A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A landau-type construction concerning (L)′ ⊂ L

Let 1 ? p < ∞ and 1/p + 1/q = 1. For a locally finite measure space (X, S, μ) and a measurable complex-valued function f ∉ L^q functions g ∈ L^p may be constructed explicitly which satisfy

     

The principle of general localization on unit sphere

In this paper we study the general localization principle for Fourier-Laplace series on unit sphere SN⊂RN+1. Weak type (1,1) property of maximal functions is used to establish the estimates of the maximal operators of Riesz means at critical index . The properties Jacobi polynomials are used in estimating the maximal operators of spectral expansions in L₂(SN). For extending positive results on critical line , 1?p?2, we apply interpolation theorem for the family of the linear operators of weak types. The generalized localization principle is established by the analysis of spectral expansions in L₂. We have proved the sufficient conditions for the almost everywhere convergence of Fourier-Laplace series by Riesz means on the critical line.     

Derivatives of the L-cosine transform

The L^p-cosine transform of an even, continuous function is defined by

     

Maximal operators and singular integral operators along submanifolds

In this paper we prove, for certain values of p, the L^p boundedness of the maximal operator

     

The L-L estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

We first obtain the L^p-L^q estimates of solutions to the Cauchy problem for one-dimensional damped wave equation

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

Estimates of the coefficients of the Jacobi expansion by measures of smoothness

For expansion by Jacobi polynomials we relate smoothness given by appropriate K-functionals in Lp, 1?p?2, to estimates on the coefficients in the ?q form. As a corollary for 1<p?2, and an the coefficients of the Legendre expansion of f∈Lp[−1,1], we obtain the estimate

     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

On a deterministic linear partial differential system

We develop an Lp theory for the Cauchy problem of linear partial differential systems of the form

     

Refinement type equations and Grincevi?jus series

We consider L¹-solutions of the following refinement type equations

     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation

     

Non-selfadjoint matrix Sturm-Liouville operators with spectral singularities

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L²(R₊,S) by the differential expression

     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

Heisenberg inequalities for Jacobi transforms

The classical Heisenberg uncertainty principle states that for f∈L²(R),