uniform approximation of holomorphic functions on bounded hartogs domains in ℂ2 on etudie l algebre a des fonctions continues sur qui sont holomorphes dans approximation uniforme des fonctions de a par des fonctions holomorphes dans un voisinage de question de la densite uniforme dans a de l algebre a des fonctions regulieres sur qui sont holomorphes dans

the diederich fornaess index i for domains of non trivial index abstract we study bounded pseudoconvex domains in complex euclidean space we define an index associated to the boundary and show this new index is equivalent to the diederich fornaess index defined in 1977 this connects the diederich fornaess index to boundary conditions and refines the levi pseudoconvexity we also prove the worm domain is of index 2 it is the first time that a precise non trivial diederich fornaess index in euclidean spaces is obtained this finding also indicates that the diederich fornaess index is a continuum in 0 1 not a discrete set the ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations

stein compacts in levi flat hypersurfaces we explore connections between geometric properties of the levi foliation of a levi flat hypersurface m and holomorphic convexity of compact sets in m or bounded in part by m applications include extendability of cauchy riemann functions solvability of the b equation approximation of cauchy riemann and holomorphic functions and global regularity of the neumann operator

a sufficient condition for global regularity of the neumann operator abstract a theory of global regularity of the neumann operator is developed which unifies the two principal approaches to date namely the one via compactness due to kohn nirenberg j j kohn l nirenberg non coercive boundary value problems comm pure appl math 18 1965 443 492 and catlin david catlin global regularity of the neumann problem in y t siu ed complex analysis of several variables in proc sympos pure math vol 41 amer math soc providence ri 1984 pp 39 49 and the one via plurisubharmonic defining functions and or vector fields that commute approximately with due to boas and the author harold p boas emil j straube sobolev estimates for the neumann operator on domains in c n admitting a defining function that is plurisubharmonic on the boundary math z 206 1 1991 81 88 harold p boas emil j straube de rham cohomology of manifolds containing the points of infinite type and sobolev estimates for the neumann problem j geom anal 3 3 1993 225 235

boundary regularity for on q pseudoconvex wedges of cn for a wedge of c n we refine the notion of weak q pseudoconvexity of g zampieri solvabil

a cascadic multigrid algorithm for the stokes equations a variant of multigrid schemes for the stokes problem is discussed in particular we propose and analyse a cascadic version for the stokes problem the analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms

sobolev estimates for the cauchy riemann complex on c1 pseudoconvex domains let omega subset mathbb c n be a bounded pseudoconvex domain with c k boundary k 1 in this paper we will prove that the cauchy riemann operator overline partial has a bounded solution operator in the sobolev space w s p q omega for all 0 leq s k frac 1 2

the bergman projection on hartogs domains in estimates in l 2 sobolev norms are proved for the bergman projection in certain smooth bounded hartogs domains in c 2 in particular 1 if the domain is pseudoconvex and nonwormlike the normal vector does not wind on a critical set in the boundary then the bergman projection is regular and 2 barrett s counter example domains with irregular bergman projection nevertheless admit a priori estimates

compact and subelliptic estimates for the overline partial neumann operator on c2 pseudoconvex domains in this paper we will show that catlin s property p implies regularity for the overline partial neumann operator in w 1 on c 2 domains we will also look at the subelliptic case and derive the natural generalization to c k domains when k 2

behavior of the bergman projection on the diederich fornaess worm w 1 introduction in this paper we show that the bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by sobolev norms let q be a smooth bounded pseudoconvex domain in c and let p t denote the orthogonal projection from l2 f onto the bergman subspace b ff l2 f3 r with respect to the weighted norm i fll j olf z l 2 e t lzll2dv l 2 let wk t denote the sobolev space consisting of functions whose derivatives of order to k g2 on the other hand there is a large collection of results implying that for certain types of domains the unweighted bergman projection p pr preserves w k for all k 0 see fk for the strictly pseudoconvex case for results on weakly pseudoconvex domains the reader may consult ko2 ca si and the recent bstl ch as well as the references cited therein most of these results are focused on the a neumann operator rather than p see bst2 for the connection also except for kol positive results in this area are typically valid for any choice of smooth positive weight function on 0 the question of whether or not p is similarly well behaved for all weakly pseudoconvex domains has remained open for many years in this paper we show that this is not the case in fact when q is the so called worm domain of diederich and forna ss then p does not map w to w k when k zr total amount of winding this latter quantity is explained in section 4 below where the construction of the worm is reviewed and the main result is proved the proof depends on computations for a piecewise levi flat model domain depending in turn on certain one dimensional computations these are treated in sections 3 and 2 respectively section 5 contains additional remarks and questions

lp regularity of the bergman projection on domains covered by the polydisc abstract if a bounded domain can be covered by the polydisc through a rational proper holomorphic map then the bergman projection is l p bounded for p in a non trivial interval depending on the ramified rational covering this result can be applied to the symmetrized polydisc and to the hartogs triangle with exponent

characterization of automorphisms on the barrett and the diederich fornaess worm domains in this paper we show that every automorphism on either the barrett or the diederick fornaess worm domains is given by a rotation in w variable in particular any automorphism on either one of these two domains can be extended smoothly up to the boundary

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

weighted and boundary l p estimates for solutions of the equation on lineally convex domains of finite type and applications we obtain sharp weighted estimates for solutions of the equation u f in a lineally convex domain of finite type precisely we obtain estimates in the spaces l p being the distance to the boundary with two different types of hypothesis on the form f first if the data f belongs to l p 1 we have a mixed gain on the index p and the exponent secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic l p estimate with weight 1 moreover we extend those results to 1 and obtain l p and bmo estimates these results allow us to extend the l p regularity results for weighted bergman projection obtained in cdm14b for convex domains to more general weights

accelerated degenerate kernel methods for linear integral equations abstract a new approach to the theory of kernel approximations is developed for the numerical solution of fredholm integral equations of the second kind the objective is to determine the solution to any desired accuracy using a degenerate kernel operator of fixed rank that is without increasing the number of terms in the approximate kernel consequently high accuracy is achieved without incurring the computational costs associated with solving large algebraic systems the procedure employs projection techniques to construct accelerated degenerate kernel schemes that give rise to a bisequence of higher order approximate solutions error expressions are obtained and a numerical example of solving a weakly singular equation is given

the cauchy riemann equations on product domains we establish the l 2 theory for the cauchy riemann equations on product domains provided that the cauchy riemann operator has closed range on each factor we deduce regularity of the canonical solution on p 1 forms in special sobolev spaces represented as tensor products of sobolev spaces on the factors of the product this leads to regularity results for smooth data