Nash-type Inequality Research Articles

On fractional inequalities on metric measure spaces with polar decomposition

Abstract In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for 1 < p ≤ q < ∞ 1<p\leq q<\infty is playing a key role in the proof. Moreover, we also prove the fractional Hardy–Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy–Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.

Laeng-Morpurgo-type uncertainty inequalities for the Weinstein transform

In this work, by combining Carlson-type and Nash-type inequalities for the Weinstein transform $\mathscr{F}_W$ on $\mathbb{K}=\mathbb{R}^{d-1}\times[0,\infty)$, we show Laeng-Morpurgo-type uncertainty inequalities. We establish also local-type uncertainty inequalities for the Weinstein transform $\mathscr{F}_W$, and we deduce a Heisenberg-Pauli-Weyl-type inequality for this transform.

Heisenberg-type uncertainty principles for the Dunkl–Weinstein transform

Let [Formula: see text], and let [Formula: see text], [Formula: see text], the Lebesgue spaces associated with the Dunkl–Weinstein operator [Formula: see text]. In this paper, we prove Heisenberg-type uncertainty inequalities for the Dunkl–Weinstein transform [Formula: see text] on [Formula: see text]; specifically, we establish a local-type uncertainty inequality for the Dunkl–Weinstein transform [Formula: see text], and we deduce Heisenberg-Pauli-Weyl-type inequality for this transform on [Formula: see text]. Next, we prove Clarkson-type and Nash-type inequalities for the Dunkl–Weinstein transform [Formula: see text] on [Formula: see text]. Finally, by combining these inequalities, we show Laeng–Morpurgo-type uncertainty inequalities for [Formula: see text] on [Formula: see text].

Fractional Hypocoercivity

This paper is devoted to kinetic equations without confinement. We investigate the large time behaviour induced by collision operators with fat tailed local equilibria. Such operators have an anomalous diffusion limit. In the appropriate scaling, the macroscopic equation involves a fractional diffusion operator so that the optimal decay rate is determined by a fractional Nash type inequality. At kinetic level we develop an \(\mathrm {L}^2\)-hypocoercivity approach and establish a rate of decay compatible with the fractional diffusion limit.

Discrete approximation to Brownian motion with varying dimension in unbounded domains

We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the restrictions in [11] (or [3]) that the underlying state space has to be bounded, and that the initial distribution of the limiting continuous process has to be its invariant distribution. The approach in this paper is that we first obtain heat kernel upper bounds for the approximating random walks that are uniform in their mesh sizes, by establishing an isoperimetric inequality and a Nash-type inequality based on their Dirichlet form characterization. Using the heat kernel upper bound, we then show the tightness of the approximating random walks via delicate analysis.

Optimal relaxation of bump-like solutions of the one-dimensional Cahn–Hilliard equation

In this article, we derive optimal relaxation rates for the Cahn-Hilliard equation on the one-dimensional torus and the line. We consider initial conditions with a finite (but not small) L 1-distance to an appropriately defined bump. The result extends the relaxation method developed previously for a single transition layer (the “kink”) to the case of two transition layers (the “bump”). As in the previous work, the tools include Nash-type inequalities, duality arguments, and Schauder estimates. For both the kink and the bump, the energy gap is translation invariant and its decay alone cannot specify to which member of the family of minimizers the solution converges. Whereas in the case of the kink, the conserved quantity singles out the longtime limit, in the case of a bump, a new argument is needed. On the torus, we quantify the (initially algebraic and ultimately exponential) convergence to the bump that is the longtime limit; on the line, the bump-like states are merely metastable and we quantify the initial algebraic relaxation behavior.

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [6] for Nash-type inequalities with power order considerably and also extend that of [26] for second order differential operators on a complete non-compact manifold.

Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric α-stable processes on ℝd with d ⩾ 1 and α ∈ (0, 2), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x) = (1 + ∣x∣)β with β > α, we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.

Harmonic analysis associated to the canonical Fourier Bessel transform

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

A variety of uncertainty principles for the Dunkl transform on ℝd

In this work, we prove Clarkson-type and Nash-type inequalities in the Dunkl setting on [Formula: see text] for [Formula: see text]-functions. By combining these inequalities, we show Heisenberg-type inequalities for the Dunkl transform on [Formula: see text], and we deduce local-type uncertainty inequalities for the Dunkl transform on [Formula: see text].

On-diagonal Heat Kernel Lower Bound for Strongly Local Symmetric Dirichlet Forms

This paper studies strongly local symmetric Dirichlet forms on general measure spaces. The underlying space is equipped with the intrinsic metric induced by the Dirichlet form, with respect to which the metric measure space does not necessarily satisfy volume-doubling property. Assuming Nash-type inequality, it is proved in this paper that outside a properly exceptional set, given a pointwise on-diagonal heat kernel upper bound in terms of the volume function, the comparable heat kernel lower bound also holds. The only assumption made on the volume growth rate is that it can be bounded by a continuous function satisfying doubling property, in other words, is not exponential.

Beyond traditional Curvature-Dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \in [n,\infty]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound, and discover a new case yielding a \emph{single} model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty]$): a (positively curved) sphere of (possibly negative) dimension $N \in (-\infty,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $N$-dimensional Cheeger isoperimetric inequality, and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincar\'e inequality uniformly for all $N \in (-\infty,1-\epsilon]$, and enjoy a two-level concentration of the type $\exp(-\min(t,t^2))$. Our main technical tool is a generalized version of the Heintze--Karcher theorem, which we extend to the range $N \in (-\infty,1)$.

A variation on uncertainty principles for the generalized q-Bessel Fourier transform

The aim of this paper is to prove new uncertainty principles for the generalized q-Fourier Bessel transform studied earlier in [9]. To do so we prove a Nash-type inequality and a Carlson-type inequality for this transformation. From this we deduce a variation on Heisenberg's uncertainty inequality and Faris's local uncertainty principle. We also prove a variation on Donoho–Stark's uncertainty principle. Our results can be applied to the q-Bessel Fourier transform [7].

The Generalized Diffusion Phenomenon and Applications

We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainders. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp.

Super-Poincaré and Nash-type inequalities for subordinated semigroups

We prove that if a super-Poincare inequality is satisfied by an infinitesimal generator \(-A\) of a symmetric contraction semigroup on \(L^2\) and that is contracting on \(L^1\), then it implies a corresponding super-Poincare inequality for \(-g(A)\) for any Bernstein function \(g\). We also study the converse of this statement. We prove similar results for Nash-type inequalities. We apply our results to Euclidean, Riemannian, hypoelliptic and Ornstein–Uhlenbeck settings.

On equivalence of super log Sobolev and Nash type inequalities

We prove the equivalence of Nash type and super logSobolev inequalities for Dirichlet forms. We also show that both inequalities are equivalent to Orlicz-Sobolev type inequalities. No ultracontractivity of the semigroup is assumed. It is known that there is no equivalence be- tween super log Sobolev or Nash type inequalities and ultracontractiv- ity. We discuss Davies-Simon’s counterexample as borderline case of this equivalence and related open problems.

Variations on uncertainty principles for integral operators

The aim of this paper is to prove new uncertainty principles for an integral operator with a bounded kernel. To do so we prove a Nash-type inequality and a Carlson-type inequality for this transformation. From this we deduce a variation on Heisenberg’s uncertainty inequality and Faris’s local uncertainty principle. We also prove a variation on Donoho-Stark’s uncertainty principle. Our results can be applied to a wide variety of integral operators, including the Fourier transform, the Fourier-Bessel transform, the generalized Fourier transform and the-transform.

ERGODICITY FOR INFINITE PARTICLE SYSTEMS WITH LOCALLY CONSERVED QUANTITIES

We analyse degenerate infinite-dimensional sub-elliptic generators, and obtain estimates on the long-time behaviour of the corresponding Markov semigroups that describe a model of heat conduction. In particular, we establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to equilibrium is polynomial. Consequently, there is no spectral gap, but a Liggett–Nash-type inequality is shown to hold.

Transition Density Estimates for a Class of Lévy and Lévy-Type Processes

We show on- and off-diagonal upper estimates for the transition densities of symmetric Lévy and Lévy-type processes. To get the on-diagonal estimates, we prove a Nash-type inequality for the related Dirichlet form. For the off-diagonal estimates, we assume that the characteristic function of a Lévy(-type) process is analytic, which allows us to apply the complex analysis technique.

Hypercontractivity, Nash inequalities and subordination for classes of nonlinear semigroups

A suitable notion of hypercontractivity for a nonlinear semigroup {T t } is shown to imply Nash-type inequalities for its generator H, provided a subhomogeneity property holds for the energy functional (u,Hu). We use this fact to prove that, for semigroups generated by operators of p-Laplacian-type, hypercontractivity implies ultracontractivity. Then we introduce the notion of subordinated nonlinear semigroups when the corresponding Bernstein function is f(x)=x α , and write an explicit formula for the associated generator. It is shown that hypercontractivity still holds for the subordinated semigroup and, hence, that Nash-type inequalities hold as well for the subordinated generator.