A priori bounds for elliptic equations

In this paper we study the relationship existing between the maximum principle and the principal eigenvalue of second order elliptic operators and the expected exit times of the corresponding diffusions. A probabilistic approach is discussed that provides a good understanding of classical results established analytically. We also obtain an Alexandrov-Bakelman-Pucci type of estimate using similar methods.     

Lower bounds for matrices on block weighted sequence spaces I

In this paper we consider some matrix operators on block weighted sequence spaces l _p (w, F). The problem is to find the lower bound of some matrix operators such as Hausdorff and Hilbert matrices on l _p (w, F). This study is an extension of papers by G. Bennett, G.J.O. Jameson and R. Lashkaripour.      

A priori bounds for approximations of Markov programs

This note determines a priori bounds for B. L. Fox's [J. Math. Anal. Appl., 34 (1971), 665–670] scheme of approximating discounted Markov programs, thus refining bounds recently obtained by D. J. White (Notes in Descision Theory No. 43, University of Manchester, 1977). The approximation scheme focuses careful attention on only a subset of the state space and uses a fixed function to characterize future returns outside the designated subset. The a priori bounds are useful to design the specific approximation, that is, to select the appropriate subset on which the approximation is based.     

A convolution operator in weighted spaces

Convolution operators in R ^m with a homogeneous function of an arbitrary complex order are considered in spaces with weighted norms. The role of the weight is played by some power of the distance to a k-dimensional plane. The fundamental content of the paper consists in the investigation of the transversal operators (TO), which arise from convolutions as a result of a Fourier transform along the mentioned plane and a change of variables. Formulas for the composition of TO are derived and the (finite-dimensional) kernels and cokernels of elliptic TO are described. Problems for TO are indicated, containing additional boundary conditions or potentials. The properties of the convolution operators are obtained from the corresponding properties of the TO.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 208–237, 1990.     

A priori bounds for periodic solutions of a Duffing equation

In this paper a well known Duffing type equation is considered. By means of a Liapunov function and careful estimation, we establish a priori bounds for periodic solutions and their derived functions. Coincidence theorems can then be applied to yield sufficient conditions for the existence of periodic solutions. Our conclusion improve several well known results in the literature.     

A priori bounds for superlinear problems involving the N-Laplacian

In this paper we establish a priori bounds for positive solutions of the equation

     

A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

A priori bounds for degenerate and singular evolutionary partial integro-differential equations

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional p-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi’s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.     

A Schur test for weighted mixed-norm spaces

Summary We prove a Schur test for mixed-norm spaces L^p,q, 1 < p,q < ∞. Also we prove another version of the Schur test for discrete weighted mixed-norm spaces l^p,q _w, 1 < p,q < ∞, and wis a weight. We show that if w ₁, and w ₂are two weight functions on the index sets Jx Iand K x Lrespectively, and A =(a _{ji, kl} ) _{j∈J, i∈I, k∈K, l∈L} is an infinite matrix, then under certain conditions, Ais a bounded operator from l^p,q _w1, 1 < p,q < ∞ to l^p,q _w2. This will be a key result in proving boundedness of important operators in our work in time-frequency analysis.</o:p>     

Approximation in weighted Hardy spaces

This paper is concerned with several approximation problems in the weighted Hardy spacesH ^p(Ω) of analytic functions in the open unit disc D of the complex plane ℂ. We prove that ifX is a relatively closed subset of D, the class of uniform limits onX of functions inH ^p(Ω) coincides, moduloH ^p(Ω), with the space of uniformly continuous functions on a certain hull ofX which are holomorphic on its interior. We also solve the simultaneous approximation problems of describing Farrell and Mergelyan sets forH ^p(Ω), giving geometric characterizations for them. By replacing approximating polynomials by polynomial multipliers of outer functions, our results lead to characterizations of the same sets with respect to cyclic vectors in the classical Hardy spacesH ^p(D), 1 ⪯p < ∞. Dedicated to Professor Nácere Hayek on the occasion of his 75th birthday.     

Approximation in weighted Orlicz spaces

We prove some direct and converse theorems of trigonometric approximation in weighted Orlicz spaces with weights satisfying so called Muckenhoupt’s A _p condition.     

A note on weighted bounds for rough singular integrals

We show that the $L^2(w)$ operator norm of the composition $M°T_{\mathrm{\Omega }}$, where M is the maximal operator and $T_{\mathrm{\Omega }}$ is a rough homogeneous singular integral with angular part $\mathrm{\Omega }\in L^{\infty }(S^{n?1})$, depends quadratically on ${[w]}_{A_2}$, and that this dependence is sharp.     

A priori bounds for an indefinite superlinear elliptic equation with exponential growth

This paper considers positive solutions to problem