Sharp Conditions for the BBM Formula and Asymptotics of Heat Content-Type Energies

We study L r (or L r, ∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \({\mathbb {R}^n}\). We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \({\mathbb R^{2n}}\), while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L 1 to L 1 and from L 1 to L 1, ∞ are equivalent properties, boundedness from L 1 × L 1 to L 1/2 and from L 1 × L 1 to L 1/2, ∞ may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.

On linear integral operators with nonnegative kernels

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Given an operator T bounded on a weighted Lp space, the factorization technique of Rubio de Francia forces strong conditions on the weight.This algorithm is extended to two weight problems, and is shown to yield not just necessary but sufficient conditions in a wide range of settings.A nonnegative weight w . Ap for some p > 1 if for all intervals (or cubes) I, mlwiI)dx(wJr~m'"f 'sc' where C is a universal constant independent of I. We will use the notation w(I) for fj w(x) dx and I(w) for w(I)/\I\.Muckenhoupt studied these weights in detail, establishing, among other facts, that the Hardy-Littlewood maximal operator / -> /* is bounded on Lp(wdx) if and only if w G Ap [5].A weight w Ai provided I(w) < Cessinf w for all intervals I, or equivalently, w*(x) = supxeII(w) < Cw(x) a.e.Peter Jones [4] showed that w Ap if and only if there exist weights u and v Ai with w = uv1~p.Recently, Jose L. Rubio de Francia applied Maurey's theory of factorization of operators to such weighted norm inequalities and obtained a very elegant proof of the Jones' factorization theorem [7].This technique, which I have dubbed the Rubio de Francia Algorithm, or the RdFA, yields factorization theorems for many types f operators.Guido Weiss [10] observed that in the unweighted case, when the operator is an integral operator with nonnegative kernel, the condition forced by the RdFA corresponds to the hypotheses in Schur's lemma.So, the RdFA produces necessary and sufficient conditions for the boundedness of an operator.In 1, we will apply the RdFA to the two-weight problem for such integral operators, obtaining necessary and sufficient conditions.In 2, we will look at a number of applications.1.The RdFA.Let X denote an appropriate measure space, 7?" for n > 1 or the unit circle in the complex plane.Let p and A be nonnegative weight functions.Lp(p) will denote LP(X,p(x)dx).THEOREM (THE RDFA).Let Q(x,y) > 0 and let T be the operator Tf(x) = Q(x,y)f(y)dy.Let p > 1, with l/p + 1/q = 1.Then T: Lp(p) -> LP(X) is a

This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting. In this paper, we follow this direction and show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions u in W^{s,2} in fact belong to W^{t,p}_{loc} for anys le t < min {2s,1}, where p>2 reflects the amount of integrability gained. In other words, our gain of differentiability does not depend on the amount of integrability we are able to gain. This extends numerous results in previous works, where either continuity of the coefficient was required or only an in general smaller gain of differentiability was proved.

Sharp results for the Weyl product on modulation spaces

Local set stability and target control of probabilistic Boolean networks

La tesi � dedicata allo studio di varie propriet� qualitative possedute dalle soluzioni di equazioni ellittiche poste nello spazio euclideo. L'attenzione principale del lavoro � rivolta a soluzioni intere di equazioni anisotrope/eterogenee che mostrano qualche genere di propriet� di simmetria e, in particolare, che posseggono simmetria unidimensionale. L'elaborato � diviso in due parti. La prima parte � riservata ad equazioni alle derivate parziali locali, mentre la seconda si concentra su di una classe meno usuale di equazioni non locali, determinate da operatori integrali.

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

This manuscript uses Eringen's nonlocal thermoelasticity theory to study Rayleigh wave propagation in a homogeneous piezo‐thermoelastic orthotropic half‐space. This subject is discussed using the three‐phase‐lag (TPL) model of hyperbolic thermoelasticity and the state‐space method via the Cayley‐Hamilton theorem. The fundamental constitutive relations for nonlocal piezo‐thermoelastic orthotropic media have been established. Thermally insulated, or isothermal, surfaces and stress‐free, open, or closed circuits are instances of boundary conditions. The Rayleigh wave's frequency equations are obtained in various kinds of instances. Certain special and particular instances have been deduced and compared with earlier studies. The particle path on the surface during Rayleigh wave propagation is elliptical. To illustrate and validate the analytical developments, the frequency equations, specific loss factor, eccentricities, and inclination of particle trajectories with wave normals are numerically solved. The computer‐simulated results are then shown graphically. The various Rayleigh wave properties, such as propagation speed, attenuation coefficient, penetration depth, and specific loss against wave number, are graphically compared for various conditions in the presence and absence of nonlocal parameters. The effects of nonlocal parameters on various Rayleigh wave characteristics and a comparison of three hyperbolic thermoelasticity models including Lord‐Shulman (LS), dual‐phase‐lag (DPL), and three‐phase‐lag (TPL) are presented. Subsequently, a comparison of the GN‐III and GN‐II models has been shown in both local and nonlocal settings. Propagation speed and specific loss exhibit the opposite tendency when comparing the GN‐II and GN‐III models. Additionally, the famous Rayleigh wave frequency in classical elasticity has been derived as a result of this investigation.

A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon loss, which is typically the main obstacle to photonic based quantum information processing. Overall this approach opens a novel path towards quantum networks with atomic nodes and photonic links.

A rigorous mathematical framework is provided for a substructuring-based domain-decomposition approach for nonlocal problems that feature interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation problems is used in which a computational domain is subdivided into non-overlapping subdomains. In the nonlocal setting, this approach is substructuring-based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains Key results include the equivalence between the global, single-domain nonlocal problem and its multi-domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal domain-decomposition methods.

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods.

Enhancing the sufficient condition of sparsity pattern recovery