Heat Semigroup Research Articles

Boundedness and Asymptotic Behavior to a Chemotaxis System with Indirect Signal Generation and Singular Sensitivity

In this paper, we consider the following indirect signal generation and singular sensitivity n t = Δ n + χ ∇ ⋅ n / φ c ∇ c , x ∈ Ω , t > 0 , c t = Δ c − c + w , x ∈ Ω , t > 0 , w t = Δ w − w + n , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N N = 2 , 3 with smooth boundary ∂ Ω . Under the nonflux boundary conditions for n , c , and w , we first eliminate the singularity of φ c by using the Neumann heat semigroup and then establish the global boundedness and rates of convergence for solution.

Boundedness of partial difference transforms for heat semigroups generated by discrete Laplacian

Boundedness of partial difference transforms for heat semigroups generated by discrete Laplacian

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

Boundedness of differential transforms for the heat semigroup generated by biharmonic operator

In this paper we analyze the convergence of the following type seriesTNf(x)=∑j=N1N2vj(e−aj+1Δ2f(x)−e−ajΔ2f(x)),x∈Rn, where {e−tΔ2}t>0 is the heat semigroup of the biharmonic operator Δ2 with Δ being the classical laplacian, N=(N1,N2)∈Z2 with N1<N2, {vj}j∈Z is a bounded real sequences and {aj}j∈Z is a ρ-lacunary sequence of positive numbers, that is, 1<ρ≤aj+1/aj,for allj∈Z. Our analysis will consist in the boundedness, in Lp(Rn) and in BMO(Rn), of the operators TN and its maximal operator T⁎f(x)=supN⁡|TNf(x)|. The proofs of these results need the language of semigroups in an essential way.

Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including the Laplace-Beltrami operator, Schrödinger operators and divergence form elliptic operators on Riemannian manifolds or domains. We prove \(L^p\) Bernstein inqualities as well as a “reverse inequality” which is new even for compact manifolds (with or without boundary). Such a reverse inequality can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques. For example, once reformulating Bernstein inequalities in a semi-classical fashion we prove and use weak factorization of smooth functions à la Dixmier–Malliavin and BMO–\(L^\infty \) multiplier results (in contrast to the usual \(L^\infty \)–BMO ones). Also, our approach reveals a link between the \(L^p\)-Bernstein inequality and the boundedness on \(L^p\) of the Riesz transform.

A Note on Gaussian BV Function and Its Heat Semigroup Characterization

A Note on Gaussian BV Function and Its Heat Semigroup Characterization

Characterizations of vanishing Campanato classes related to Schrödinger operators via fractional semigroups on stratified groups

Let be a Schrödinger operator on stratified Lie groups, where is the sub‐Laplacian on and V belongs to the reverse Hölder class. In this paper, we introduce a new Campanato‐type space of vanishing mean oscillation associated with L. By Carleson measures related to fractional heat semigroups, we establish an equivalent characterization of . As an application, we prove that the dual of is , where is the completeness of .

Global existence of solutions to a weakly coupled critical parabolic system in two-dimensional exterior domains

This paper is concerned with the existence and non-existence of global-in-time solutions of the initial-boundary value problem of the following weakly coupled parabolic system{∂tu(x,t)−Δu(x,t)=v(x,t)p,(x,t)∈Ω×(0,∞),∂tv(x,t)−Δv(x,t)=u(x,t)q,(x,t)∈Ω×(0,∞),u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×(0,∞),u(x,0)=f(x)≥0,v(x,0)=g(x)≥0,x∈Ω, where Ω is an exterior domain in R2 having a smooth boundary ∂Ω. The given pair (p,q) with 0<q≤p describes the effect of weakly coupled nonlinearity and (f,g) is given initial data. We determine the respective regions for existence and nonexistence of global-in-time solutions to the problem. In the case of whole space R2 (without boundary condition) Escobedo–Herrero (1991) found the global existence for 2+p−pq<0 and non-existence for 2+p−pq≥0. We emphasize that in the case of exterior domain, the critical case 2+p−pq=0 with (p,q)≠(2,2) belongs to the global existence in the contrast of the case of whole space. This difference comes from the behavior of linear two-dimensional heat semigroup etΔΩ in exterior domains.

Characterizations of Product Hardy Spaces in Bessel Setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

Heat kernels of the discrete Laguerre operators

For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

Well-posedness of the Cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces

We consider the well-posedness of the Cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces $L^r_{\text{uloc}}(\mathbb R^n)$. In our setting, an initial function that is spatially periodic or converges to a nonzero constant at infinity is admitted. Our result is applicable to the one dimensional viscous Burgers equation. For the proof, we use the $L^p_{\text{uloc}}- L^q_{\text{uloc}}$ estimate for the heat semigroup obtained by Maekawa--Terasawa [20], the Banach fixed point theorem, and the comparison principle.

Mapping properties of fundamental harmonic analysis operators in the exotic Bessel framework

We prove sharp power-weighted Lp, weak type and restricted weak type inequalities for the heat semigroup maximal operator and Riesz transforms associated with the Bessel operator Bν in the exotic range of the parameter −∞<ν<1. Moreover, in the same framework, we characterize basic mapping properties for other fundamental harmonic analysis operators, including the heat semigroup based vertical g-function and fractional integrals (Riesz potential operators).

Asymptotics of Smoothed Wasserstein Distances

We investigate contraction of the Wasserstein distances on \(\mathbb {R}^{d}\) under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space—where the heat semigroup corresponds to smoothing the measures by Gaussian convolution—the situation is more subtle. We prove precise asymptotics for the 2-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the p-Wasserstein distances for p≠ 2 as well as the χ2 divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.

Transportation inequalities for Markov kernels and their applications

We study the relationship between functional inequalities for a Markov kernel on a metric space X and inequalities of transportation distances on the space of probability measures P(X). Extending results of Luise and Savaré on Hellinger–Kantorovich contraction inequalities for the particular case of the heat semigroup on an RCD(K,∞) metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the “dynamic dual” formulation of the Hellinger–Kantorovich distance to define a new family of divergences on P(X) which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.

Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials

<p style='text-indent:20px;'>Some weighted inequalities for the maximal operator with respect to the discrete diffusion semigroups associated with exceptional Jacobi and Jacobi-Dunkl polynomials are given. This setup allows to extend the corresponding results obtained for discrete heat semigroup recently to richer class of differential-difference operators.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

Heat semigroups on Weyl algebra

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators ∇i± forming the Lie algebra [∇j±,∇k±]=iRjk± and [∇j+,∇k−]=i12(Rjk++Rjk−) with some anti-symmetric matrices Rij± and define the corresponding Laplacians Δ±=g±ij∇i±∇j± with some positive matrices g±ij. We show that the heat semigroups exp(tΔ±) can be represented as a Gaussian average of the operators expξ,∇± and use these representations to compute the product of the semigroups, exp(tΔ+)exp(sΔ−) and the corresponding heat kernel.

Maximal functions for Weinstein operator

Abstract In the present paper, we study in the harmonic analysis associated to the Weinstein operator, the boundedness on L p of the uncentered maximal function. First, we establish estimates for the Weinstein translation of characteristic function of a closed ball with radius ɛ centered at 0 on the upper half space ℝd–1× ]0, +∞ [. Second, we prove weak-type L 1-estimates for the uncentered maximal function associated with the Weinstein operator and we obtain the L p -boundedness of this operator for 1 &lt; p ≤ + ∞. As application, we define a large class of operators such that each operator of this class satisfies these L p -inequalities. In particular, the maximal operator associated respectively with the Weinstein heat semigroup and the Weinstein-Poisson semigroup belong to this class.

Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups

Let L = − Δ ℍ n + V be a Schrödinger operator on the Heisenberg group ℍ n , where Δ ℍ n is the sub-Laplacian on ℍ n and the nonnegative potential V belongs to the reverse Hölder class B q with q ∈ Q / 2 , ∞ . Here, Q = 2 n + 2 is the homogeneous dimension of ℍ n . Assume that e − t L t &gt; 0 is the heat semigroup generated by L . The semigroup maximal function related to the Schrödinger operator L is defined by T L ∗ f u ≔ sup t &gt; 0 e − t L f u . The Riesz transform associated with the operator L is defined by R L = ∇ ℍ n L − 1 / 2 , and the dual Riesz transform is defined by R L ∗ = L − 1 / 2 ∇ ℍ n , where ∇ ℍ n is the gradient operator on ℍ n . In this paper, the author first introduces a class of Morrey spaces associated with the Schrödinger operator L on ℍ n . Then, by using some pointwise estimates of the kernels related to the nonnegative potential, the author establishes the boundedness properties of these operators T L ∗ , R L , and R L ∗ acting on the Morrey spaces. In addition, it is shown that the Riesz transform R L = ∇ ℍ n L − 1 / 2 is of weak-type 1 , 1 . It can be shown that the same conclusions are also true for these operators on generalized Morrey spaces.

Global Existence for an Attraction–Repulsion Chemotaxis-Fluid System in a Framework of Besov–Morrey type

We consider an attraction-repulsion chemotaxis model in the whole space $${\mathbb {R}}^3$$ with logistic source coupled with the incompressible Navier–Stokes equations. By means of a contraction argument, we obtain the existence and uniqueness of global mild solutions in a framework of Besov type, namely Besov spaces based on Morrey spaces. In comparison with previous results for the system dealt with, that framework provides new solutions and allows us to consider larger initial data and force classes for global existence and uniqueness. To carry out our results, we prove some essential lemmas and estimates related to the heat semigroup and continuity properties for the Bony decomposition in our setting.