Estimates for imaginary powers of Laplace operator in variable Lebesgue spaces and applications

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in ? ⁿ . Using the Mellin transform argument, fromthese estimates we obtain the boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.     

Beale-Kato-Majda type condition for Burgers equation

We consider a multidimensional Burgers equation on the torus Td and the whole space Rd. We show that, in case of the torus, there exists a unique global solution in Lebesgue spaces. For a torus we also provide estimates on the large time behaviour of solutions. In the case of Rd we establish the existence of a unique global solution if a Beale-Kato-Majda type condition is satisfied. To prove these results we use the probabilistic arguments which seem to be new.     

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n?1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ? ? _n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W ₂ ¹ (Q) under the additional requirement that the boundary function belongs to the space W ₂ ^1/2 (?Q).     

Minimal self-joinings of infinite mixing actions of rank 1

We prove that measure-preserving actions of rank 1 of the groups ? ⁿ and ? ⁿ on a Lebesgue space with a σ-finite measure have minimal self-joinings.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

Typical ℤn-actions can be inserted only in injective ℝn-actions

We study actions of the groups ?ⁿ and ?ⁿ by Lebesgue space automorphisms. We prove that a typical ?ⁿ-action can be inserted only in injective actions of ?ⁿ, n ∈ ?. We give a simple proof of the fact that a typical ?²-action cannot be inserted in an ?-action.     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

Extrapolation for Classes of Weights Related to a Family of Operators and Applications

In this work we give extrapolation results on weighted Lebesgue spaces for weights associated to a family of operators. The starting point for the extrapolation can be the knowledge of boundedness on a particular Lebesgue space as well as the boundedness on the extremal case L ^∞. This analysis can be applied to a variety of operators appearing in the context of a Schrödinger operator (??Δ?+?V) where V satisfies a reverse Hölder inequality. In that case the weights involved are a localized version of Muckenhoupt weights.     

Orthogonal sets: The axiom of choice and proof of a fixed point theorem

In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L ^p space.     

Absolutely continuous measure for a jump-type Fleming-Viot process

In this paper, we prove that the random measure of the one-dimensional jump-type Fleming-Viot process is absolutely continuous with respect to the Lebesgue measure in R, provided the mutation operator satisfies certain regularity conditions. This result is an important step towards the representation of the Fleming-Viot process with jumps in terms of the solution of a stochastic partial differential equation.     

Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces Lp, with p∈[1,∞]. Local results for arbitrary initial data are also given.     

Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H¹-critical semilinear wave equation on a smooth bounded domain Ω⊂R². First, we prove an appropriate Strichartz type estimate using the Lq spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.     

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L _r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L ₃ = D(D ² ? β ²) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.     

Iterated maximal functions in variable exponent Lebesgue spaces

In this paper we study the iterated Hardy?CLittlewood maximal operator in variable exponent Lebesgue spaces with exponent allowed to reach the value 1. We use modulars where the L ^p(·)-modular is perturbed by a logarithmic-type function, and the results hold also in the more general context of such Musielak?COrlicz spaces.     

Maximal function and Riesz potential on p-adic linear spaces

For maximal function and Riesz potential on p-adic linear space ? _p ⁿ we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential I ^α(f) exists at point b, then b is L ^q Lebesgue point for some q.     

Estimates for the ergodic measure and polynomial stability of plane stochastic curve shortening flow

We establish moment estimates for the invariant measure?μ of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity on L ¹(μ) for the related Kolmogorov operator.     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

On the A-obstacle problem and the Hausdorff measure of its free boundary

In this paper, we prove existence and uniqueness of an entropy solution to the A-obstacle problem, for L ¹-data. We also extend the Lewy?CStampacchia inequalities to the general framework of L ¹-data and show convergence and stability results. We then prove that the free boundary has finite (N ? 1)-Hausdorff measure, which completes previous works on this subject by Caffarelli for the Laplace operator and by Lee and Shahgholian for the p-Laplace operator when p?>?2.     

Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories

We consider a nonlinear reaction-diffusion equation on the whole space Rd. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally bounded in L² only. Then we adapt the short trajectory method to establish the existence of the global attractor and, if d?3, we find an upper bound of its Kolmogorov's ε-entropy.