L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables

Order estimates are obtained for the best approximations of the classesB _1, ^r in the spaceL _q with 1<q< and classesB _, ^r in a uniform metric. The behavior of Kolmogorov widths of the classesB _p, ^r ,1<p, in the metric of L is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 79–92, January, 1995.     

Estimates of the Best M-Term Trigonometric Approximations of the Classes Lβ, pψ of Periodic Functions of Many Variables in the Space Lq

We obtain order estimates for the best trigonometric approximations of the classes L _{, p} of periodic functions of many variables in the space L _q for 1 < p < q 2 and 1 < q p < .     

Best Orthogonal Trigonometric Approximations of Classes of Functions of Many Variables Lβ, pψ

We obtain order estimates for the best orthogonal trigonometric approximations of classes of functions of many variables L _{, p} in the space L _q, 1 < p < q < , q > 2.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

On (k,n)-blocking sets which can be obtained as a union of conics

A family of conics in PG(2,q) is called saturated if any line LPG(2,q) is incident with at least one conic of the family. Then, if <(q+1)/2, the support of is a (k,n)-blocking set. It is shown that in this way one can get blocking sets whose character n is small compared to q; it is also shown that cannot be taken independent of q, but must necessarily increase as q does.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

Interpolation of spectra

By the M.Riesz Convexity Theorem, an operator T on the space of simple integrable functions into the measurable functions (on some measure space) which has continuous extensions to L^p() and L^q() , where 1 p q , also has continuous exten — sions to all L^r () , p r q . It is shown that, whenever (T_p) and (T_q) are o-dimensional (in particular, countable) then the spectra (T_r) (p r q) are pairwise identical. For q = , only w^*-continuous extensions are considered. An example due to Dayanithy shows that the conclusion fails in general.     

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations E_m,n(f)p with undetermined n nodes in the metric of the space L_p(0, 1) (1p) is studied in a class of functionsf(x) for which f ^m+1 (x)L_q(0, 1)1(1qt8) or var {f(m) (x); 0, 1}1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n , and asymptotic formulas are obtained for p= and 1q in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.     

Pseudo-mesures p- adiques associées aux fonctions L deQ

It is shown that some standard results concerning the p-adic L- functions, L_p(), ofQ(p-divisibilities of 1/2L_p(, s), and congruences for 1/2L_p(, t)–1/2L_p(, s), s, t_p) are direct consequences of a general structural theorem, based only on the functional properties of the p-adic pseudo-measures and distributions attached to these L_p-functions (essentially the eulerian ones). The method suggests that all such divisibilities and congruences are obtained systematically by this way, and are the best possible (in a standard point of view). In particular, these results improve significantly all the known ones.     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Imbedding of pseudo-Riemannian manifolds in a pseudo-euclidean space

It is proved that every pseudo-Riemannian manifold M _{(p, q)} ⁿ with the C^k metric (3k) has an isometric C^k imbedding in the large in E _{(p, q)} ^{n(n+1)(3n+11)/2} , p(n+1)², q(n+1)².Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 193–198, February, 1971.     

A convolution metric in the space of measures and ε-isometries in Lp

For >0, 2,4,6,... on the set of those Borel measures on , such that x^pd(x)<, one introduces a metric, characterizing the nearness of the convolutions x^p* and x^p*. From the convergence of a sequence of probability measures in this metric there follows its convergence in the Kantorovich-Rubinshtein metric. From here one derives theorems on the approximation of -isometries: if H is a finite-dimensional subspace in L_p, then there exists a continuous function _H(), such that for any linear -isometric operator THL_p there exists a linear isometry UH L_p, such that T–U<_H().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 151–156, 1987.     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

Equivalence of K-functional and modulus of smoothness of functions on the sphere

In the present note certain fundamental estimates of the constructive theory of functions on the sphere Sⁿ Rⁿ⁺¹, n 1, are sharpened on the basis of the equivalence of the K-functional and the modulus of smoothness of functions. In particular a Bernshtein-type inequality for spherical polynomials is made more precise. The estimates obtained are applied to deduce a membership criterion for the function f L_p(Sⁿ), 1 p , to the space H_r H_r L_p(Sⁿ) depending on the growth of the norm of derivatives of best approximation polynomials of the function f, which is an analog of a result found by S. B. Stechkin related to continuous periodic functions.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 123–129, September, 1992.The author wishes to express his deep gratitude to Academician S. M. Nikol'skii and Professor P. I. Lizorkin for discussion of the results of the present note.     

Linear method for the approximation of differentiable functions

This article is devoted to the problem of the approximation of functions in the metric of space L_p(0, 1), the s-th derivative of the functions being continuous and the (s + l)-th derivative belonging to the space L_q(0, 1), with p, q 1, by spline functions of order s with fixed almost uniform nodes.Translated from Matematicheskie Zametki, Vol. 7, No. 4, pp. 423–430, April, 1970.     

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in L_q(–, ) of differentiable functions whose k-th derivatives belong to L_p(–, ), by splines S_m (x) with nonfixed nodes, under the extra assumption that the norms in L_s(–, ) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 501–510, May, 1971.     

Trigonometric Widths of the Classes B p,θ Ω of Periodic Functions of Many Variables

We obtain exact order estimates for the trigonometric widths of the classes of periodic functions of many variables in the space L _q, 1 < p 2 q < p/(p – 1).     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Perturbation of Dirichlet forms by measures

Perturbations of a Dirichlet form by measures are studied. The perturbed form –_–+₊ is defined for _– in a suitable Kato class and ₊ absolutely continuous with respect to capacity. L _p-properties of the corresponding semigroups are derived by approximating _– by functions. For treating ₊, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L _p -L _q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ₁ the same is shown to be true for the perturbed semigroup, for a large class of measures.