A number of statements similar to the Marcinkiewicz interpolation theorem are presented. The difference from the classical forms of this theorem is that the spaces of integrable functions are replaced by certain classes of functions that are extensions of various Hardy spaces.
Let (E) be a domain in (mathbb R^d). We investigate the regularity of the characteristic function (mathcal X_E) depending on the behavior of the (delta)-neighborhoods of the boundary of (E). The regularity is measured in terms of the Nikol’skii–Besov and Lizorkin–Triebel spaces.
A. N. Kolmogorov’s famous theorem of 1925 implies that the partial sums of the Fourier series of any integrable function (f) of one variable converge to it in (L^p) for all (pin(0,1)). It is known that this does not hold true for functions of several variables. In this paper we prove that, nevertheless, for any function of several variables there is a subsequence of Pringsheim partial sums that converges to the function in (L^p) for all (pin(0,1)). At the same time, in a fairly general case, when we take the partial sums of the Fourier series of a function of several variables over an expanding system of index sets, there exists a function for which the absolute values of a certain subsequence of these partial sums tend to infinity almost everywhere. This is so, in particular, for a system of dilations of a fixed bounded convex body and for hyperbolic crosses.
The paper deals with some recent assertions about truncations (fmapsto |f|) and compositions (fmapsto gcirc f) in the spaces (A^s_{p,q}(mathbb R^n)), (Ain{B,F}).
We consider the operator (H=L+V) that is a perturbation of the Taibleson–Vladimirov operator (L=mathfrak{D}^alpha) by a potential (V(x)=b|x|^{-alpha}), where (alpha>0) and (bgeq b_*). We prove that the operator (H) is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value (b_*) depends on (alpha)). While the operator (H) is nonnegative definite, the potential (V(x)) may well take negative values as (b_*<0) for all (0<alpha<1). The equation (Hu=v) admits a Green function (g_H(x,y)), that is, the integral kernel of the operator (H^{-1}). We obtain sharp lower and upper bounds on the ratio of the Green functions (g_H(x,y)) and (g_L(x,y)).
In this paper we fix (1le p<infty) and consider ((Omega,d,mu)) to be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure (mu) supporting a (p)-Poincaré inequality such that (Omega) is a uniform domain in its completion (overlineOmega). We realize the trace of functions in the Dirichlet–Sobolev space (D^{1,p}(Omega)) on the boundary (partialOmega) as functions in the homogeneous Besov space (Hkern-1pt B^alpha_{p,p}(partialOmega)) for suitable (alpha); here, (partialOmega) is equipped with a non-atomic Borel regular measure (nu). We show that if (nu) satisfies a (theta)-codimensional condition with respect to (mu) for some (0<theta<p), then there is a bounded linear trace operator (T colon, D^{1,p}(Omega)to Hkern-1pt B^{1-theta/p}(partialOmega)) and a bounded linear extension operator (E colon, Hkern-1pt B^{1-theta/p}(partialOmega)to D^{1,p}(Omega)) that is a right-inverse of (T).
We prove embedding theorems for spaces of functions of positive smoothness defined on a Hölder domain of (n)-dimensional Euclidean space.
Given a one-parameter family of operators on the manifold (mathbb R^ntimesmathbb T^m), we solve the problem of the best recovery of an operator for a given value of the parameter from inaccurate data on the operators for other values of the parameter from a certain compact set. We construct a family of best recovery methods. As a consequence, we obtain families of best recovery methods for the solutions of the heat equation and the Dirichlet problem for a half-space.
It is known that results on universal sampling discretization of the square norm are useful in sparse sampling recovery with error measured in the square norm. In this paper we demonstrate how known results on universal sampling discretization of the uniform norm and recent results on universal sampling representation allow us to provide good universal methods of sampling recovery for anisotropic Sobolev and Nikol’skii classes of periodic functions of several variables. The sharpest results are obtained in the case of functions of two variables, where the Fibonacci point sets are used for recovery.
We study the stability of solutions to nonlinear equations in finite-dimensional spaces. Namely, we consider an equation of the form (F(x)=overline{y}) in the neighborhood of a given solution (overline{x}). For this equation we present sufficient conditions under which the equation (F(x)+g(x)=y) has a solution close to (overline{x}) for all (y) close to (overline{y}) and for all continuous perturbations (g) with sufficiently small uniform norm. The results are formulated in terms of (lambda)-truncations and contain applications to necessary optimality conditions for a constrained optimization problem with equality-type constraints. We show that these results on (lambda)-truncations are also meaningful in the case of degeneracy of the linear operator (F'(overline{x})).
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