Estimation of the uniform modulus of continuity of the generalized Bessel potential

We study generalized Bessel potentials constructed by means of convolutions of functions with kernels that generalize the classical Bessel-Macdonald kernels. In contrast to the classical case, nonpower singularities of kernels in a neighborhood of the origin are admitted. The integral properties of functions are characterized in terms of decreasing rearrangements. The differential properties of potentials are described by the kth-order moduli of continuity in the uniform norm. An order-sharp upper estimate is established for the modulus of continuity of a potential. Such estimates play an important role in the theory of function spaces. They allow one to establish sharp embedding theorems for potentials, find majorants of the moduli of continuity, and estimate the approximation numbers of embedding operators.     

Estimates of the uniform modulus of continuity for Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the classical Bessel and Riesz potentials as well as for their generalizations. Bessel potentials are determined by the convolutions of functions with Bessel-MacDonald kernels G _α. In this paper we characterize the integral properties of functions by their decreasing rearrangements. The differential properties of potentials are characterized by their modulus of continuity of order k in the uniform norm. Estimates of such type were obtained by A. Gogatishvili, J. Neves, and B. Opic in the case k > α. Here, we remove this restriction and obtain the results for all values k ∈ N. We find order-sharp estimates from above for moduli of continuity and construct the examples confirming the sharpness. On the base of these results we obtain the order-sharp estimates for continuity envelope function in the space of potentials, and give estimates for the approximation numbers of the embedding operator.     

Two-sided estimate for the modulus of continuity of a convolution

We study the differential properties of the convolution of functions with a generalized Bessel-Macdonald kernel. The integral properties of a function are characterized in terms of its decreasing permutation. The differential properties of the convolution are described in terms of its modulus of continuity of arbitrary order in the uniform norm. We obtain order-sharp estimates for the modulus of continuity of the convolution. By way of application, we present two-sided estimates for the modulus of continuity of the classical Bessel potential.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Bessel fusion multipliers

In this paper we study properties of a Bessel multiplier when the symbol involved belongs to lp. Furthermore, we introduce the concept of Bessel fusion multiplier which generalizes a Bessel multiplier for Bessel fusion sequences. We study their behavior when the symbol belongs to lp and some continuity properties.     

An Index Integral and Convolution Operator Related to the Kontorovich-Lebedev and Mehler-Fock Transforms

We deal with an index integral involving the product of the modified Bessel functions and associated Legendre functions. It was discovered by Ferrell (Nucl Instrum Methods Phys Res B 96:483?C485, 1995) while comparing solutions of the Laplace equation in different coordinate systems in his study of the so-called surface plasmons in various condensed matter samples. This integral is quite interesting from the pure mathematical point of view and it is absent in famous reference books for series and integrals. We give a rigorous proof of this formula and discuss its particular cases. We also construct a convolution operator associated with this integral, which is related to the classical Kontorovich-Lebedev and Mehler-Fock transforms. Mapping properties and the norm estimates in weighted L _p -spaces, 1 ?? p ?? 2 are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation

The classical solution of the Dirichlet problem with a continuous boundary function for a linear elliptic equation with Hölder continuous coefficients and right-hand side satisfies the interior Schauder estimates describing the possible increase of the solution smoothness characteristics as the boundary is approached, namely, of the solution derivatives and their difference ratios in the corresponding Hölder norm. We prove similar assertions for the generalized solution with some other smoothness characteristics. In contrast to the interior Schauder estimates for classical solutions, our established estimates for the differential characteristics imply the continuity of the generalized solution in a sense natural for the problem (in the sense of (n-1)-dimensional continuity) up to the boundary of the domain in question. We state the global properties in terms of the boundedness of the integrals of the square of the difference between the solution values at different points with respect to especially normalized measures in a certain class.     

Monotonic sequences related to zeros of Bessel functions

In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J ₀ at multiples of π, i.e., at the zeros of J _1/2. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

Growth properties of semigroups generated by fractional powers of certain linear operators

Certain semigroups are generated by powers ?(?A)^a, for closed operators A in Banach space and 0 < a < 1. Properties of extent of the resolvent set and size of the resolvent operator of A correspond to properties relating to the sectors of holomorphy of the semigroups, and their growth near the origin and infinity. In this paper, we deal with semigroups having two different types of growth properties. In the first instance, the semigroup grows near the origin as r^?t, 0 < t < 1. We show that such semigroups are fractional-power semi-groups of operators A, whose resolvents decay as r^?s, 0 < s < 1, in subsectors of the right-hand half-plane. In the second instance, the semigroups are bounded near the origin, and admit special estimates on growth at the periphery of their sectors of definition. We show that for the corresponding A, the resolvent is defined and admits special growth estimates in a region which contains every subsector of the right half-plane; and in these subsectors, the resolvent decays as r^?1.     

Convolution of Rayleigh functions with respect to the Bessel index

A convolution of Rayleigh functions with respect to the Bessel index can be treated as a special function in its own right. It appears in constructing global-in-time solutions for some semilinear evolution equations in circular domains and may control the smoothing effect due to nonlinearity. An explicit representation for it is derived which involves the special function ψ(x) (the logarithmic derivative of the Γ-function). The properties of the convolution in question are established. Asymptotic expansions for small and large values of the argument are obtained and the graph is presented.     

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.     

A note on uniform strong convergence of bivariate density estimates

In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form ${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$ for arbitrary time t and arbitrary λ>0, where ψ is a smooth bump function supported in [?2,2] if λ<1 and supported in [1,2] if λ≥1. This generalizes previous results in Müller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.     

Nonlinear potential theory for Sobolev spaces on Carnot groups

Considering Bessel kernels on a Carnot group, we establish the main facts of nonlinear potential theory: a Wolff-type inequality, capacity estimates, and a strong capacity inequality. Deriving corollaries, we give an inequality of Sobolev-Adams type and relations between the capacity and Hausdorff measure, as well as lower bounds on the Teichmüller capacity. These yield the continuity of monotone functions of a Sobolev class and some estimates applicable to studying the fine properties of functions.     

System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.     

Generating differential equations satisfied by products of functions

Based on the coefficients of two homogeneous linear differential equations, a method is proposed to construct a third homogeneous linear differential equations which is satisfied by all products of the form uv, where u and v satisfy, respectively, the first and the second given differential equation. The method was used recently in the computation of rapidly oscillatory integrals with kernels which are products of Bessel functions and their variants.     

Galerkin alternating-direction method for a kind of second-order quasi-linear hyperbolic problems

A kind of second-order quasi-linear hyperbolic equation is firstly transformed into a first-order system of equations, then the Galerkin alternating-direction procedure for the system is derived. The optimal order estimates in H¹ norm and L² norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. The numerical experiment is also given to support the theoretical analysis. Comparing the results of numerical example with the theoretical analysis, they are uniform.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.