L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

Norm inequalities for convolution operators

We study norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O'Neil's inequality for the convolution operators and prove corresponding estimate from below. Second, we obtain Young–O'Neil-type estimate in the Lorentz spaces for the limit value parameters, i.e., $6K1f{6}_{L(p,h_1)\to L(p,h_2)}$. Finally, similar estimates in the weighted Lorentz spaces are presented. To cite this article: E. Nursultanov et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Approximation by convolution operators

слЕДУь п. к. сИккЕМА, Мы ИсслЕДУЕМ АппРОксИМ АцИОННыЕ сВОИстВА ОпЕРАтОРОВ $$u_\varrho ^\beta (f,x) = \frac{1}{{\beta _\varrho }}\int\limits_{ - \infty }^\infty {f(x - t)\beta ^\varrho (t) dt(\varrho \to \infty ).} $$ жДЕсьΒ — НЕОтРИцАтЕл ьНАь сУММИРУЕМАь ФУН кцИь, \(\beta _\varrho = \int\limits_{ - \infty }^\infty {\beta ^\varrho (t) dt} \) И ВыпОлНЕНы УслОВИь: (i)Β(0)=1 ИΒ НЕпРЕРыВНА В тО ЧкЕt=0, (ii) \(\mathop {\sup }\limits_{\left| t \right| > \delta } \beta (t)< 1\) Дль кАжДОгОδ>0. ДОкАжАНО, ЧтО ЁкспОНЕ НцИАльНыИ пОРьДОк Ап пРОксИМАцИИ МОжЕт Быть ДОстИгНУт тОлькО Дль ФУНкцИИ ВИДА (fx)=ax+b И (fx)=ae ^bx+c. ЁтО — ИсклУЧИтЕльНыЕ слУЧАИ, пОскОлькУ УкАжАННыЕ ФУНкцИИ ьВ льУтсь ЕДИНстВЕННыМ И НЕпОДВИжНыМИ тОЧкАМ И Дль ОпЕРАтОРОВU _β ^? . ДОкАжАНО тАкжЕ, ЧтО пР И УДАЧНОМ ВыБОРЕΒ МО жНО ДОБИтьсь «пОЧтИ Ёксп ОНЕНцИАльНОгО» пОРьДкА АппРОксИМАц ИИ. НАкОНЕц, В пОслЕДНЕИ т ЕОРЕМЕ УтВЕРжДАЕтсь, ЧтО сУЩЕстВУУт тАкИЕΒ Иf, ЧтОU _β ^? (f,x) пРИp→∞ РАсхОДьтсь НА МНОжЕстВЕ пОлОжИтЕл ьНОИ МЕРы.     

On integral convolution operators

We study integral convolutions defined on functions ofn variables in symmetric spaces and obtain new additive estimates for the mean value of the nonincreasing permutation(K ^* f) ^** (t) of the absolute value of the integral convolution on the interval [0,t] for anyt>0. As an example of application of the estimates obtained, we prove the boundedness of the integral convolution operator acting from the intersection of symmetric spaces into the Marcinkiewicz space. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 551–555, October, 1999.     

Hypercyclic algebras for convolution and composition operators

We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\text{cos}(D)$, $D{e}^D$, or $e^D?aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras.     

Asymptotic formulae for bivariate Mellin convolution operators

In this paper some Voronovskaya approximation formulae for a class of Mellin convolution operators of the type (Tw f)(x,y) = ∫R2+ Kw(tx-1,vy-1)f(t,v)tvdtdv are given. Moreover, various examples are discussed.     

Anisotropic estimates for sub-elliptic operators

In the 1970's,Folland and Stein studied a family of subelliptic scalar operators L_λwhich arise naturally in the(?)_b-complex.They introduced weighted Sobolev spaces as the natural spaces for this complex,and then obtained sharp estimates for(?)b in these spaces using integral kernels and approximate inverses.In the 1990's,Rumin introduced a differential complex for compact contact manifolds,showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator,and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper,we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.     

Hardy-type estimates for Dirac operators

We prove some Hardy type inequalities related to the Dirac operator by elementary methods, for a large class of potentials, which even includes measure valued potentials. Optimality is achieved by the Coulomb potential. When potentials are smooth enough, our estimates provide some spectral information on the operator.     

Isocapacity estimates for Hessian operators

Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou–Wang's Sobolev type inequality and Tian–Wang's Moser–Trudinger type inequality for the fully nonlinear 1≤k≤n/2$1\le k\le n/2$ Hessian operators.