On the integral modulus of continuity of Fourier series

We study L ^p -integrability (1<p<??) of a sum ?? of trigonometric series under the assumptions that the sequence of coefficients of ?? belongs to the class $\overline{\mathrm{GM}}_{\theta}^{r}$ . Then we discuss the relations between the properties of ?? and the properties of the sequence (?? _n )??GM(??,r), and deduce an estimate for modulus of continuity of ?? in L ^p norm.     

On the modulus of continuity of solid derivatives of a Cauchy-type integral

We establish sufficient conditions for the existence of solid derivatives of a continuous extension of a Cauchy-type integral onto the closure of a domain and obtain an estimate for the moduli of continuity of these derivatives. We prove that the Newton-Leibniz formula is true for certain classes of Jordan rectifiable curves. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 476–484, April, 1998.     

An estimate for the modulus of continuity of a quaternion singular Cauchy integral

We establish an upper bound for the modulus of continuity of a quaternion singular Cauchy integral in terms of the modulus of continuity of the integrand and a metric characteristic of a curve.     

On the modulus of continuity of the solution to the Dirichlet problem at a regular boundary point

A linear elliptic equation of second order with coefficients satisfying a Dini condition is considered in the paper. The modulus of continuity of a solution at a regular boundary point is investigated. An estimate for the modulus of continuity in terms of the Wiener capacity is obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 67–72, July, 1972.     

On estimate for the modulus of continuity of the Cauchy-type integral having a Lipschitz-continuous density

The Cauchy-type integral having a Lipschitz-continuous density is under investigation. It is considered in a Jordan region that has a piecewise regular boundary without cusps and therefore it can be continuously extended over the closure of the region. Then the boundary values form a function, whose modulus of continuity is to be estimated. One parameter-dependent estimate is obtained and one algorithm for its evaluating is developed. The algorithm is demonstrated on some examples.     

Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

We estimate the modulus of continuity of a Cauchy-type integral in a closed domain and its limit values on the boundary in the case where the boundary of the domain is an arbitrary closed rectifiable Jordan curve.     

Lower bounds of the integral modulus of continuity of a function in terms of its Fourier coefficients

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 107–112, November, 1992.     

Some properties of the modulus of continuity

Necessary and sufficient conditions are derived for the continuity and semiadditivity of the modulus of continuity of a functionf(x) given on a compact in n-dimensional euclidean space.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 495–500, May, 1971.     

Path continuity of fractional Dirichlet functionals

We prove that the quasi continuous version of a functional in E^p_r is continuous along the sample paths of the Dirichlet process provided that p>2, 0<r?1 and pr>2, without assuming the Meyer equivalence. Parallel results for multi-parameter processes are also obtained. Moreover, for 1<p<2, we prove that a n parameter Dirichlet process does not touch a set of (p,2n)-zero capacity. As an example, we also study the quasi-everywhere existence of the local times of martingales on path space.     

Asymptotical representation of singular integral with the Hilbert kernel near a point of weak continuity of density

We derive an asymptotical representation for singular integral with the Hilbert kernel near a fixed point where its density vanishes as a negative power of module of logarithm of distance from this point.     

Subexponential behaviour of the Dirichlet heat kernel

We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for large time in terms of the survival probability of a Brownian motion, under the assumption that the latter decays subexponentially for large time. We also obtain a lower bound for the Dirichlet heat kernel for arbitrary open and connected sets in Euclidean space.     

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Estimates for the best approximation and integral modulus of continuity of a function in terms of its Fourier coefficients

In the integral metric, lower bounds are obtained for the best approximation and the modulus of continuity of a function in terms of its Fourier coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 4, pp. 496–503, April, 1998.