Bessel Potential Research Articles

The Green formula and layer potentials

The explicit form of all possible variants of the Green formula is described for a boundary value problem when the “basic” operator is an arbitrary partial differential operator with variable matrix coefficients and the “boundary” operators are quasi-normal with vector-coefficients. If the system possesses a fundamental solution, a representation formula for the solution is derived and boundedness properties of the relevant layer potentials, mapping function spaces on the boundary (Bessel potential, Besov, Zygmund spaces) into appropriate weighted function spaces on the domain are established. We conclude by discussing some closely related topics: traces of functions from weighted spaces, traces of potential-type functions, Plemelji formulae, Calderon projections, and minimal smoothness requirements for the surface and coefficients.

Pseudodifferentia Equations on Manifolds with Boundary: Fredholm Property and Asymptotic

The main purpose of the present paper is the investigation of systems of pseudo-differential equations (PsDEs) with symbols from extended Hormander classes on a manifold with smooth boundary. Equations are treated in anisotropic Bessel potential spaces with weight (BPSwW). Theorem about factorization of symbols, proved earlier by E. Shamir, R. Duduchava and E. Shargorodsky is revised and general criteria is obtained for PsDEs in BPSwW on manifolds with smooth boundary to possess the Fredholm property. It is proved that the criteria is invariant with respect to the weight exponents and the conormal smoothness parameter, which participate in the definition of the spaces. In the second part of the paper results of G. Eskin and J. Bennish on asymptotic of solutions to systems of PsDEs (L2–theory) are extended and complete asymptotic expansion of a solution to near the boundary is obtained (Lp–theory). More precise description of exponents and of logarithmic terms of the expansion is presented. Investigations are carried out in connection with problems arising in elasticity (crack problems) and some other fields of mathematical physics when the potential method is applied. In forthcoming papers asymptotic of a function represented by a potential will be presented when asymptotic of a density on the boundary of the domain is known.

On the constants for some Sobolev imbeddings

We consider the imbedding inequalit), I1.(). sr,,,dll IIn,fnu; H (R) is the Sobolev space (or Bessel potential space) ofL type and (integer or fa6tional) order n. We write down upper bounds for the constants Sr,n,a, using an argument previously applied in the literature in particular cases. We prove that the upper bounds computed in this way are in fact the sharp constants if (r= 2 or) n > d/2, r= oe, and exhibit the maximising functions. Furthermore, using convenient trial functions, we derive lower bounds on Sr,n,d for n > d/2, 2 < r < oo; in many cases these are close to the previous upper bounds, as illustrated by a number ofexamples, thus characterizing the sharp constants with little uncertainty.

Capacitary type estimates in strongly nonlinear potential theory and applications

In this article a general result on smooth truncation of Riesz and Bessel potentials in Orlicz-Sobolev spaces is given and a capacitary type estimate is presented. We construct also a space of uasicontinuous functions and an alternative characterization of this space and a description of its dual are established. For the Riesz kernel Rm, we prove that operators of strong type (A, A), are also of capacitaries strong and weak types (m, A).

Fractional diffusion and fractional heat equation

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Boundary Problems of Thermopiezoelectricity in Domains with Cuspidal Edges

Abstract Dirichlet- and Neumann-type boundary value problems of statics are considered in three-dimensional domains with cuspidal edges filled with a homogeneous anisotropic medium. Using the method of the theory of a potential and the theory of pseudodifferential equations on manifolds with boundary, we prove the existence and uniqueness theorems in Besov and Bessel-potential spaces, and study the smoothness and a complete asymptotics of solutions near the cuspidal edges.

The Wiener-Hopf integral equation for fractional Riesz-Bessel motion

AbstractThis paper gives an approximate solution to the Wiener-Hopf integral equation for filtering fractional Riesz-Bessel motion. This is obtained by showing that the corresponding covariance operator of the integral equation is a continuous isomorphism between appropriate fractional Sobolev spaces. The proof relies on properties of the Riesz and Bessel potentials and the theory of fractional Sobolev spaces.

Nonlinear Approximation with Local Fourier Bases

It is shown that local Fourier bases are unconditional bases for the modulation spaces on R, including the Bessel potential spaces and the Segal algebra S0 . As a consequence, the abstract function spaces, that are defined by the approximation properties with respect to a local Fourier basis, are precisely the modulation space s.

Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,Hs,p, which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S − 1/p ∉ ℤ. For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.

Optimality of embeddings of logarithmic Bessel potential spaces

Optimality of embeddings of logarithmic Bessel potential spaces

A relation between convolution type operators on intervals in sobolev spaces

We construct an operator relation between convolution type operators with or without a reflection on a union of finite intervals and corresponding Wiener-Hopf operators. This relation is the reult for several other relations between intermediate operators constructed for that purpose. The presented relations are obtained with the help of different extension methods that annulate particular actions of the related operators. All the operators are defined in Bessel potential spaces or Sobolev spaces. In particular, the final relation enables us to derive properties from a Wiener-Hopf operator to the intial one. An example of application of the presented results is done in a differaction problem by a union of two strips

On Relations Between Bessel Potential Spaces and Riesz Potential Spaces

We present a relation between the Bessel potential spaces and the Riesz potential spaces. The ideas of the proof are to characterize each potential spaces and to give a correspondence between individual Bessel potentials and Riesz potentials.

Fractional diffusion and fractional heat equation

This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Sharp Estimates in Smoothing Theorems for Schrödinger Semigroups

The smoothing properties of Schrödinger semigroups, e−tH, H=−12Δ+V, on the scale of Bessel potential spaces Lp, α are studied. We strengthen the (Lp−Lp, α)-smoothing theorem due to M. A. Kon and the author. The new version of this theorem contains a sharp time-estimate for the norm of the semigroup e−tH. We also get an estimate for the constants arising in the form-boundedness inequality for the Kato class potentials.

Linear extension operators for restrictions of function spaces to irregular open sets

Let an open set $Ω ⊂ ℝ^n$ satisfy for some 0≤d≤n and ε > 0 the condition: the $d$-Hausdorff content $H_d(Ω∩B) ≥ ε|B|^{d/n}$ for any ball B centered in Ω of volume |B|≤1. Let $H_p^s$ denote the Bessel potential space on $ℝ^n$ 1 < p < ∞,s > 0, and let $H_p^

On the Prandtl Equation

Abstract The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis is proved in the Sobolev space and Bessel potential spaces under certain restrictions on 𝑝 and 𝑠.

Inequalities related toH p smoothness of Sobolev type

By exploiting a class of maximal functions and Littlewood-Paley theory, a list of embedding inequalities onHp-Sobolev spaces andHp boundedness results for Riesz and Bessel potentials are obtained at one stroke.

The Characterization of the Regularity of the Jacobian Determinant in the Framework of Bessel Potential Spaces on Domains

Let 2 ⩽ m ⩽ n. The paper gives necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from ℋ p 1 s 1 × ℋ p 2 s 2 × … × ℋ p m s m into S′. Here all spaces are defined on ℝn or on domains Ω⊂ℝn. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

Possible long-range dependence in fractional random fields

Abstract Many existing stochastic models have been developed for description and analysis of Markov diffusion. This paper outlines the new concept of α -duality which lays the foundation for an extension of Markov diffusion to fractional diffusion. The theory of Riesz and Bessel potentials and the corresponding potential spaces play a key role in this new approach. We establish the existence of an important subclass of fractional random fields, namely that of fractional Riesz–Bessel motions, which extends the class of fractional Brownian motions. As a result, the scope of Markov diffusion is widened to cover random fields with long-range dependence.

Quantum Harmonic Analysis and Geometric Invariants

We develop two topics in parallel and show their inter-relation. The first centers on the notion of a fractional-differentiable structure on a commutative or a non-commutative space. We call this studyquantum harmonic analysis. The second concerns homotopy invariants for these spaces and is an aspect of non-commutative geometry. We study an algebra A, which will be a Banach algebra with unit, represented as an algebra of operators on a Hilbert space H. In order to obtain a geometric interpretation of A, we define a derivative on elements of A. We do this in a Hilbert space context, takingdaas a commutatorda=[Q,a]. HereQis a basic self-adjoint operator with discrete spectrum, increasing sufficiently rapidly that exp(−βQ)2has a trace wheneverβ>0. We can define fractional differentiability of orderμ, with 0<μ⩽1, by the boundedness of (Q2+I)μ/2a(Q2+I)−μ/2. Alternatively we can require the boundedness of an appropriate smoothing (Bessel potential) ofda. We find that it is convenient to assume the boundedness of (Q2+I)−β/2da(Q2+I)−α/2, where we chooseα,β⩾0 such thatα+β<1. We show that this also ensures a fractional derivative of orderμ=1−βin the first sense. We define a family of interpolation spaces Jβ,α. Each such space is a Banach algebra of operator, whose elements have a fractional derivative of orderμ=1−β>0. We concentrate on subalgebras A of Jβ,αwhich have certain additional covariance properties under a group Z2×G acting on H by a unitary representationγ×U(g). In addition, the derivativeQis assumed to be G-invariant. The geometric interpretation flows from the assumption that elements of A possess an arbitrarily small fractional derivative. We study homotopy invariants of A in terms of equivariant, entire cyclic cohomology. In fact, the existence of a fractional derivative on A allows the construction of the cochainτJLO, which plays the role of the integral of differential forms. We give a simple expression for a homotopy invariant ZQ(a;g), determined by pairingτJLO, with a G-invariant elementa∈A, such thatais a square root of the identity. This invariant is ZQ(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−Q2+it da)dt. This representation of the pairing is reminiscent of the heat-kernel representation for an index. In fact this quantity is an invariant, in the following sense. We isolate a simple condition on a familyQ(λ) of differentiations that yields a continuously-differentiable familyτJLO(λ) of cochains. Since ZQ(a;g) need not be an integer, continuity ofτJLO(λ) inλis insufficient to prove the constancy of the pairing. However the existence of the derivative leads to the existence of the homotopy. As τ,a vanishes forτa coboundary, and asdτJLO(λ)/dλis a coboundary, our condition onQ(λ) ensures that ZQ(λ)(a;g) is independent ofλ. Hence it is a homotopy invariant. The theory of ZQ(a;g) reduces to the study of the Radon transform of sequences of certain functions. The fractional differentiability properties of elements of A translate into properties of the asymptotics of the sequences of Radon transforms. The condition thatτJLOfit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study, and in particular the assumptionα+β<1. Thus the study of fractionally- differentiable structures dove-tails naturally with the theory of homotopy invariants. In our study of quantum harmonic analysis, we introduce spaces T(−β,α) of operator-valued distributions. These spaces are bounded, linear operators between Sobolev spaces. The elements of the interpolation spaces, the Banach algebras Jβ,αhave derivativesdawhich belong to the spaces T(−β,α). For a certain range ofβandα, we extend the theory of the Radon transform from products of regularized, bounded operators to products of regularized, operator-valued distributions. We sometimes wish to evaluate such an invariant at the endpoint of an interval such asλ∈(0,1], where ZQ(λ)(a;g) becomes singular asλ→0. We discuss in brief a procedure to regularize the endpoint, and a method to recover ZQ(λ)(a;g) fully from certain partial information at the endpoint. Finally, we generalize this approach to cover the case whenQcan be split into the sum of “independent” partsQ1+Q2, such that (Q1+Q2)2=(Q1)2+(Q2)2. Here we assume thatQ1and (Q2)2are G-invariant, but not necessarilyQ2. With further assumptions ona, the most important being that (Q1)2−(Q2)2commutes witha, we obtain a modified formula for an invariant, namely Z{Qj}(a;g)=(1/π)∫∞−∞e−t2Tr(γU(g)ae−(Q21+Q22)/2+itd1a)dt.