uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

tangential boundary limits and exceptional sets for holomorphic functions in dirichlet type spaces according to the well known theorem of fatou the harmonic extension u to the upper half space r x of a function f in lp rn 1 1 in this paper we study the analogous question in the unit ball b of c recall that the correct analogue of fatou s theorem for b is the theorem of kor myi k stating that invariant harmonic extensions poisson szeg6 integrals of functions in lp s 1 p oo have admissible limits at almost every point of the unit sphere s in particular this is true for holomorphic functions in the hardy space hp b as is well known korfinyi s admissible regions allow parabolic tangential approach along the complex tangential directions these admissible regions are best possible as far as the order of tangency to the boundary is concerned even if only bounded holomorphic functions are considered as shown in h s however they are not best possible in a stronger sense su 1 by analogy with the real variable situation it seems reasonable to expect kor myi s regions not to be optimal for holomorphic functions whose admissible boundary values are in some sense bessel type potentials of lp s functions thus we consider holomorphic functions in the unit ball which are given by fractional cauchy integrals of lp s functions these are obtained by modifying the exponent in the denominator of the cauchy kernel of b a procedure suggested by the

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

estimates and structure of α harmonic functions we prove a uniform boundary harnack inequality for nonnegative harmonic functions of the fractional laplacian on arbitrary open set d this yields a unique representation of such functions as integrals against measures on dc satisfying an integrability condition the corresponding martin boundary of d is a subset of the euclidean boundary determined by an integral test

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

weighted norm inequalities for oscillatory integrals with finite type phases on the line abstract we obtain two weighted l 2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type the conditions imposed on the weights involve geometrically defined maximal functions and the inequalities are best possible in the sense that they imply the full l p r l q r mapping properties of the oscillatory integrals our results build on work of carbery soria vargas and the first author

singular solutions for semi linear parabolic equations on nonsmooth domains we prove the existence of positive singular solutions for the semi linear parabolic equation xu tu up 0 on d 0 where p 1 d is a bounded nta domain in rn n 2 and is in a general class of signed radon measures on d covering the elliptic kato class of potentials adopted by zhang and zhao a new proof of the result based on a simple fixed point theorem is also given

boundary behavior of positive solutions of the heat equation on a semi infinite slab on etudie le comportement limite des solutions positives de l equation de la chaleur sur la plaque semi infinie x r n 1 r 0 t on introduit le concept de limite semi fine et on obtient des relations entre les divers types de limites

boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains in a uniform domain we investigate the boundary behavior of positive superharmonic functions u satisfying the nonlinear inequality u x 0 and p 0 where is the laplacian and x is the distance from a point x to the boundary of in particular we present a fatou type theorem concerning the existence of nontangential limits and a littlewood type theorem concerning the nonexistence of tangential limits

global anomalies on orbifolds we consider global anomalies for heterotic string theory formulated on orbifolds the vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies for abelian orbifolds level matching implies these cohomology conditions so suffices for the absence of anomalies for nonabelian orbifolds level matching does not suffice and there are additional constraints we give some examples to illustrate these new constraints

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

multiparameter maximal functions along dilation invariant hypersurfaces consider the hypersurface sn l ll x7 i in rn l the asso ciated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles 0 1 an analogous result is proved for a quadratic surface in r3

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

on the differentiability of lipschitz besov functions lr and ordinary differentiability is proved for functions in the lipschitz besov spaces b q 1 0 using certain maximal operators measuring smoothness these techniques allow also the study of lacunary directional differentiability and of tangential convergence of poisson integrals

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

the fundamental group of affine curves in positive characteristic abstract it is shown that the commutator subgroup of the fundamental group of a smooth irreducible affine curve over an uncountable algebraically closed field k of positive characteristic is a profinite free group of rank equal to the cardinality of k

thin sets at the boundary in his letter 18 hayman raised the following questions if u is negative subharmonic in the unit disk then for almost all u z has a limit as z tends to e nontangentially and outside a near thin set 17 chapter 7 there ought to be a global result i e not in an angle outside a suitable exceptional set what should it be it might be easier for green potentials where the limit is always zero of course the questions also exist in higher dimensions the purpose of this paper is to give an answer to hayman s questions we shall show that there are exceptional sets suitable for the description of especially tangential boundary behavior of superharmonic functions as a result we shall prove that the above nontangential restriction can be relaxed see corollary 1 1 let us first recall the notion of minimal thinness this notion can be defined in a very general framework let d be a martin space with martin boundary md a set e d is called minimally thin at a minimal boundary point x if the reduced function over e of the martin kernel at x is a potential the fatou naim doob theorem describes the boundary behavior of nonnegative superharmonic functions or equivalently nonpositive subharmonic functions with minimally thin sets

extending sobolev functions with partially vanishing traces from locally e δ domains and applications to mixed boundary problems we prove that given any k n for each open set rn and any closed subset d of such that is locally an e domain near d there exists a linear and bounded extension operator ek d mapping for each p 1 the space wdk p into wdk p rn here with o denoting either or rn the space wdk p o is defined as the completion in the classical sobolev space wk p o of restrictions to o of functions from cc rn whose supports are disjoint from d in turn this result is used to develop a functional analytic theory for the class wdk p including intrinsic characterizations boundary traces and extensions results interpolation theorems among other things which is then employed in the treatment of mixed boundary value problems formulated in locally e domains finally we also prove extension results on the scales of besov and bessel potential spaces on e domains with partially vanishing traces on ahlfors regular sets and explore some of the implications of such extension results

explicit construction of nontrivial elements for homotopy groups of classical lie groups nontrivial elements of homotopy groups for unitary orthogonal and symplectic groups are given explicitly in particular a representatives of generators of nontrivial homotopy groups of stable special unitary orthogonal and symplectic groups are constructed using clifford algebras b the values for winding numbers for stable su so and sp are calculated for generators of homotopy groups and c representatives of generators of homotopy groups n 2 o n 1 2n 2 u n 1 4n 2 sp n 1 are given

a gradient bound for free boundary graphs we prove an analogue for a one phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation it follows in particular that every energy minimizing free boundary that is a graph is also smooth the method we use also leads to a new proof of the classical minimal surface gradient bound 2010 wiley periodicals inc

on the picard principle for δ μ given a local kato measure on mathbb r d setminus 0 d ge 2 let mathcal h 0 delta mu u be the convex cone of all continuous real solutions u 0 to the equation u u 0 on the punctured unit ball u satisfying lim x to 1 u x 0 it is shown that mathcal h 0 delta mu u ne 0 if and only if the operator f mapsto int u g cdot y f y d mu y where g denotes the green function on u is bounded on mathcal l 2 u mu and has a norm which is at most one moreover extremal rays in mathcal h 0 delta mu u are characterized and it is proven that satisfies the picard principle on u that is that mathcal h 0 delta mu u consists of one ray provided there exists a suitable sequence of shells in u such that on these shells is either small or not too far from being radial further it is shown that the verification of the picard principle can be localized several results on l2 sub eigenfunctions and 3g inequalities which are used in the paper but may be of independent interest are proved at the end of the paper

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions we show that ii a set e is dispersely decomposed into subsets then the capacity of e is comparable to the summation of tile capacities of the subsets from this fact it is derived that the lebesgue measure of a certain expanded set is estimated by the ca

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

global anomalies and algebraic topology the algebraic topology aspect of the global pure gauge anomaly calculation is investigated in particular the use of a cohomology sequence clarifies the method initiated by witten nucl phys 223 422 433 1983 and elitzur and nair nucl phys b 243 205 1984 examples in su n sp n and so n are discussed

null sets of harmonic measure on nta domains lipschitz approximation revisited we show the david jerison construction of big pieces of lipschitz graphs inside a corkscrew domain does not require surface measure be upper ahlfors regular thus we can study absolute continuity of harmonic measure and surface measure on nta domains of locally finite perimeter using lipschitz approximations a partial analogue of the f and m riesz theorem for simply connected planar domains is obtained for nta domains in space as one consequence every wolff snowflake has infinite surface measure

patching and thickening problems in recent years much progress has been made on the structure of fundamental groups of algebraic curves by means of patching techniques in formal and rigid geometry a number of these results have concerned curves over algebraically closed fields of characteristic p e g the proofs of the abhyankar conjecture ra ha5 and of the geometric case of the shafarevich conjecture po1 ha6 and the realization of galois groups over projective curves sa1 st1 while the rigid approach to patching is often regarded as more intuitive than the formal approach its foundations are less well established but constructions involving the formal approach have tended to be technically more cumbersome the purpose of the current paper is to build on previous formal patching results in order to create a framework in which such constructions are facilitated in the process we prove a result asserting that singular curves over a field k can be thickened to curves over k t with prescribed behavior in a formal neighborhood of the singular locus and similarly for covers of curves afterwards we obtain applications to fundamental groups of curves over large fields the structure of the paper is as follows section 1 concerns patching problems for projective curves x over a power series ring r k t1 tn it is shown theorem 1 that giving a coherent projective module over x is equivalent to giving such modules compatibly on a formal neighborhood of each singular point of the closed fibre x and on the formal thickening along the complement of the singular locus s of x section 2 applies this to thickening problems in both the absolute and relative senses namely it shows theorem 3 that such an x can be constructed from its closed fibre x and from complete local thickenings near s such that x is a trivial deformation away from s moreover theorem 2 given a morphism x z and a thickening z of z such that the local thickenings near s are compatible with that of z there is a unique such thickening x of x that is compatible with the given data in section 3 these results are applied to the problem of thickening and deforming covers theorem 4 there combines the results of sections 1 and 2 to show that covers of reducible k curves can be thickened to covers of curves over k t and theorem 5 then shows how this can be used over large fields to

families of wildly ramified covers of curves in this paper i investigate wildly ramified g galois covers of curves inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i branched at exactly one point over an algebraically closed field k of characteristic p i answer a question of m raynaud by showing that proper families of such covers of a twisted projective line are isotrivial the method is to construct an affine moduli space for covers whose inertia group is of the form inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i there are two other applications of this space in the case that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i the first uses formal patching to compute the dimension of the space of nonisotrivial deformations of in terms of the lower jump of the filtration of higher inertia groups the second gives necessary criteria for good reduction of families of such covers these results will be used in a future paper to prove the existence of such covers with specified ramification data

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

subdyadic square functions and applications to weighted harmonic analysis abstract through the study of novel variants of the classical littlewood paley stein g functions we obtain pointwise estimates for broad classes of highly singular fourier multipliers on r d satisfying regularity hypotheses adapted to fine subdyadic scales in particular this allows us to efficiently bound such multipliers by geometrically defined maximal operators via general weighted l 2 inequalities in the spirit of a well known conjecture of stein our framework applies to solution operators for dispersive pde such as the time dependent free schrodinger equation and other highly oscillatory convolution operators that fall well beyond the scope of the calderon zygmund theory

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

a stein conjecture for the circle we discuss the manner in which one might expect directional maximal functions to control the fourier extension operator via l 2 weighted inequalities we prove a general inequality of this type for the extension operator restricted to circles in the plane

two sided green function estimates for killed subordinate brownian motions a subordinate brownian motion is a levy process that can be obtained by replacing the time of the brownian motion by an independent subordinator the infinitesimal generator of a subordinate brownian motion is where is the laplace exponent of the subordinator in this paper we consider a large class of subordinate brownian motions without diffusion component and with comparable to a regularly varying function at infinity this class of processes includes symmetric stable processes relativistic stable processes sums of independent symmetric stable processes sums of independent relativistic stable processes and much more we give sharp two sided estimates on the green functions of these subordinate brownian motions in any bounded fat open set d w hend is a bounded c 1 1 open set we establish an explicit form of the estimates in terms of the distance to the boundary as a consequence of such sharp green function estimates we obtain a boundary harnack principle in c 1 1 open sets with explicit rate of decay

rigidity reduction and ramification in this paper we consider wildly ramified g galois covers of curves f y 1k branched at exactly one point over an algebraically closed field k of characteristic p for g equal to ap or psl2 p we prove abhyankar s inertia conjecture that all possible inertia groups occur over infinity for such covers f in addition we prove that the set of conductors that can be realized depends on the group the method we use is to compute the reduction of galois covers of mathbb p 1 overline mathbb q branched at 3 points we observe that the existence of covers with given inertia in characteristic p is closely related to the arithmetic of covers in characteristic zero

estimates on green functions and schrodinger type equations for non symmetric diffusions with measure valued drifts abstract in this paper we establish sharp two sided estimates for the green functions of non symmetric diffusions with measure valued drifts in bounded lipschitz domains as consequences of these estimates we get a 3g type theorem and a conditional gauge theorem for these diffusions in bounded lipschitz domains informally the schrodinger type operators we consider are of the form l where l is a uniformly elliptic second order differential operator is a vector valued signed measure belonging to k d 1 and is a signed measure belonging to k d 2 in this paper we establish two sided estimates for the heat kernels of schrodinger type operators in bounded c 1 1 domains and a scale invariant boundary harnack principle for the positive harmonic functions with respect to schrodinger type operators in bounded lipschitz domains

global comparison of perturbed green functions given a green function g e g with respect to 2 0 2 on a region x where g has a generalized triangle property and given a g bounded signed measure on x necessary and sufficient conditions are given for the existence of a perturbed green function which is comparable with g this is done in the general setting of measurable spaces applications to c1 1 regions in rd and for the classical case 2 to finitely connected regions in 2 are given

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

champagne subdomains with unavoidable bubbles abstract a champagne subdomain of a connected open set u 0 in r d d 2 is obtained by omitting pairwise disjoint closed balls b x r x x x the bubbles where x is an infinite locally finite set in u the union a of these balls may be unavoidable that is brownian motion starting in u a and killed when leaving u may hit a almost surely or equivalently a may have harmonic measure 1 for u a recent publications by gardiner and ghergu d 3 and by pres d 2 give rather sharp answers to the question of how small such a set a may be when u is the unit ball in this paper using a totally different approach optimal results are obtained which hold also for arbitrary connected open sets u

boundary tangential convergence on spaces of homogeneous type we study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity our results contain those already known for the euclidean case and give new ones for stratified nilpotent lie groups and for solutions of the dirichlet problem on lipschitz domains

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

on maximal functions and poisson szego integrals we study a class of maximal functions of hardy littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type 1 1 as a consequence we show that poisson szego integrals of lp functions possess certain boundary limits which are not implied by koranyi s theorem introduction the classical theorem of fatou is concerned with the existence of nontangential boundary limits of the poisson integral of a function defined on rn or on the unit circle such limits exist almost everywhere if is in lp rn 1 0 if is locally integrable on rn the maximal function mn is defined by ain io sup j ii

green function estimates for subordinate brownian motions stable and beyond a subordinate brownian motion x is a l evy process which can be obtained by replacing the time of the brownian motion by an independent subordinator in this paper when the laplace exponent phi of the corresponding subordinator satisfies some mild conditions we first prove the scale invariant boundary harnack inequality for x on arbitrary open sets then we give an explicit form of sharp two sided estimates of the green functions of these subordinate brownian motions in any bounded c 1 1 open set as a consequence we prove the boundary harnack inequality for x on any c 1 1 open set with explicit decay rate unlike cite ksv2 ksv4 our results cover geometric stable processes and relativistic geometric stable process i e the cases when the subordinator has the laplace exponent phi lambda log 1 lambda alpha 2 0 alpha and phi lambda log 1 lambda m alpha 2 2 alpha m 0 0 d 2

generalized local fatou theorems and area integrals let x be a space of homogeneous type and w a subset of x x 0 oo then under minimal conditions on w we obtain a relationship between two modes of convergence at the boundary x for functions defined on w this result gives new local fatou theorems of the carleson type for solutions of laplace parabolic and laplace beltrami equations as immediate consequences of the classical results lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained

poisson integrals of regular functions tangential convergence of poisson integrals is proved for certain spaces of regular functions which contain the spaces of bessel potentials of lp functions 1 1

marcinkiewicz integrals with rough kernels onlp 1 p 2 we prove that for dimension 3 the generalized marcinkiewicz integral with certain rough kernels is bounded onlp rn 1 p 2 and it is also of weak type 1 1

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

unavoidable collections of balls for censored stable processes abstract we study avoidability of collections of balls in bounded c 1 1 opens sets for censored stable processes 1 2 the results are analogous to the ones obtained for brownian motion in gardiner and ghergu 2010 11 on the way we derive a wiener aikawa type criterion for minimal thinness with respect to the censored stable processes

interpolating sequences for besov spaces abstract we characterise the interpolating sequences for the besov spaces b p and for their multiplier spaces we also construct linear operators of interpolation

the boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations we investigate the boundary growth of positive superharmonic functions u on a bounded domain in mathbb r n n 3 satisfying the nonlinear elliptic inequality 0 le delta u le c delta omega x alpha u p quad rm in omega where c 0 0 and p 0 are constants and delta omega x is the distance from x to the boundary of the result is applied to show a harnack inequality for such superharmonic functions also we study the existence of positive solutions with singularity on the boundary of the nonlinear elliptic equation delta u vu f x u quad rm in omega where v and f are borel measurable functions conditioned by the generalized kato class

admissible convergence of poisson integrals in symmetric spaces let f be an lp function p 1 on the distinguished boundary of a furstenberg satake compactification of a symmetric space we prove that the poisson integral pf of f converges admissibly at almost all components of each boundary of the compactification this was known previously only for large p in particular pf converges admissibly to f almost everywhere at the distinguished boundary a similar result is obtained for the normalized x poisson integral 9jf the method of proof uses maximal functions and it also gives a new proof of almost everywhere restricted convergence of pf and 9 xf for f e l

the oort conjecture on lifting covers of curves we show that the conjecture of oort on lifting covers of curves is true the main ingredients in the proof are a deformation argument in characteristic p and a special case of a very recent result by obus wewers a kind of boundedness result is given as well