Square area integral estimates for subharmonic functions in NTA domains

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic functionu in a bounded NTA domainD inR ⁿ . We achieve this by showing that a weighted Riesz measure ofu is a Carleson measure, with the Carleson norm bounded by a constant independent ofu. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions inD.     

Carleson embedding theorem on convex finite type domains

This paper concerns certain geometric aspects of function theory on smoothly bounded convex domains of finite type in Cn. Specifically, we prove the Carleson-Hörmander inequality for this class of domains and provide examples of Carleson measures improving a known result concerning such measures associated to bounded holomorphic functions.     

Existenz und Eindeutigkeit für Lösungen des Neumann—Tricomi-Problems im ℝ2

In a bounded simply-connected domainG \( \subseteq \) ?² a boundary value problem for a linear partial differential equation of second orderLu=f is studied. The operatorL is elliptic inG?{y>0}, parabolic forG?{y=0} and hyperbolic inG?{y<0}. The boundary value problem consists in findingu satisfyingLu=f inG, d _n u=φ on the elliptic part of the boundary ofG, u=ψ on the noncharacteristic part (which is not empty) of the hyperbolic part of the boundary ofG.d _n u denotes the conormal (with respect toL) derivative ofu. It is proved that the problem has a generalized solution in anL ²-weight space. Uniqueness is otained in the class of quasiregular solutions. In order to get the results apriori estimates are proved; theorems from functional analysis are used.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

Invariant subspaces in Bergman spaces and Hedenmalm's boundary value problem

A functionG in a Bergman spaceA ^p , 0<p<∞, in the unit diskD is calledA ^p -inner if |G| ^p ?1 annihilates all bounded harmonic functions inD. Extending a recent result by Hedenmalm forp=2, we show (Thm. 2) that the unique compactly-supported solution Φ of the problem $$\Delta \Phi = \chi _D (|G|^p - 1),$$ where χ _D denotes the characteristic function ofD andG is an arbitraryA ^p -inner function, is continuous inC, and, moreover, has a vanishing normal derivative in a weak sense on the unit circle. This allows us to extend all of Hedenmalm's results concerning the invariant subspaces in the Bergman spaceA ² to a generalA ^p -setting.     

Characterization of Carleson measures by the Hausdorff-Young property

It is shown that the Laplace transform of an L ^p (1 < p ≤ 2) function defined on the positive semiaxis satisfies the Hausdorff-Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure. In addition, Carleson’s weighted L ^p inequality for the harmonic extension is given with a numeric constant.     

Sur la mesure de sortie du super mouvement brownien

Summary We study some properties of the exit measure of super Brownian motion from a smooth domainD inR ^d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of u=u ² inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of u=u ² inD that tends to on an open subsetO of D and to 0 on the complement in D of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

It is proved that if Ω ⊂ Rⁿ {R^n}  is a bounded Lipschitz domain, then the inequality || u ||₁ \leqslant c(n)\textdiam( W)ò_W | e^D(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here e^D(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and ò_W | e^D(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure e^D(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.     

Quasi-Nearly Subharmonic Functions and Quasiconformal Mappings

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mapping of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if u?°?f is quasi-nearly subharmonic for all quasi-nearly subharmonic u and f satisfying some additional conditions, then f is quasiconformal. Similar results are further established for the class of regularly oscillating functions.     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

W 2,1 regularity for solutions of the Monge–Ampère equation

In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampère equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D ² u, namely D ² u∈Llog ^k L for any k∈?.     

On the solutions to certain laplace inequalities with applications to geometry of submanifolds

LetM be a complete Riemannian manifold with Ricci curvature bounded from below. We give an explicit estimate for the size of the negative sets of solutions to the differential inequality Δu ≥λu where Δ is the Laplacian and λ is a negative constant. This inequality arises naturally when we study the lengthH of the mean curvature of an isometric immersionf of M into another Riemannian manifoldN with curvature bounded above by some constantκ. Suppose that the image f(M) does not meet the cut locus of some pointo ∈ N. As a consequence of our estimate, we prove that, givenρ > 0, if supH is less than a certain explicit expression μ(κ, ρ) inρ andκ on any domainU that contains an inscribed ball of radius greater than an explicitly computable numberR, then the diameter of the setf(U) inN must exceed 2ρ. Moreover, if supH = μ(κ, ρ) onM and the diameter off(M) inN equals 2ρ, thenf is a minimal immersion into a distance sphere of radiusρ inN.     

Generalized Cranston-McConnell inequalities and Martin boundaries of unbounded domains

LetD be a domain inR ² with the Green functionG(x, y) for the Laplace equation. We give a generalization of the Cranston-McConnell inequality concerning the integrability of positive harmonic functions onD. A typical new inequality is {fx137-01} whereu andv ₁,..., v_nare positive superharmonic functions onD andc _nis a constant depending only onn. The generalized Cranston-McConnell inequality is used for the determination of the Martin boundary of a certain unbounded domain.     

On the inverse problem of the product of a semi-classical form by a polynomial

A form (linear functional) u is called regular if there exists a sequence of polynomials {P_n}_n⩾0, deg P_n=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials Φ and Ψ such that D(Φu) + Ψu = 0, where D designs the derivative operator.On certain regularity conditions, the product of a semi-classical form by a polynomial, gives a semi-classical form. In this paper, we consider the inverse problem: given a semi-classical form v, find all regular forms u which satisfy the relation x²u = −λv, $\text{λ ∈}\text{C}{}^1$. We give the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relatively to u. An example is treated with a nonsymmetric form v.     

Estimates for k-Hessian operator and some applications

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F _k [u] = 0, where F _k [u] is the elementary symmetric function of order k, 1 ? ? 6 n, of the eigenvalues of the Hessian matrix D ² u. For example, F ₁[u] is the Laplacian Δu and F _n [u] is the real Monge-Ampère operator detD ² u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k-convex functions, and give several estimates for the mixed k-Hessian operator. Applications of these estimates to the k-Green functions are also established.     

Weighted bounds for the carleson maximal operator inR n

The Carleson maximal operator is shown to be bounded inL ^p (w) for certain values ofp and certain radial weightsw when acting on products of radial functions and homogeneous harmonic polynomials. Partially supported by DGICYT (MEC Spain). PB 92/187.     

Bergman Spaces and Carleson Measures on Homogeneous Isotropic Trees

Hastings studied Carleson measures for non-negative subharmonic functions on the polydisk and characterized them by a certain geometric condition relative to Lebesgue measure σ. Cima & Wogen and Luecking proved analogous results for weighted Bergman spaces on the unit ball and other open subsets of \(\mathbb {C}^{n}\). We consider a similar problem on a homogeneous tree, and study how the characterization and properties of Carleson measures for various function spaces depend on the choice of reference measure σ.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Interpolating and orthogonal polynomials on fractals

A special class of closed subsetsF ofR ⁿ , referred to as sets preserving Markov's inequality, are considered. Typically,F may be a fractal such as the Cantor set or von Koch's curve, butF may also be a closed Lipschitz domain. We investigate interpolation to smooth functions onF where the points of interpolation belong toF. We also consider orthogonal polynomials onF∩B, whereB is a ball with center inF, and their relation to spaces of smooth functions onF.