.     

Unitary Equivalence of Proper Extensions of a Symmetric Operator and the Weyl Function

Let A be a densely defined simple symmetric operator in ${\mathfrak{H}}$ , let ${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$ be a boundary triplet for A ^* and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A ₀}, uniquely up to the unitary similarity. Here ${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric ${A \subset A^*}$ and a special boundary triplet ${\widetilde{\Pi}}$ for{A, A} such that the corresponding Weyl function is ${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$ , where B is a non-self-adjoint bounded operator in ${\mathcal{H}}$ . We are interested in the problem whether the result on the unitary similarity remains valid for ${\widetilde{M}(\cdot)}$ in place of M(·). We indicate some sufficient conditions in terms of the operators A ₀ and ${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in ${L^2(\mathbb{R}_+)}$ , we show that ${\widetilde{M}(\cdot)}$ defined in ${\Omega_+ \subset \mathbb{C}_+}$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function ${\widetilde{M}(\cdot)}$ does not determine A _B even up to the similarity.     

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)² is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L _V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V ₀ ≤ V ₁ and ${\mathcal{L}}$ a second order elliptic operator.     

Equilibrium measures on saddle sets of holomorphic maps on \mathbb{P }^2

We consider the case of hyperbolic basic sets $\Lambda $ of saddle type for holomorphic maps $f:{\mathbb{P }}^2{\mathbb{C }}\rightarrow {\mathbb{P }}^2{\mathbb{C }}$ . We study equilibrium measures $\mu _\phi $ associated to a class of Hölder potentials $\phi $ on $\Lambda $ , and find the measures $\mu _\phi $ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $\delta _{\mu _\phi }$ of $\mu _\phi $ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $\mu _\phi $ in the case when the preimage counting function is constant on $\Lambda $ . For terminal/minimal saddle sets we prove that an invariant measure $\nu $ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the restriction $f|_\Lambda $ . This allows then to obtain formulas for the measure $\nu $ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $\nu $ .     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Separable forms of {{\mathbb{G}}_m} -ACTIONS ON {{\mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ and show that they are linearizable. In particular, $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ are linearizable, and k-forms of $ {\mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.     

On Weak*-Convergence in \boldsymbol{H^1_L(\boldsymbol{\mathbb R}^d)}

Let L?=???Δ?+?V be a Schrödinger operator on $\mathbb R^d$ , d?≥?3, where V is a nonnegative function, $V\ne 0$ , and belongs to the reverse Hölder class RH _d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space $H^1_L(\mathbb R^d)$ .     

Tangent-Point Repulsive Potentials for a Class of Non-smooth m-dimensional Sets in ℝ n . Part I: Smoothing and Self-avoidance Effects

We consider repulsive potential energies $\mathcal {E}_{q}(\Sigma)$ , whose integrand measures tangent-point interactions, on a large class of non-smooth m-dimensional sets Σ in ? ⁿ . Finiteness of the energy $\mathcal {E}_{q}(\Sigma)$ has three sorts of effects for the set Σ: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of Σ onto suitable m-planes and therefore large m-dimensional Hausdorff measure of Σ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey–Sobolev embedding theorem: Any admissible set Σ with finite $\mathcal {E}_{q}$ -energy, for any exponent q>2m, is, in fact, a C ¹-manifold whose tangent planes vary in a Hölder continuous manner with the optimal Hölder exponent μ=1?(2m)/q. Moreover, the patch size of the local C ^1,μ -graph representations is uniformly controlled from below only in terms of the energy value $\mathcal {E}_{q}(\Sigma)$ .     

Singular Integral Operators on an Open Arc in Spaces with Weight

The aim of this work is to study a singular integral operator ${\mathbf{A}=aI+bS_\Gamma}$ A = a I + b S Γ with the Cauchy operator S _Γ (SIO) and Hölder continuous coefficients a, b in the space ${\mathbb{H}^0_\mu(\Gamma,\rho)}$ H μ 0 ( Γ , ρ ) of Hölder continuous functions with an power “Khvedelidze” weight. The underlying curve is an open arc. It is well known, that such operator is Fredholm if and only if, along with the ellipticity condition ${a^2(t)-b^2(t)\not=0,\, t\in\Gamma}$ a 2 ( t ) - b 2 ( t ) ≠ 0 , t ∈ Γ , the “Gohberg–Krupnik arc condition” is fulfilled (see Duduchava, in Dokladi Akademii Nauk SSSR 191:16–19, 1970). Based on the Poincare–Beltrami formula for a composition of singular integral operators and the celebrated Muskhelishvili formula describing singularities of Cauchy integral, the formula for a composition of weighted singular integral operators is proved. Using the obtained composition formula and the localization, the Fredholm criterion of the SIO is derived in a natural way, by looking for the regularizer of the operator ${\mathbf{A}}$ A and equating to 0 non-compact operators. The approach is space-independent and this is demonstrated on similar results obtained for SIOs with continuous coefficients in the Lebesgue spaces with a “Khvedelidze” weight ${\mathbb{L}_p(\Gamma,\rho)}$ L p ( Γ , ρ ) , investigated earlier by Gohberg and Krupnik (Studia Mathematica 31:347–362, 1968; One Dimensional Singular Integral Operators II, Operator Theory, Advances and Applications, vol. 54, chapter IX, 1979) with a different approach.     

Symmetry and quasi-centrality of the operator space projective tensor product

For C*-algebras A and B, the operator space projective tensor product ${A\widehat{\otimes}B}$ and the Banach space projective tensor product ${A\otimes_{\gamma}B}$ are shown to be symmetric. We also show that ${A\widehat{\otimes}B}$ is a weakly Wiener algebra. Finally, quasi-centrality and the unitary group of ${A\widehat{\otimes}B}$ are discussed.     

Bernoulli coding map and almost sure invariance principle for endomorphisms of {\mathbb P^k}

Let f be an holomorphic endomorphism of ${\mathbb{P}^k}$ and μ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems ${(\mathbb{P}^k,f,\mu)}$ . Our class ${\mathcal {U}}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map ${\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)}$ . We obtain the invariance principle for an observable ψ on ${(\mathbb{P}^k,f,\mu)}$ by applying Philipp–Stout’s theorem for ${\chi = \psi \circ \omega}$ on (Σ, s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class ${\mathcal {U}}$ . As an application, we give a direct proof of the absolute continuity of the measure μ when it satisfies Pesin’s formula. This approach relies on the central limit theorem for the unbounded observable log ${{\tt Jac}\, f \in \mathcal{U}}$ .     

Stochastics and thermodynamics for equilibrium measures of holomorphic endomorphisms on complex projective spaces

It was proved by Urbański and Zdunik (Fund Math 220:23–69, 2013) that for every holomorphic endomorphism $f:{{\mathbb { P}}}^k\rightarrow {{\mathbb { P}}}^k$ of a complex projective space ${{\mathbb { P}}}^k,k\ge 1$ , there exists a positive number $\kappa _f>0$ such that if $J$ is the Julia set of $f$ (i.e. the support of the maximal entropy measure) and $\phi :J\rightarrow {\mathbb R}$ is a Hölder continuous function with $\sup (\phi )-\inf (\phi )<\kappa _f$ (pressure gap), then $\phi $ admits a unique equilibrium state $\mu _\phi $ on $J$ . In this paper we prove that the dynamical system ( $f,\mu _\phi $ ) enjoys exponential decay of correlations of Hölder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. We also show that the topological pressure function $t\mapsto P(t\phi )$ is real-analytic throughout the open set of all parameters $t$ for which the potentials $t\phi $ have pressure gaps.