On the Boundedness of Singular Integral Operators in Symmetric Spaces

We consider the integral convolution operators \varepsilon } {k\left( {x - y} \right)f\left( y \right)dy}$$ " align="middle" border="0"> defined on spaces of functions of several real variables. For the kernels k(x) satisfying the Hörmander condition, we establish necessary and sufficient conditions under which the operators {T } are uniformly bounded from Lorentz spaces into Marcinkiewicz spaces.     

L ∞-solutions for some nonlinear degenerate elliptic and parabolic equations

In this paper we prove the existence of bounded solutions for equations whose prototype is:

Convolution quadrature and discretized operational calculus. I

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals

Formal and convergent solutions of singular partial differential equations

We consider the problem of the existence of convergent series solutions for partial differential operators of the form . We give first conditions for P such that the linear equation Pu=f has an analytic solution, then we solve nonlinear equations of the form Pu=F(x,u). For applications we treat also cases with parameters and give a proof of a theorem of S. Kaplan [5]. In the last section we consider a case where small denominators occur.     

On the Similarity of Some Differential Operators to Self-Adjoint Ones

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form , in the space with weight . As is well known, the answer to this problem in the case is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions .     

Application of Integrals of Scalar Functions with Respect to a Vector Measure to Some Problems of Functional Analysis and the Theory of Linear Integral Equations

We prove some statements on the decomposition of indefinite integrals of scalar functions with respect to a vector measure. We also consider continuous linear operators acting from the fundamental Banach space to a Hilbert space H. This gives a representation theorem for continuous linear operators from X to H. These results are applied to most general linear integral equations of the form . Such equations are equivalent to certain infinite systems of scalar integral equations and to infinite systems of linear algebraic equations. Bibliography: 11 titles.     

Small-time sharp bounds for kernels of convolution semigroups

We study small-time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in R ^d , d ≥ 1, including the processes with jump measures which are exponentially and subexponentially localized at ∞. For a large class of Lévy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at ∞. In case of Lévy measures that are symmetric and absolutely continuous with densities g such that g(x) ? f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form

$${p_t}\left( x \right) \asymp h{\left( t \right)^{ - d}} \cdot {1_{\left\{ {\left| x \right| \leqslant \theta h\left( t \right)} \right\}}} + tg\left( x \right) \cdot {1_{\left\{ {\left| x \right| \geqslant \theta h\left( t \right)} \right\}}},t \in \left( {0,{t_0}} \right),{t_0} > 0,x \in {\mathbb{R}^d}.$$

This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at ∞ for transition densities of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

     

On Hankel Transform and Hankel Convolution of Beurling Type Distributions Having Upper Bounded Support

In this paper we study Beurling type distributions in the Hankel setting. We consider the space of Beurling type distributions on (0, ) having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space . We also establish Paley Wiener type theorems for Hankel transformations of distributions in .     

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if      

Theorems on imbedding in anisotropic spaces of vector-valued functions

Imbedding theorems are proved for abstract anisotropic spaces of Sobolev type. In particular, it is proved that if G is a bounded set satisfying thel horn condition, then there holds the imbedding where , H is a Hilbert space, and A is a self-adjoint positive operator.Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 297–301, August, 1977.     

Sharp Pointwise Interpolation Inequalities for Derivatives

We prove new pointwise inequalities involving the gradient of a function , the modulus of continuity of the gradient , and a certain maximal function and show that these inequalities are sharp. A simple particular case corresponding to and is the Landau type inequality , where the constant 8/3 is best possible and

On Compactness of Regular Integral Operators in the Space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

In this paper we obtain a sufficient condition for quite continuity of Fredholm type integral operators in the space L₁(a, b). Uniform approximations by operators with degenerate kernels of horizontally striped structures are constructed. A quantitative error estimate is obtained. We point out the possibility of application of the obtained results to second kind integral equations, including convolution equations on a finite interval, equations with polar kernels, one-dimensional equations with potential type kernels, and some transport equations in non-homogeneous layers.     

Hoffman’s Least Error Bounds for Systems of Linear Inequalities

Let E be a normed space, and . Let . We give some exact formulas for 7#x03C4;.     

Second-Order Subelliptic Operators on Lie Groups I: Complex Uniformly Continuous Principal Coefficients

We consider second-order subelliptic operators with complex coefficients over a connected Lie group G. If the principal coefficients are right uniformly continuous then we prove that the operators generate strongly continuous holomorphic semigroups with kernels K satisfying Gaussian bounds. Moreover, the kernels are Hölder continuous and for each 0, 1 and > 0 one has estimates

Equivalence and symmetries of first order differential equations

In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations . That means, the transformed unknown function is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind is compared to similar results obtained by means of auxiliary functional equations.     

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form

On the global bifurcation for a class of degenerate equations

We consider the nonlinear Dirichlet boundary value problems for the second order equation

The Banach--Mazur Theorem for Spaces with Asymmetric Norm

We establish an analog of the Banach—Mazur theorem for real separable linear spaces with asymmetric norm: every such space can be linearly and isometrically embedded in the space of continuous functions f on the interval [0,1] equipped with the asymmetric norm . This assertion is used to obtain nontrivial representations of an arbitrary convex closed body , an arbitrary compact set , and an arbitrary continuous function .     

On the essential approximate point spectrum of operators

If belongs to the essential approximate point spectrum of a Banach space operatorTB(X) and is a sequence of positive numbers with lim ^j a _j =0, then there existsxX such that for every polynomialp. This result is the best possible — if for some constantc>0 thenT has already a non-trivial invariant subspace, which is not true in general.     

Inequalities for Moduli of Continuity in Abstract Banach Spaces

Suppose that X is a Banach space, K denotes the set of real numbers R or the set of nonnegative real numbers R _{+}, is a family of linear operators from X into X such that T _0=I is the identity operator in X, for all , and there exists M such that for all . The expression is called the rth order modulus of continuity of an element x with step h in the space X with respect to the family A(K). The properties of are studied. Bibliography: 3 titles.