On a singular integral on product spaces

We show that if K(x,y)=Ω(x,y)/|x|ⁿ|y|^m is a Calder n-Zygmund kerned on Rⁿ×R^m, where Ω∈L²(Sⁿ⁻¹×S^m−1) and b(x,y) is any bounded function which is radial with x∈Rⁿ and y∈R^m respectively, then b(x,y)K(x,y) is the kernel of a convolution operator which is bounded on L^p(Rⁿ×R^m) for 1<p<∞ and n≧2, m≧2. Project supported by NSFC     

Strong approximation ofGSBV functions by piecewise smooth functions

Let Ω be an open and bounded subset ofR ⁿ with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R ^m ) whose jump setS _vis essentially closed and polyhedral and which are of classW ^{k, ∞} (S _v,R ^m) for every integerk are strongly dense inGSBV ^p(Ω,R ^m ), in the sense that every functionu inGSBV ^p(Ω,R ^m ) is approximated inL ^p(Ω,R ^m ) by a sequence of functions {v _k{_j∈N with the described regularity such that the approximate gradients ∇v _jconverge inL ^p(Ω,R ^nm ) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS _{v j} converges to the (n−1)-dimensional measure ofS _u. The structure ofS _v can be further improved in casep≤2.

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Sunto Sia Ω un aperto limitato diR ⁿ con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R ^m ) con insieme di saltoS _v essenzialmente chiuso e poliedrale che sono di classeW ^{k, ∞} (S _v,R ^m ) per ogni interok sono fortemente dense inGSBV ^p(Ω,R ^m ), nel senso che ogni funzioneu∈GSBV ^p(Ω,R ^m ) è approssimata inL ^p(Ω,R ^m ) da una successione di funzioni {v _j}_j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v _jconvergono inL ^p(Ω,R ^nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS _{v j}converge alla misura (n−1)-dimensionale diS _u. La struttura diS _vpuó essere migliorata nel caso in cuip≤2.

     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

On oscillating kernels in the Besov space

In this paper we consider a convolution operator Tf=p.v. Ω * f with Ω(x)=K(x)×e^iλh(x), λ>0, where K(x) is a weak Calderón-Zygmund kernel and h(x) is a real-valued differentiable function. We give a boundedness criterion for such an operator to map the Besov space B ₁ ^0.1 (Rⁿ) into itself. This research was partially supported by NNSF and NEC in P. R. China.     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.     

Weighted endpoint estimates for commutators of Marcinkiewicz integrals

Let μΩ^mb be the commutator generalized by μΩ, the n-dimensional Marcinkiewicz integral, and b ∈ BMO（R^n）. The author establishes the weighted weak LlogL-type estimates for μΩ^mb when Ω satisfies a kind of Dini conditions, which improves the known result essentially.     

Planar algebra of the subgroup-subfactor

We give an identification between the planar algebra of the subgroupsubfactor R ⋊ H ⊂ R ⋊ G and the G-invariant planar subalgebra of the planar algebra of the bipartite graph ★ _n , where n = [G: H]. The crucial step in this identification is an exhibition of a model for the basic construction tower, and thereafter of the standard invariant of R ⋊ H ⊂ R ⋊ G in terms of operator matrices. We also obtain an identification between the planar algebra of the fixed algebra subfactor R ^G ⊂ R ^H and the G-invariant planar subalgebra of the planar algebra of the ‘flip’ of ★ _n .     

Boundary Harnack principle for second order elliptic equations with unbounded drift

We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a _ij D _ij u + b _i D _i u = 0 in a bounded domain Ω ⊂  \mathbb R {\mathbb R} ⁿ . We assume that b _i ∈ L ⁿ (Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1]. Based on this result, we derive the H?lder regularity of u/v for uniform domains. Bibliography: 27 titles.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

The space Ω m p (R d ) and some properties

Let m be a v-moderate function defined on R ^d and let g ∈ L ²(R ^d ). In this work, we defineΩ _m ^p (R ^d ) to be the vector space of f ∈ L _m ² (R ^d ) such that the Gabor transform V ^f belongs to L ^p (R ^2d ), where 1 ≤ p < ∞. We equip it with a norm and show that it is a Banach space with this norm. We also study some preliminary properties of Ω _m ^p (R ^d ). We also discuss inclusion properties and obtain the dual space of Ω _m ^p (R ^d ). At the end of this work, we study multipliers from L _w ¹ (R ^d ) into Ω _w ^p (R ^d ) and from Ω _w ^p (R ^d ) into L _w−1 ^∞ (R ^d ), where w is the Beurling weight function. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 139–145, January, 2006.     

Perturbazione dello spettro di un operatore ellittico di tipo variazionale,in relazione ad una variazione del campo

Summary Let Ω be an open bounded set in R^m and let E_Ω be an elliptic self-adjoint variational operator with Dirichlet's data zero. Changing suitably the shape of Ω, we obtain a new open set Ω* such that E_Ω* has only simple eigenvalues. Moreover choosing a suitable metric for the family of the open sets diffromorphic to Ω, the set of all Ω* such that E_Ω* has multiple eigenvalues, is of the first category.

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Entrata in Redazione il 20 dicembre 1972.

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Lavoro eseguito nell'ambito del contratto di ricerca del C.N.R. comitato per la matematica.     

On a method for interpolating functions on chaotic nets

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x| ^m form odd,K(x)=|x| ^m In |x| form even (x∈ℝ ⁿ ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝ ⁿ . We construct a function of the form

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

A note on the multilinear singular integral operators

Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈ Lipα(Sn-1) (0 ＜α≤ 1) and implies the Lp(Rn) (1 ＜ p ＜ oo) boundedness for the operator TA. Some endpoint estimates are also established.     

A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

To any finite symmetric subsetR ⊂ O₃ corresponds a Hecke operatorT _R on L²(S ²) which leaves the eigenspaces ℋ_n (n ≥ 0) of the Laplacian invariant. We compute the trace ofT _R | ℋ_n and prove that the sum of the positive eigenvalues ofT _R on ⊕_k=0 ^n-1ℋ_k prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator , and renormalizing the traces ofT _{R m} | ℋ _n slightly, we obtain sequences of Fourier coefficients of modular forms on Γ₀(4). Dedicated to R. Remmert on the occasion of his seventieth birthday     

A uniqueness theorem and the myrberg phenomenon for a Zalcman domain

LetR=Δ₀\∪_nΔ_n be a Zalcman domain (or L-domain), where Δ₀ : 0<|z| <1, Δ_n : |z-c _n|≤r _n,c_n ↘0, Δ_n ⊂ Δ₀ and Δ_n ∩ Δ_m= φ(n≠m). 0217 0115 V 3 For an unlimited two-sheeted covering with the branch points {φ^-1(c _n)}, set . In the casec _n=2⁻ⁿ , it was proved that if a uniqueness theorem is valid forH ^∞ (R) atz=0, then the Myrberg phenomenon occurs. One might suspect that the converse also holds. In this paper, contrary to this intuition, we show that the converse of this previous result is not true. In addition, we generalize the previous result for more general sequences {c _n}. By this generalization we can even partly simplify the previous proof. To complete the present work the first and second (third, resp.) named authors were supported in part by Grant-in-Aid for Scientific Research, No. 10304010 (10640190, 11640187, resp.), Japanese Ministry of Education, Science and Culture.     

Universal PI-algebras and algebras of generic matrices

Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM _n (Ω[ξ]) generated by the matricesY _k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X ₁, …,X _m) is a polynomial linear in all theX _i but one (similar to Formanek’s central polynomials for matrix rings) andf ² is central forM _n (Ω), thenf is central forM _n (Ω). (The existence of a polynomial not central forM _n (Ω), but whose square is central forM _n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.     

Sharp stability results for almost conformal maps in even dimensions

Let Ω, ⊂R ⁿ and n ≥ 4 be even. We show that if a sequence {u^j} in W^1,n/2(Ω;R ⁿ) is almost conformal in the sense that dist (∇u^j,R ⁺SO(n)) converges strongly to 0 in L^n/2 and if u^j converges weakly to u in W^1,n/2, then u is conformal and ∇u^j → ∇u strongly in L _loc ^q for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + |A|^n/2) and vanishes exactly onR ⁺ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L¹ estimates for Hodge decompositions.